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GNU Scientific Library
**********************

     GSL 2.7, May 27, 2021

     The GSL Team

     Copyright © 1996-2021 The GSL Team

* Menu:

* Introduction::
* Using the Library::
* Error Handling::
* Mathematical Functions::
* Complex Numbers::
* Polynomials::
* Special Functions::
* Vectors and Matrices::
* Permutations::
* Combinations::
* Multisets::
* Sorting::
* BLAS Support::
* Linear Algebra::
* Eigensystems::
* Fast Fourier Transforms (FFTs): Fast Fourier Transforms FFTs.
* Numerical Integration::
* Random Number Generation::
* Quasi-Random Sequences::
* Random Number Distributions::
* Statistics::
* Running Statistics::
* Moving Window Statistics::
* Digital Filtering::
* Histograms::
* N-tuples::
* Monte Carlo Integration::
* Simulated Annealing::
* Ordinary Differential Equations::
* Interpolation::
* Numerical Differentiation::
* Chebyshev Approximations::
* Series Acceleration::
* Wavelet Transforms::
* Discrete Hankel Transforms::
* One Dimensional Root-Finding::
* One Dimensional Minimization::
* Multidimensional Root-Finding::
* Multidimensional Minimization::
* Linear Least-Squares Fitting::
* Nonlinear Least-Squares Fitting::
* Basis Splines::
* Sparse Matrices::
* Sparse BLAS Support::
* Sparse Linear Algebra::
* Physical Constants::
* IEEE floating-point arithmetic::
* Debugging Numerical Programs::
* Contributors to GSL::
* Autoconf Macros::
* GSL CBLAS Library::
* GNU General Public License::
* GNU Free Documentation License::
* Index::

 — The Detailed Node Listing —

Introduction

* Routines available in GSL::
* GSL is Free Software::
* Obtaining GSL::
* No Warranty::
* Reporting Bugs::
* Further Information::
* Conventions used in this manual::

Using the Library

* An Example Program::
* Compiling and Linking::
* Shared Libraries::
* ANSI C Compliance::
* Inline functions::
* Long double::
* Portability functions::
* Alternative optimized functions::
* Support for different numeric types::
* Compatibility with C++::
* Aliasing of arrays::
* Thread-safety::
* Deprecated Functions::
* Code Reuse::

Compiling and Linking

* Linking programs with the library::
* Linking with an alternative BLAS library::

Error Handling

* Error Reporting::
* Error Codes::
* Error Handlers::
* Using GSL error reporting in your own functions::
* Examples::

Mathematical Functions

* Mathematical Constants::
* Infinities and Not-a-number::
* Elementary Functions::
* Small integer powers::
* Testing the Sign of Numbers::
* Testing for Odd and Even Numbers::
* Maximum and Minimum functions::
* Approximate Comparison of Floating Point Numbers::

Complex Numbers

* Representation of complex numbers::
* Complex number macros::
* Assigning complex numbers::
* Properties of complex numbers::
* Complex arithmetic operators::
* Elementary Complex Functions::
* Complex Trigonometric Functions::
* Inverse Complex Trigonometric Functions::
* Complex Hyperbolic Functions::
* Inverse Complex Hyperbolic Functions::
* References and Further Reading::

Polynomials

* Polynomial Evaluation::
* Divided Difference Representation of Polynomials::
* Quadratic Equations::
* Cubic Equations::
* General Polynomial Equations::
* Examples: Examples<2>.
* References and Further Reading: References and Further Reading<2>.

Special Functions

* Usage::
* The gsl_sf_result struct::
* Modes::
* Airy Functions and Derivatives::
* Bessel Functions::
* Clausen Functions::
* Coulomb Functions::
* Coupling Coefficients::
* Dawson Function::
* Debye Functions::
* Dilogarithm::
* Elementary Operations::
* Elliptic Integrals::
* Elliptic Functions (Jacobi): Elliptic Functions Jacobi.
* Error Functions::
* Exponential Functions::
* Exponential Integrals::
* Fermi-Dirac Function::
* Gamma and Beta Functions::
* Gegenbauer Functions::
* Hermite Polynomials and Functions::
* Hypergeometric Functions::
* Laguerre Functions::
* Lambert W Functions::
* Legendre Functions and Spherical Harmonics::
* Logarithm and Related Functions::
* Mathieu Functions::
* Power Function::
* Psi (Digamma) Function: Psi Digamma Function.
* Synchrotron Functions::
* Transport Functions::
* Trigonometric Functions::
* Zeta Functions::
* Examples: Examples<3>.
* References and Further Reading: References and Further Reading<3>.

Airy Functions and Derivatives

* Airy Functions::
* Derivatives of Airy Functions::
* Zeros of Airy Functions::
* Zeros of Derivatives of Airy Functions::

Bessel Functions

* Regular Cylindrical Bessel Functions::
* Irregular Cylindrical Bessel Functions::
* Regular Modified Cylindrical Bessel Functions::
* Irregular Modified Cylindrical Bessel Functions::
* Regular Spherical Bessel Functions::
* Irregular Spherical Bessel Functions::
* Regular Modified Spherical Bessel Functions::
* Irregular Modified Spherical Bessel Functions::
* Regular Bessel Function—Fractional Order::
* Irregular Bessel Functions—Fractional Order::
* Regular Modified Bessel Functions—Fractional Order::
* Irregular Modified Bessel Functions—Fractional Order::
* Zeros of Regular Bessel Functions::

Coulomb Functions

* Normalized Hydrogenic Bound States::
* Coulomb Wave Functions::
* Coulomb Wave Function Normalization Constant::

Coupling Coefficients

* 3-j Symbols::
* 6-j Symbols::
* 9-j Symbols::

Dilogarithm

* Real Argument::
* Complex Argument::

Elliptic Integrals

* Definition of Legendre Forms::
* Definition of Carlson Forms::
* Legendre Form of Complete Elliptic Integrals::
* Legendre Form of Incomplete Elliptic Integrals::
* Carlson Forms::

Error Functions

* Error Function::
* Complementary Error Function::
* Log Complementary Error Function::
* Probability functions::

Exponential Functions

* Exponential Function::
* Relative Exponential Functions::
* Exponentiation With Error Estimate::

Exponential Integrals

* Exponential Integral::
* Ei(x): Ei x.
* Hyperbolic Integrals::
* Ei_3(x): Ei_3 x.
* Trigonometric Integrals::
* Arctangent Integral::

Fermi-Dirac Function

* Complete Fermi-Dirac Integrals::
* Incomplete Fermi-Dirac Integrals::

Gamma and Beta Functions

* Gamma Functions::
* Factorials::
* Pochhammer Symbol::
* Incomplete Gamma Functions::
* Beta Functions::
* Incomplete Beta Function::

Hermite Polynomials and Functions

* Hermite Polynomials::
* Derivatives of Hermite Polynomials::
* Hermite Functions::
* Derivatives of Hermite Functions::
* Zeros of Hermite Polynomials and Hermite Functions::

Legendre Functions and Spherical Harmonics

* Legendre Polynomials::
* Associated Legendre Polynomials and Spherical Harmonics::
* Conical Functions::
* Radial Functions for Hyperbolic Space::

Mathieu Functions

* Mathieu Function Workspace::
* Mathieu Function Characteristic Values::
* Angular Mathieu Functions::
* Radial Mathieu Functions::

Psi (Digamma) Function

* Digamma Function::
* Trigamma Function::
* Polygamma Function::

Trigonometric Functions

* Circular Trigonometric Functions::
* Trigonometric Functions for Complex Arguments::
* Hyperbolic Trigonometric Functions::
* Conversion Functions::
* Restriction Functions::
* Trigonometric Functions With Error Estimates::

Zeta Functions

* Riemann Zeta Function::
* Riemann Zeta Function Minus One::
* Hurwitz Zeta Function::
* Eta Function::

Vectors and Matrices

* Data types::
* Blocks::
* Vectors::
* Matrices::

Blocks

* Block allocation::
* Reading and writing blocks::
* Example programs for blocks::

Vectors

* Vector allocation::
* Accessing vector elements::
* Initializing vector elements::
* Reading and writing vectors::
* Vector views::
* Copying vectors::
* Exchanging elements::
* Vector operations::
* Finding maximum and minimum elements of vectors::
* Vector properties::
* Example programs for vectors::

Matrices

* Matrix allocation::
* Accessing matrix elements::
* Initializing matrix elements::
* Reading and writing matrices::
* Matrix views::
* Creating row and column views::
* Copying matrices::
* Copying rows and columns::
* Exchanging rows and columns::
* Matrix operations::
* Finding maximum and minimum elements of matrices::
* Matrix properties::
* Example programs for matrices::
* References and Further Reading: References and Further Reading<4>.

Permutations

* The Permutation struct::
* Permutation allocation::
* Accessing permutation elements::
* Permutation properties::
* Permutation functions::
* Applying Permutations::
* Reading and writing permutations::
* Permutations in cyclic form::
* Examples: Examples<4>.
* References and Further Reading: References and Further Reading<5>.

Combinations

* The Combination struct::
* Combination allocation::
* Accessing combination elements::
* Combination properties::
* Combination functions::
* Reading and writing combinations::
* Examples: Examples<5>.
* References and Further Reading: References and Further Reading<6>.

Multisets

* The Multiset struct::
* Multiset allocation::
* Accessing multiset elements::
* Multiset properties::
* Multiset functions::
* Reading and writing multisets::
* Examples: Examples<6>.

Sorting

* Sorting objects::
* Sorting vectors::
* Selecting the k smallest or largest elements::
* Computing the rank::
* Examples: Examples<7>.
* References and Further Reading: References and Further Reading<7>.

BLAS Support

* GSL BLAS Interface::
* Examples: Examples<8>.
* References and Further Reading: References and Further Reading<8>.

GSL BLAS Interface

* Level 1::
* Level 2::
* Level 3::

Linear Algebra

* LU Decomposition::
* QR Decomposition::
* QR Decomposition with Column Pivoting::
* LQ Decomposition::
* QL Decomposition::
* Complete Orthogonal Decomposition::
* Singular Value Decomposition::
* Cholesky Decomposition::
* Pivoted Cholesky Decomposition::
* Modified Cholesky Decomposition::
* LDLT Decomposition::
* Tridiagonal Decomposition of Real Symmetric Matrices::
* Tridiagonal Decomposition of Hermitian Matrices::
* Hessenberg Decomposition of Real Matrices::
* Hessenberg-Triangular Decomposition of Real Matrices::
* Bidiagonalization::
* Givens Rotations::
* Householder Transformations::
* Householder solver for linear systems::
* Tridiagonal Systems::
* Triangular Systems::
* Banded Systems::
* Balancing::
* Examples: Examples<9>.
* References and Further Reading: References and Further Reading<9>.

QR Decomposition

* Level 2 Interface::
* Triangle on Top of Rectangle::
* Triangle on Top of Triangle::
* Triangle on Top of Trapezoidal::
* Triangle on Top of Diagonal::

Banded Systems

* General Banded Format::
* Symmetric Banded Format::
* Banded LU Decomposition::
* Banded Cholesky Decomposition::
* Banded LDLT Decomposition::

Eigensystems

* Real Symmetric Matrices::
* Complex Hermitian Matrices::
* Real Nonsymmetric Matrices::
* Real Generalized Symmetric-Definite Eigensystems::
* Complex Generalized Hermitian-Definite Eigensystems::
* Real Generalized Nonsymmetric Eigensystems::
* Sorting Eigenvalues and Eigenvectors::
* Examples: Examples<10>.
* References and Further Reading: References and Further Reading<10>.

Fast Fourier Transforms (FFTs)

* Mathematical Definitions::
* Overview of complex data FFTs::
* Radix-2 FFT routines for complex data::
* Mixed-radix FFT routines for complex data::
* Overview of real data FFTs::
* Radix-2 FFT routines for real data::
* Mixed-radix FFT routines for real data::
* References and Further Reading: References and Further Reading<11>.

Numerical Integration

* Introduction: Introduction<2>.
* QNG non-adaptive Gauss-Kronrod integration::
* QAG adaptive integration::
* QAGS adaptive integration with singularities::
* QAGP adaptive integration with known singular points::
* QAGI adaptive integration on infinite intervals::
* QAWC adaptive integration for Cauchy principal values::
* QAWS adaptive integration for singular functions::
* QAWO adaptive integration for oscillatory functions::
* QAWF adaptive integration for Fourier integrals::
* CQUAD doubly-adaptive integration::
* Romberg integration::
* Gauss-Legendre integration::
* Fixed point quadratures::
* Error codes::
* Examples: Examples<11>.
* References and Further Reading: References and Further Reading<12>.

Introduction

* Integrands without weight functions::
* Integrands with weight functions::
* Integrands with singular weight functions::

Examples

* Adaptive integration example::
* Fixed-point quadrature example::

Random Number Generation

* General comments on random numbers::
* The Random Number Generator Interface::
* Random number generator initialization::
* Sampling from a random number generator::
* Auxiliary random number generator functions::
* Random number environment variables::
* Copying random number generator state::
* Reading and writing random number generator state::
* Random number generator algorithms::
* Unix random number generators::
* Other random number generators::
* Performance::
* Examples: Examples<12>.
* References and Further Reading: References and Further Reading<13>.
* Acknowledgements::

Quasi-Random Sequences

* Quasi-random number generator initialization::
* Sampling from a quasi-random number generator::
* Auxiliary quasi-random number generator functions::
* Saving and restoring quasi-random number generator state::
* Quasi-random number generator algorithms::
* Examples: Examples<13>.
* References::

Random Number Distributions

* Introduction: Introduction<3>.
* The Gaussian Distribution::
* The Gaussian Tail Distribution::
* The Bivariate Gaussian Distribution::
* The Multivariate Gaussian Distribution::
* The Exponential Distribution::
* The Laplace Distribution::
* The Exponential Power Distribution::
* The Cauchy Distribution::
* The Rayleigh Distribution::
* The Rayleigh Tail Distribution::
* The Landau Distribution::
* The Levy alpha-Stable Distributions::
* The Levy skew alpha-Stable Distribution::
* The Gamma Distribution::
* The Flat (Uniform) Distribution: The Flat Uniform Distribution.
* The Lognormal Distribution::
* The Chi-squared Distribution::
* The F-distribution::
* The t-distribution::
* The Beta Distribution::
* The Logistic Distribution::
* The Pareto Distribution::
* Spherical Vector Distributions::
* The Weibull Distribution::
* The Type-1 Gumbel Distribution::
* The Type-2 Gumbel Distribution::
* The Dirichlet Distribution::
* General Discrete Distributions::
* The Poisson Distribution::
* The Bernoulli Distribution::
* The Binomial Distribution::
* The Multinomial Distribution::
* The Negative Binomial Distribution::
* The Pascal Distribution::
* The Geometric Distribution::
* The Hypergeometric Distribution::
* The Logarithmic Distribution::
* The Wishart Distribution::
* Shuffling and Sampling::
* Examples: Examples<14>.
* References and Further Reading: References and Further Reading<14>.

Statistics

* Mean, Standard Deviation and Variance: Mean Standard Deviation and Variance.
* Absolute deviation::
* Higher moments (skewness and kurtosis): Higher moments skewness and kurtosis.
* Autocorrelation::
* Covariance::
* Correlation::
* Weighted Samples::
* Maximum and Minimum values::
* Median and Percentiles::
* Order Statistics::
* Robust Location Estimates::
* Robust Scale Estimates::
* Examples: Examples<15>.
* References and Further Reading: References and Further Reading<15>.

Robust Location Estimates

* Trimmed Mean::
* Gastwirth Estimator::

Robust Scale Estimates

* Median Absolute Deviation (MAD): Median Absolute Deviation MAD.
* S_n Statistic::
* Q_n Statistic::

Running Statistics

* Initializing the Accumulator::
* Adding Data to the Accumulator::
* Current Statistics::
* Quantiles::
* Examples: Examples<16>.
* References and Further Reading: References and Further Reading<16>.

Moving Window Statistics

* Introduction: Introduction<4>.
* Handling Endpoints::
* Allocation for Moving Window Statistics::
* Moving Mean::
* Moving Variance and Standard Deviation::
* Moving Minimum and Maximum::
* Moving Sum::
* Moving Median::
* Robust Scale Estimation::
* User-defined Moving Statistics::
* Accumulators::
* Examples: Examples<17>.
* References and Further Reading: References and Further Reading<17>.

Robust Scale Estimation

* Moving MAD::
* Moving QQR::
* Moving S_n::
* Moving Q_n::

Examples

* Example 1::
* Example 2; Robust Scale: Example 2 Robust Scale.
* Example 3; User-defined Moving Window: Example 3 User-defined Moving Window.

Digital Filtering

* Introduction: Introduction<5>.
* Handling Endpoints: Handling Endpoints<2>.
* Linear Digital Filters::
* Nonlinear Digital Filters::
* Examples: Examples<18>.
* References and Further Reading: References and Further Reading<18>.

Linear Digital Filters

* Gaussian Filter::

Nonlinear Digital Filters

* Standard Median Filter::
* Recursive Median Filter::
* Impulse Detection Filter::

Examples

* Gaussian Example 1::
* Gaussian Example 2::
* Square Wave Signal Example::
* Impulse Detection Example::

Histograms

* The histogram struct::
* Histogram allocation::
* Copying Histograms::
* Updating and accessing histogram elements::
* Searching histogram ranges::
* Histogram Statistics::
* Histogram Operations::
* Reading and writing histograms::
* Resampling from histograms::
* The histogram probability distribution struct::
* Example programs for histograms::
* Two dimensional histograms::
* The 2D histogram struct::
* 2D Histogram allocation::
* Copying 2D Histograms::
* Updating and accessing 2D histogram elements::
* Searching 2D histogram ranges::
* 2D Histogram Statistics::
* 2D Histogram Operations::
* Reading and writing 2D histograms::
* Resampling from 2D histograms::
* Example programs for 2D histograms::

N-tuples

* The ntuple struct::
* Creating ntuples::
* Opening an existing ntuple file::
* Writing ntuples::
* Reading ntuples::
* Closing an ntuple file::
* Histogramming ntuple values::
* Examples: Examples<19>.
* References and Further Reading: References and Further Reading<19>.

Monte Carlo Integration

* Interface::
* PLAIN Monte Carlo::
* MISER::
* VEGAS::
* Examples: Examples<20>.
* References and Further Reading: References and Further Reading<20>.

Simulated Annealing

* Simulated Annealing algorithm::
* Simulated Annealing functions::
* Examples: Examples<21>.
* References and Further Reading: References and Further Reading<21>.

Examples

* Trivial example::
* Traveling Salesman Problem::

Ordinary Differential Equations

* Defining the ODE System::
* Stepping Functions::
* Adaptive Step-size Control::
* Evolution::
* Driver::
* Examples: Examples<22>.
* References and Further Reading: References and Further Reading<22>.

Interpolation

* Introduction to 1D Interpolation::
* 1D Interpolation Functions::
* 1D Interpolation Types::
* 1D Index Look-up and Acceleration::
* 1D Evaluation of Interpolating Functions::
* 1D Higher-level Interface::
* 1D Interpolation Example Programs::
* Introduction to 2D Interpolation::
* 2D Interpolation Functions::
* 2D Interpolation Grids::
* 2D Interpolation Types::
* 2D Evaluation of Interpolating Functions::
* 2D Higher-level Interface::
* 2D Interpolation Example programs::
* References and Further Reading: References and Further Reading<23>.

Numerical Differentiation

* Functions::
* Examples: Examples<23>.
* References and Further Reading: References and Further Reading<24>.

Chebyshev Approximations

* Definitions::
* Creation and Calculation of Chebyshev Series::
* Auxiliary Functions::
* Chebyshev Series Evaluation::
* Derivatives and Integrals::
* Examples: Examples<24>.
* References and Further Reading: References and Further Reading<25>.

Series Acceleration

* Acceleration functions::
* Acceleration functions without error estimation::
* Examples: Examples<25>.
* References and Further Reading: References and Further Reading<26>.

Wavelet Transforms

* Definitions: Definitions<2>.
* Initialization::
* Transform Functions::
* Examples: Examples<26>.
* References and Further Reading: References and Further Reading<27>.

Transform Functions

* Wavelet transforms in one dimension::
* Wavelet transforms in two dimension::

Discrete Hankel Transforms

* Definitions: Definitions<3>.
* Functions: Functions<2>.
* References and Further Reading: References and Further Reading<28>.

One Dimensional Root-Finding

* Overview::
* Caveats::
* Initializing the Solver::
* Providing the function to solve::
* Search Bounds and Guesses::
* Iteration::
* Search Stopping Parameters::
* Root Bracketing Algorithms::
* Root Finding Algorithms using Derivatives::
* Examples: Examples<27>.
* References and Further Reading: References and Further Reading<29>.

One Dimensional Minimization

* Overview: Overview<2>.
* Caveats: Caveats<2>.
* Initializing the Minimizer::
* Providing the function to minimize::
* Iteration: Iteration<2>.
* Stopping Parameters::
* Minimization Algorithms::
* Examples: Examples<28>.
* References and Further Reading: References and Further Reading<30>.

Multidimensional Root-Finding

* Overview: Overview<3>.
* Initializing the Solver: Initializing the Solver<2>.
* Providing the function to solve: Providing the function to solve<2>.
* Iteration: Iteration<3>.
* Search Stopping Parameters: Search Stopping Parameters<2>.
* Algorithms using Derivatives::
* Algorithms without Derivatives::
* Examples: Examples<29>.
* References and Further Reading: References and Further Reading<31>.

Multidimensional Minimization

* Overview: Overview<4>.
* Caveats: Caveats<3>.
* Initializing the Multidimensional Minimizer::
* Providing a function to minimize::
* Iteration: Iteration<4>.
* Stopping Criteria::
* Algorithms with Derivatives::
* Algorithms without Derivatives: Algorithms without Derivatives<2>.
* Examples: Examples<30>.
* References and Further Reading: References and Further Reading<32>.

Linear Least-Squares Fitting

* Overview: Overview<5>.
* Linear regression::
* Multi-parameter regression::
* Regularized regression::
* Robust linear regression::
* Large dense linear systems::
* Troubleshooting::
* Examples: Examples<31>.
* References and Further Reading: References and Further Reading<33>.

Linear regression

* Linear regression with a constant term::
* Linear regression without a constant term::

Large dense linear systems

* Normal Equations Approach::
* Tall Skinny QR (TSQR) Approach: Tall Skinny QR TSQR Approach.
* Large Dense Linear Systems Solution Steps::
* Large Dense Linear Least Squares Routines::

Examples

* Simple Linear Regression Example::
* Multi-parameter Linear Regression Example::
* Regularized Linear Regression Example 1::
* Regularized Linear Regression Example 2::
* Robust Linear Regression Example::
* Large Dense Linear Regression Example::

Nonlinear Least-Squares Fitting

* Overview: Overview<6>.
* Solving the Trust Region Subproblem (TRS): Solving the Trust Region Subproblem TRS.
* Weighted Nonlinear Least-Squares::
* Tunable Parameters::
* Initializing the Solver: Initializing the Solver<3>.
* Providing the Function to be Minimized::
* Iteration: Iteration<5>.
* Testing for Convergence::
* High Level Driver::
* Covariance matrix of best fit parameters::
* Troubleshooting: Troubleshooting<2>.
* Examples: Examples<32>.
* References and Further Reading: References and Further Reading<34>.

Solving the Trust Region Subproblem (TRS)

* Levenberg-Marquardt::
* Levenberg-Marquardt with Geodesic Acceleration::
* Dogleg::
* Double Dogleg::
* Two Dimensional Subspace::
* Steihaug-Toint Conjugate Gradient::

Examples

* Exponential Fitting Example::
* Geodesic Acceleration Example 1::
* Geodesic Acceleration Example 2::
* Comparing TRS Methods Example::
* Large Nonlinear Least Squares Example::

Basis Splines

* Overview: Overview<7>.
* Initializing the B-splines solver::
* Constructing the knots vector::
* Evaluation of B-splines::
* Evaluation of B-spline derivatives::
* Working with the Greville abscissae::
* Examples: Examples<33>.
* References and Further Reading: References and Further Reading<35>.

Sparse Matrices

* Data types: Data types<2>.
* Sparse Matrix Storage Formats::
* Overview: Overview<8>.
* Allocation::
* Accessing Matrix Elements::
* Initializing Matrix Elements::
* Reading and Writing Matrices::
* Copying Matrices::
* Exchanging Rows and Columns::
* Matrix Operations::
* Matrix Properties::
* Finding Maximum and Minimum Elements::
* Compressed Format::
* Conversion Between Sparse and Dense Matrices::
* Examples: Examples<34>.
* References and Further Reading: References and Further Reading<36>.

Sparse Matrix Storage Formats

* Coordinate Storage (COO): Coordinate Storage COO.
* Compressed Sparse Column (CSC): Compressed Sparse Column CSC.
* Compressed Sparse Row (CSR): Compressed Sparse Row CSR.

Sparse BLAS Support

* Sparse BLAS operations::
* References and Further Reading: References and Further Reading<37>.

Sparse Linear Algebra

* Overview: Overview<9>.
* Sparse Iterative Solvers::
* Examples: Examples<35>.
* References and Further Reading: References and Further Reading<38>.

Sparse Iterative Solvers

* Overview: Overview<10>.
* Types of Sparse Iterative Solvers::
* Iterating the Sparse Linear System::

Physical Constants

* Fundamental Constants::
* Astronomy and Astrophysics::
* Atomic and Nuclear Physics::
* Measurement of Time::
* Imperial Units::
* Speed and Nautical Units::
* Printers Units::
* Volume, Area and Length: Volume Area and Length.
* Mass and Weight::
* Thermal Energy and Power::
* Pressure::
* Viscosity::
* Light and Illumination::
* Radioactivity::
* Force and Energy::
* Prefixes::
* Examples: Examples<36>.
* References and Further Reading: References and Further Reading<39>.

IEEE floating-point arithmetic

* Representation of floating point numbers::
* Setting up your IEEE environment::
* References and Further Reading: References and Further Reading<40>.

Debugging Numerical Programs

* Using gdb::
* Examining floating point registers::
* Handling floating point exceptions::
* GCC warning options for numerical programs::
* References and Further Reading: References and Further Reading<41>.

GSL CBLAS Library

* Level 1: Level 1<2>.
* Level 2: Level 2<2>.
* Level 3: Level 3<2>.
* Examples: Examples<37>.



File: gsl-ref.info,  Node: Introduction,  Next: Using the Library,  Prev: Top,  Up: Top

1 Introduction
**************

The GNU Scientific Library (GSL) is a collection of routines for
numerical computing.  The routines have been written from scratch in C,
and present a modern Applications Programming Interface (API) for C
programmers, allowing wrappers to be written for very high level
languages.  The source code is distributed under the GNU General Public
License.

* Menu:

* Routines available in GSL::
* GSL is Free Software::
* Obtaining GSL::
* No Warranty::
* Reporting Bugs::
* Further Information::
* Conventions used in this manual::


File: gsl-ref.info,  Node: Routines available in GSL,  Next: GSL is Free Software,  Up: Introduction

1.1 Routines available in GSL
=============================

The library covers a wide range of topics in numerical computing.
Routines are available for the following areas,

Complex Numbers                 Roots of Polynomials            Special Functions
                                                                
                                                                
Vectors and Matrices            Permutations                    Combinations
                                                                
                                                                
Sorting                         BLAS Support                    Linear Algebra
                                                                
                                                                
CBLAS Library                   Fast Fourier Transforms         Eigensystems
                                                                
                                                                
Random Numbers                  Quadrature                      Random Distributions
                                                                
                                                                
Quasi-Random Sequences          Histograms                      Statistics
                                                                
                                                                
Monte Carlo Integration         N-Tuples                        Differential Equations
                                                                
                                                                
Simulated Annealing             Numerical Differentiation       Interpolation
                                                                
                                                                
Series Acceleration             Chebyshev Approximations        Root-Finding
                                                                
                                                                
Discrete Hankel Transforms      Least-Squares Fitting           Minimization
                                                                
                                                                
IEEE Floating-Point             Physical Constants              Basis Splines
                                                                
                                                                
Wavelets                        Sparse BLAS Support             Sparse Linear Algebra
                                                                

The use of these routines is described in this manual.  Each chapter
provides detailed definitions of the functions, followed by example
programs and references to the articles on which the algorithms are
based.

Where possible the routines have been based on reliable public-domain
packages such as FFTPACK and QUADPACK, which the developers of GSL have
reimplemented in C with modern coding conventions.


File: gsl-ref.info,  Node: GSL is Free Software,  Next: Obtaining GSL,  Prev: Routines available in GSL,  Up: Introduction

1.2 GSL is Free Software
========================

The subroutines in the GNU Scientific Library are “free software”; this
means that everyone is free to use them, and to redistribute them in
other free programs.  The library is not in the public domain; it is
copyrighted and there are conditions on its distribution.  These
conditions are designed to permit everything that a good cooperating
citizen would want to do.  What is not allowed is to try to prevent
others from further sharing any version of the software that they might
get from you.

Specifically, we want to make sure that you have the right to share
copies of programs that you are given which use the GNU Scientific
Library, that you receive their source code or else can get it if you
want it, that you can change these programs or use pieces of them in new
free programs, and that you know you can do these things.

To make sure that everyone has such rights, we have to forbid you to
deprive anyone else of these rights.  For example, if you distribute
copies of any code which uses the GNU Scientific Library, you must give
the recipients all the rights that you have received.  You must make
sure that they, too, receive or can get the source code, both to the
library and the code which uses it.  And you must tell them their
rights.  This means that the library should not be redistributed in
proprietary programs.

Also, for our own protection, we must make certain that everyone finds
out that there is no warranty for the GNU Scientific Library.  If these
programs are modified by someone else and passed on, we want their
recipients to know that what they have is not what we distributed, so
that any problems introduced by others will not reflect on our
reputation.

The precise conditions for the distribution of software related to the
GNU Scientific Library are found in the GNU General Public License(1).
Further information about this license is available from the GNU Project
webpage Frequently Asked Questions about the GNU GPL(2).

The Free Software Foundation also operates a license consulting service
for commercial users (contact details available from
‘http://www.fsf.org’.

   ---------- Footnotes ----------

   (1) 
https://www.gnu.org/software/gsl/manual/html_node/GNU-General-Public-License.html#GNU-General-Public-License

   (2) http://www.gnu.org/copyleft/gpl-faq.html


File: gsl-ref.info,  Node: Obtaining GSL,  Next: No Warranty,  Prev: GSL is Free Software,  Up: Introduction

1.3 Obtaining GSL
=================

The source code for the library can be obtained in different ways, by
copying it from a friend, purchasing it on CDROM or downloading it from
the internet.  A list of public ftp servers which carry the source code
can be found on the GNU website, ‘http://www.gnu.org/software/gsl/’.

The preferred platform for the library is a GNU system, which allows it
to take advantage of additional features in the GNU C compiler and GNU C
library.  However, the library is fully portable and should compile on
most systems with a C compiler.

Announcements of new releases, updates and other relevant events are
made on the <info-gsl@gnu.org> mailing list.  To subscribe to this
low-volume list, send an email of the following form:

     To: info-gsl-request@gnu.org
     Subject: subscribe

You will receive a response asking you to reply in order to confirm your
subscription.


File: gsl-ref.info,  Node: No Warranty,  Next: Reporting Bugs,  Prev: Obtaining GSL,  Up: Introduction

1.4 No Warranty
===============

The software described in this manual has no warranty, it is provided
“as is”.  It is your responsibility to validate the behavior of the
routines and their accuracy using the source code provided, or to
purchase support and warranties from commercial redistributors.  Consult
the GNU General Public License(1) for further details.

   ---------- Footnotes ----------

   (1) 
https://www.gnu.org/software/gsl/manual/html_node/GNU-General-Public-License.html#GNU-General-Public-License


File: gsl-ref.info,  Node: Reporting Bugs,  Next: Further Information,  Prev: No Warranty,  Up: Introduction

1.5 Reporting Bugs
==================

A list of known bugs can be found in the ‘BUGS’ file included in the GSL
distribution or online in the GSL bug tracker.  (1) Details of
compilation problems can be found in the ‘INSTALL’ file.

If you find a bug which is not listed in these files, please report it
to <bug-gsl@gnu.org>.

All bug reports should include:

   - The version number of GSL

   - The hardware and operating system

   - The compiler used, including version number and compilation options

   - A description of the bug behavior

   - A short program which exercises the bug

It is useful if you can check whether the same problem occurs when the
library is compiled without optimization.  Thank you.

Any errors or omissions in this manual can also be reported to the same
address.

   ---------- Footnotes ----------

   (1) (1) ‘http://savannah.gnu.org/bugs/?group=gsl’


File: gsl-ref.info,  Node: Further Information,  Next: Conventions used in this manual,  Prev: Reporting Bugs,  Up: Introduction

1.6 Further Information
=======================

Additional information, including online copies of this manual, links to
related projects, and mailing list archives are available from the
website mentioned above.

Any questions about the use and installation of the library can be asked
on the mailing list <help-gsl@gnu.org>.  To subscribe to this list, send
an email of the following form:

     To: help-gsl-request@gnu.org
     Subject: subscribe

This mailing list can be used to ask questions not covered by this
manual, and to contact the developers of the library.

If you would like to refer to the GNU Scientific Library in a journal
article, the recommended way is to cite this reference manual, e.g.:

     M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.

If you want to give a url, use “‘http://www.gnu.org/software/gsl/’”.


File: gsl-ref.info,  Node: Conventions used in this manual,  Prev: Further Information,  Up: Introduction

1.7 Conventions used in this manual
===================================

This manual contains many examples which can be typed at the keyboard.
A command entered at the terminal is shown like this:

     $ command

The first character on the line is the terminal prompt, and should not
be typed.  The dollar sign $ is used as the standard prompt in this
manual, although some systems may use a different character.

The examples assume the use of the GNU operating system.  There may be
minor differences in the output on other systems.  The commands for
setting environment variables use the Bourne shell syntax of the
standard GNU shell (‘bash’).


File: gsl-ref.info,  Node: Using the Library,  Next: Error Handling,  Prev: Introduction,  Up: Top

2 Using the Library
*******************

This chapter describes how to compile programs that use GSL, and
introduces its conventions.

* Menu:

* An Example Program::
* Compiling and Linking::
* Shared Libraries::
* ANSI C Compliance::
* Inline functions::
* Long double::
* Portability functions::
* Alternative optimized functions::
* Support for different numeric types::
* Compatibility with C++::
* Aliasing of arrays::
* Thread-safety::
* Deprecated Functions::
* Code Reuse::


File: gsl-ref.info,  Node: An Example Program,  Next: Compiling and Linking,  Up: Using the Library

2.1 An Example Program
======================

The following short program demonstrates the use of the library by
computing the value of the Bessel function J_0(x) for x=5:

     #include <stdio.h>
     #include <gsl/gsl_sf_bessel.h>

     int
     main (void)
     {
       double x = 5.0;
       double y = gsl_sf_bessel_J0 (x);
       printf ("J0(%g) = %.18e\n", x, y);
       return 0;
     }

The output is shown below, and should be correct to double-precision
accuracy (1),

     J0(5) = -1.775967713143382642e-01

The steps needed to compile this program are described in the following
sections.

   ---------- Footnotes ----------

   (1) (1) The last few digits may vary slightly depending on the
compiler and platform used—this is normal


File: gsl-ref.info,  Node: Compiling and Linking,  Next: Shared Libraries,  Prev: An Example Program,  Up: Using the Library

2.2 Compiling and Linking
=========================

The library header files are installed in their own ‘gsl’ directory.
You should write any preprocessor include statements with a ‘gsl/’
directory prefix thus:

     #include <gsl/gsl_math.h>

If the directory is not installed on the standard search path of your
compiler you will also need to provide its location to the preprocessor
as a command line flag.  The default location of the ‘gsl’ directory is
‘/usr/local/include/gsl’.  A typical compilation command for a source
file ‘example.c’ with the GNU C compiler ‘gcc’ is:

     $ gcc -Wall -I/usr/local/include -c example.c

This results in an object file ‘example.o’.  The default include path
for ‘gcc’ searches ‘/usr/local/include’ automatically so the ‘-I’ option
can actually be omitted when GSL is installed in its default location.

* Menu:

* Linking programs with the library::
* Linking with an alternative BLAS library::


File: gsl-ref.info,  Node: Linking programs with the library,  Next: Linking with an alternative BLAS library,  Up: Compiling and Linking

2.2.1 Linking programs with the library
---------------------------------------

The library is installed as a single file, ‘libgsl.a’.  A shared version
of the library ‘libgsl.so’ is also installed on systems that support
shared libraries.  The default location of these files is
‘/usr/local/lib’.  If this directory is not on the standard search path
of your linker you will also need to provide its location as a command
line flag.

To link against the library you need to specify both the main library
and a supporting CBLAS library, which provides standard basic linear
algebra subroutines.  A suitable CBLAS implementation is provided in the
library ‘libgslcblas.a’ if your system does not provide one.  The
following example shows how to link an application with the library:

     $ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm

The default library path for ‘gcc’ searches ‘/usr/local/lib’
automatically so the ‘-L’ option can be omitted when GSL is installed in
its default location.

The option ‘-lm’ links with the system math library.  On some systems it
is not needed.  (1)

For a tutorial introduction to the GNU C Compiler and related programs,
see “An Introduction to GCC” (ISBN 0954161793).  (2)

   ---------- Footnotes ----------

   (1) (2) It is not needed on MacOS X

   (2) (3) ‘http://www.network-theory.co.uk/gcc/intro/’


File: gsl-ref.info,  Node: Linking with an alternative BLAS library,  Prev: Linking programs with the library,  Up: Compiling and Linking

2.2.2 Linking with an alternative BLAS library
----------------------------------------------

The following command line shows how you would link the same application
with an alternative CBLAS library ‘libcblas.a’:

     $ gcc example.o -lgsl -lcblas -lm

For the best performance an optimized platform-specific CBLAS library
should be used for ‘-lcblas’.  The library must conform to the CBLAS
standard.  The ATLAS package provides a portable high-performance BLAS
library with a CBLAS interface.  It is free software and should be
installed for any work requiring fast vector and matrix operations.  The
following command line will link with the ATLAS library and its CBLAS
interface:

     $ gcc example.o -lgsl -lcblas -latlas -lm

If the ATLAS library is installed in a non-standard directory use the
‘-L’ option to add it to the search path, as described above.

For more information about BLAS functions see *note BLAS Support: 11.


File: gsl-ref.info,  Node: Shared Libraries,  Next: ANSI C Compliance,  Prev: Compiling and Linking,  Up: Using the Library

2.3 Shared Libraries
====================

To run a program linked with the shared version of the library the
operating system must be able to locate the corresponding ‘.so’ file at
runtime.  If the library cannot be found, the following error will
occur:

     $ ./a.out
     ./a.out: error while loading shared libraries:
     libgsl.so.0: cannot open shared object file: No such file or directory

To avoid this error, either modify the system dynamic linker
configuration (1) or define the shell variable ‘LD_LIBRARY_PATH’ to
include the directory where the library is installed.

For example, in the Bourne shell (‘/bin/sh’ or ‘/bin/bash’), the library
search path can be set with the following commands:

     $ LD_LIBRARY_PATH=/usr/local/lib
     $ export LD_LIBRARY_PATH
     $ ./example

In the C-shell (‘/bin/csh’ or ‘/bin/tcsh’) the equivalent command is:

     % setenv LD_LIBRARY_PATH /usr/local/lib

The standard prompt for the C-shell in the example above is the percent
character %, and should not be typed as part of the command.

To save retyping these commands each session they can be placed in an
individual or system-wide login file.

To compile a statically linked version of the program, use the ‘-static’
flag in ‘gcc’:

     $ gcc -static example.o -lgsl -lgslcblas -lm

   ---------- Footnotes ----------

   (1) (4) ‘/etc/ld.so.conf’ on GNU/Linux systems


File: gsl-ref.info,  Node: ANSI C Compliance,  Next: Inline functions,  Prev: Shared Libraries,  Up: Using the Library

2.4 ANSI C Compliance
=====================

The library is written in ANSI C and is intended to conform to the ANSI
C standard (C89).  It should be portable to any system with a working
ANSI C compiler.

The library does not rely on any non-ANSI extensions in the interface it
exports to the user.  Programs you write using GSL can be ANSI
compliant.  Extensions which can be used in a way compatible with pure
ANSI C are supported, however, via conditional compilation.  This allows
the library to take advantage of compiler extensions on those platforms
which support them.

When an ANSI C feature is known to be broken on a particular system the
library will exclude any related functions at compile-time.  This should
make it impossible to link a program that would use these functions and
give incorrect results.

To avoid namespace conflicts all exported function names and variables
have the prefix ‘gsl_’, while exported macros have the prefix ‘GSL_’.


File: gsl-ref.info,  Node: Inline functions,  Next: Long double,  Prev: ANSI C Compliance,  Up: Using the Library

2.5 Inline functions
====================

The ‘inline’ keyword is not part of the original ANSI C standard (C89)
so the library does not export any inline function definitions by
default.  Inline functions were introduced officially in the newer C99
standard but most C89 compilers have also included ‘inline’ as an
extension for a long time.

To allow the use of inline functions, the library provides optional
inline versions of performance-critical routines by conditional
compilation in the exported header files.  The inline versions of these
functions can be included by defining the macro ‘HAVE_INLINE’ when
compiling an application:

     $ gcc -Wall -c -DHAVE_INLINE example.c

If you use ‘autoconf’ this macro can be defined automatically.  If you
do not define the macro ‘HAVE_INLINE’ then the slower non-inlined
versions of the functions will be used instead.

By default, the actual form of the inline keyword is ‘extern inline’,
which is a ‘gcc’ extension that eliminates unnecessary function
definitions.  If the form ‘extern inline’ causes problems with other
compilers a stricter autoconf test can be used, see *note Autoconf
Macros: 16.

When compiling with ‘gcc’ in C99 mode (‘gcc -std=c99’) the header files
automatically switch to C99-compatible inline function declarations
instead of ‘extern inline’.  With other C99 compilers, define the macro
‘GSL_C99_INLINE’ to use these declarations.


File: gsl-ref.info,  Node: Long double,  Next: Portability functions,  Prev: Inline functions,  Up: Using the Library

2.6 Long double
===============

In general, the algorithms in the library are written for double
precision only.  The ‘long double’ type is not supported for actual
computation.

One reason for this choice is that the precision of ‘long double’ is
platform dependent.  The IEEE standard only specifies the minimum
precision of extended precision numbers, while the precision of ‘double’
is the same on all platforms.

However, it is sometimes necessary to interact with external data in
long-double format, so the vector and matrix datatypes include
long-double versions.

It should be noted that in some system libraries the ‘stdio.h’ formatted
input/output functions ‘printf’ and ‘scanf’ are not implemented
correctly for ‘long double’.  Undefined or incorrect results are avoided
by testing these functions during the ‘configure’ stage of library
compilation and eliminating certain GSL functions which depend on them
if necessary.  The corresponding line in the ‘configure’ output looks
like this:

     checking whether printf works with long double... no

Consequently when ‘long double’ formatted input/output does not work on
a given system it should be impossible to link a program which uses GSL
functions dependent on this.

If it is necessary to work on a system which does not support formatted
‘long double’ input/output then the options are to use binary formats or
to convert ‘long double’ results into ‘double’ for reading and writing.


File: gsl-ref.info,  Node: Portability functions,  Next: Alternative optimized functions,  Prev: Long double,  Up: Using the Library

2.7 Portability functions
=========================

To help in writing portable applications GSL provides some
implementations of functions that are found in other libraries, such as
the BSD math library.  You can write your application to use the native
versions of these functions, and substitute the GSL versions via a
preprocessor macro if they are unavailable on another platform.

For example, after determining whether the BSD function ‘hypot()’ is
available you can include the following macro definitions in a file
‘config.h’ with your application:

     /* Substitute gsl_hypot for missing system hypot */

     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

The application source files can then use the include command ‘#include
<config.h>’ to replace each occurrence of ‘hypot()’ by *note
gsl_hypot(): 1a. when ‘hypot()’ is not available.  This substitution can
be made automatically if you use ‘autoconf’, see *note Autoconf Macros:
16.

In most circumstances the best strategy is to use the native versions of
these functions when available, and fall back to GSL versions otherwise,
since this allows your application to take advantage of any
platform-specific optimizations in the system library.  This is the
strategy used within GSL itself.


File: gsl-ref.info,  Node: Alternative optimized functions,  Next: Support for different numeric types,  Prev: Portability functions,  Up: Using the Library

2.8 Alternative optimized functions
===================================

The main implementation of some functions in the library will not be
optimal on all architectures.  For example, there are several ways to
compute a Gaussian random variate and their relative speeds are
platform-dependent.  In cases like this the library provides alternative
implementations of these functions with the same interface.  If you
write your application using calls to the standard implementation you
can select an alternative version later via a preprocessor definition.
It is also possible to introduce your own optimized functions this way
while retaining portability.  The following lines demonstrate the use of
a platform-dependent choice of methods for sampling from the Gaussian
distribution:

     #ifdef SPARC
     #define gsl_ran_gaussian gsl_ran_gaussian_ratio_method
     #endif
     #ifdef INTEL
     #define gsl_ran_gaussian my_gaussian
     #endif

These lines would be placed in the configuration header file ‘config.h’
of the application, which should then be included by all the source
files.  Note that the alternative implementations will not produce
bit-for-bit identical results, and in the case of random number
distributions will produce an entirely different stream of random
variates.


File: gsl-ref.info,  Node: Support for different numeric types,  Next: Compatibility with C++,  Prev: Alternative optimized functions,  Up: Using the Library

2.9 Support for different numeric types
=======================================

Many functions in the library are defined for different numeric types.
This feature is implemented by varying the name of the function with a
type-related modifier—a primitive form of C++ templates.  The modifier
is inserted into the function name after the initial module prefix.  The
following table shows the function names defined for all the numeric
types of an imaginary module ‘gsl_foo’ with function ‘fn()’:

     gsl_foo_fn               double
     gsl_foo_long_double_fn   long double
     gsl_foo_float_fn         float
     gsl_foo_long_fn          long
     gsl_foo_ulong_fn         unsigned long
     gsl_foo_int_fn           int
     gsl_foo_uint_fn          unsigned int
     gsl_foo_short_fn         short
     gsl_foo_ushort_fn        unsigned short
     gsl_foo_char_fn          char
     gsl_foo_uchar_fn         unsigned char

The normal numeric precision ‘double’ is considered the default and does
not require a suffix.  For example, the function *note gsl_stats_mean():
1d. computes the mean of double precision numbers, while the function
‘gsl_stats_int_mean()’ computes the mean of integers.

A corresponding scheme is used for library defined types, such as
‘gsl_vector’ and ‘gsl_matrix’.  In this case the modifier is appended to
the type name.  For example, if a module defines a new type-dependent
struct or typedef ‘gsl_foo’ it is modified for other types in the
following way:

     gsl_foo                  double
     gsl_foo_long_double      long double
     gsl_foo_float            float
     gsl_foo_long             long
     gsl_foo_ulong            unsigned long
     gsl_foo_int              int
     gsl_foo_uint             unsigned int
     gsl_foo_short            short
     gsl_foo_ushort           unsigned short
     gsl_foo_char             char
     gsl_foo_uchar            unsigned char

When a module contains type-dependent definitions the library provides
individual header files for each type.  The filenames are modified as
shown in the below.  For convenience the default header includes the
definitions for all the types.  To include only the double precision
header file, or any other specific type, use its individual filename:

     #include <gsl/gsl_foo.h>               All types
     #include <gsl/gsl_foo_double.h>        double
     #include <gsl/gsl_foo_long_double.h>   long double
     #include <gsl/gsl_foo_float.h>         float
     #include <gsl/gsl_foo_long.h>          long
     #include <gsl/gsl_foo_ulong.h>         unsigned long
     #include <gsl/gsl_foo_int.h>           int
     #include <gsl/gsl_foo_uint.h>          unsigned int
     #include <gsl/gsl_foo_short.h>         short
     #include <gsl/gsl_foo_ushort.h>        unsigned short
     #include <gsl/gsl_foo_char.h>          char
     #include <gsl/gsl_foo_uchar.h>         unsigned char


File: gsl-ref.info,  Node: Compatibility with C++,  Next: Aliasing of arrays,  Prev: Support for different numeric types,  Up: Using the Library

2.10 Compatibility with C++
===========================

The library header files automatically define functions to have ‘extern
"C"’ linkage when included in C++ programs.  This allows the functions
to be called directly from C++.

To use C++ exception handling within user-defined functions passed to
the library as parameters, the library must be built with the additional
‘CFLAGS’ compilation option ‘-fexceptions’.


File: gsl-ref.info,  Node: Aliasing of arrays,  Next: Thread-safety,  Prev: Compatibility with C++,  Up: Using the Library

2.11 Aliasing of arrays
=======================

The library assumes that arrays, vectors and matrices passed as
modifiable arguments are not aliased and do not overlap with each other.
This removes the need for the library to handle overlapping memory
regions as a special case, and allows additional optimizations to be
used.  If overlapping memory regions are passed as modifiable arguments
then the results of such functions will be undefined.  If the arguments
will not be modified (for example, if a function prototype declares them
as ‘const’ arguments) then overlapping or aliased memory regions can be
safely used.


File: gsl-ref.info,  Node: Thread-safety,  Next: Deprecated Functions,  Prev: Aliasing of arrays,  Up: Using the Library

2.12 Thread-safety
==================

The library can be used in multi-threaded programs.  All the functions
are thread-safe, in the sense that they do not use static variables.
Memory is always associated with objects and not with functions.  For
functions which use `workspace' objects as temporary storage the
workspaces should be allocated on a per-thread basis.  For functions
which use `table' objects as read-only memory the tables can be used by
multiple threads simultaneously.  Table arguments are always declared
‘const’ in function prototypes, to indicate that they may be safely
accessed by different threads.

There are a small number of static global variables which are used to
control the overall behavior of the library (e.g.  whether to use
range-checking, the function to call on fatal error, etc).  These
variables are set directly by the user, so they should be initialized
once at program startup and not modified by different threads.


File: gsl-ref.info,  Node: Deprecated Functions,  Next: Code Reuse,  Prev: Thread-safety,  Up: Using the Library

2.13 Deprecated Functions
=========================

From time to time, it may be necessary for the definitions of some
functions to be altered or removed from the library.  In these
circumstances the functions will first be declared `deprecated' and then
removed from subsequent versions of the library.  Functions that are
deprecated can be disabled in the current release by setting the
preprocessor definition ‘GSL_DISABLE_DEPRECATED’.  This allows existing
code to be tested for forwards compatibility.


File: gsl-ref.info,  Node: Code Reuse,  Prev: Deprecated Functions,  Up: Using the Library

2.14 Code Reuse
===============

Where possible the routines in the library have been written to avoid
dependencies between modules and files.  This should make it possible to
extract individual functions for use in your own applications, without
needing to have the whole library installed.  You may need to define
certain macros such as ‘GSL_ERROR’ and remove some ‘#include’ statements
in order to compile the files as standalone units.  Reuse of the library
code in this way is encouraged, subject to the terms of the GNU General
Public License.


File: gsl-ref.info,  Node: Error Handling,  Next: Mathematical Functions,  Prev: Using the Library,  Up: Top

3 Error Handling
****************

This chapter describes the way that GSL functions report and handle
errors.  By examining the status information returned by every function
you can determine whether it succeeded or failed, and if it failed you
can find out what the precise cause of failure was.  You can also define
your own error handling functions to modify the default behavior of the
library.

The functions described in this section are declared in the header file
‘gsl_errno.h’.

* Menu:

* Error Reporting::
* Error Codes::
* Error Handlers::
* Using GSL error reporting in your own functions::
* Examples::


File: gsl-ref.info,  Node: Error Reporting,  Next: Error Codes,  Up: Error Handling

3.1 Error Reporting
===================

The library follows the thread-safe error reporting conventions of the
POSIX Threads library.  Functions return a non-zero error code to
indicate an error and ‘0’ to indicate success:

     int status = gsl_function (...)

     if (status) { /* an error occurred */
       .....
       /* status value specifies the type of error */
     }

The routines report an error whenever they cannot perform the task
requested of them.  For example, a root-finding function would return a
non-zero error code if could not converge to the requested accuracy, or
exceeded a limit on the number of iterations.  Situations like this are
a normal occurrence when using any mathematical library and you should
check the return status of the functions that you call.

Whenever a routine reports an error the return value specifies the type
of error.  The return value is analogous to the value of the variable
‘errno’ in the C library.  The caller can examine the return code and
decide what action to take, including ignoring the error if it is not
considered serious.

In addition to reporting errors by return codes the library also has an
error handler function ‘gsl_error()’.  This function is called by other
library functions when they report an error, just before they return to
the caller.  The default behavior of the error handler is to print a
message and abort the program:

     gsl: file.c:67: ERROR: invalid argument supplied by user
     Default GSL error handler invoked.
     Aborted

The purpose of the ‘gsl_error()’ handler is to provide a function where
a breakpoint can be set that will catch library errors when running
under the debugger.  It is not intended for use in production programs,
which should handle any errors using the return codes.


File: gsl-ref.info,  Node: Error Codes,  Next: Error Handlers,  Prev: Error Reporting,  Up: Error Handling

3.2 Error Codes
===============

The error code numbers returned by library functions are defined in the
file ‘gsl_errno.h’.  They all have the prefix ‘GSL_’ and expand to
non-zero constant integer values.  Error codes above 1024 are reserved
for applications, and are not used by the library.  Many of the error
codes use the same base name as the corresponding error code in the C
library.  Here are some of the most common error codes,

 -- Variable: int GSL_EDOM

     Domain error; used by mathematical functions when an argument value
     does not fall into the domain over which the function is defined
     (like ‘EDOM’ in the C library)

 -- Variable: int GSL_ERANGE

     Range error; used by mathematical functions when the result value
     is not representable because of overflow or underflow (like
     ‘ERANGE’ in the C library)

 -- Variable: int GSL_ENOMEM

     No memory available.  The system cannot allocate more virtual
     memory because its capacity is full (like ‘ENOMEM’ in the C
     library).  This error is reported when a GSL routine encounters
     problems when trying to allocate memory with ‘malloc()’.

 -- Variable: int GSL_EINVAL

     Invalid argument.  This is used to indicate various kinds of
     problems with passing the wrong argument to a library function
     (like ‘EINVAL’ in the C library).

The error codes can be converted into an error message using the
function *note gsl_strerror(): 2c.

 -- Function: const char *gsl_strerror (const int gsl_errno)

     This function returns a pointer to a string describing the error
     code *note gsl_errno: 2c.  For example:

          printf ("error: %s\n", gsl_strerror (status));

     would print an error message like ‘error: output range error’ for a
     status value of *note GSL_ERANGE: 29.


File: gsl-ref.info,  Node: Error Handlers,  Next: Using GSL error reporting in your own functions,  Prev: Error Codes,  Up: Error Handling

3.3 Error Handlers
==================

The default behavior of the GSL error handler is to print a short
message and call ‘abort()’.  When this default is in use programs will
stop with a core-dump whenever a library routine reports an error.  This
is intended as a fail-safe default for programs which do not check the
return status of library routines (we don’t encourage you to write
programs this way).

If you turn off the default error handler it is your responsibility to
check the return values of routines and handle them yourself.  You can
also customize the error behavior by providing a new error handler.  For
example, an alternative error handler could log all errors to a file,
ignore certain error conditions (such as underflows), or start the
debugger and attach it to the current process when an error occurs.

All GSL error handlers have the type ‘gsl_error_handler_t’, which is
defined in ‘gsl_errno.h’,

 -- Type: gsl_error_handler_t

     This is the type of GSL error handler functions.  An error handler
     will be passed four arguments which specify the reason for the
     error (a string), the name of the source file in which it occurred
     (also a string), the line number in that file (an integer) and the
     error number (an integer).  The source file and line number are set
     at compile time using the ‘__FILE__’ and ‘__LINE__’ directives in
     the preprocessor.  An error handler function returns type ‘void’.
     Error handler functions should be defined like this:

          void handler (const char * reason,
                        const char * file,
                        int line,
                        int gsl_errno)

To request the use of your own error handler you need to call the
function *note gsl_set_error_handler(): 2f. which is also declared in
‘gsl_errno.h’,

 -- Function: *note gsl_error_handler_t: 2e. *gsl_set_error_handler
          (gsl_error_handler_t *new_handler)

     This function sets a new error handler, *note new_handler: 2f, for
     the GSL library routines.  The previous handler is returned (so
     that you can restore it later).  Note that the pointer to a user
     defined error handler function is stored in a static variable, so
     there can be only one error handler per program.  This function
     should be not be used in multi-threaded programs except to set up a
     program-wide error handler from a master thread.  The following
     example shows how to set and restore a new error handler:

          /* save original handler, install new handler */
          old_handler = gsl_set_error_handler (&my_handler);

          /* code uses new handler */
          .....

          /* restore original handler */
          gsl_set_error_handler (old_handler);

     To use the default behavior (‘abort()’ on error) set the error
     handler to ‘NULL’:

          old_handler = gsl_set_error_handler (NULL);

 -- Function: *note gsl_error_handler_t: 2e. *gsl_set_error_handler_off
          ()

     This function turns off the error handler by defining an error
     handler which does nothing.  This will cause the program to
     continue after any error, so the return values from any library
     routines must be checked.  This is the recommended behavior for
     production programs.  The previous handler is returned (so that you
     can restore it later).

The error behavior can be changed for specific applications by
recompiling the library with a customized definition of the ‘GSL_ERROR’
macro in the file ‘gsl_errno.h’.


File: gsl-ref.info,  Node: Using GSL error reporting in your own functions,  Next: Examples,  Prev: Error Handlers,  Up: Error Handling

3.4 Using GSL error reporting in your own functions
===================================================

If you are writing numerical functions in a program which also uses GSL
code you may find it convenient to adopt the same error reporting
conventions as in the library.

To report an error you need to call the function ‘gsl_error()’ with a
string describing the error and then return an appropriate error code
from ‘gsl_errno.h’, or a special value, such as ‘NaN’.  For convenience
the file ‘gsl_errno.h’ defines two macros which carry out these steps:

 -- Macro: GSL_ERROR (reason, gsl_errno)

     This macro reports an error using the GSL conventions and returns a
     status value of ‘gsl_errno’.  It expands to the following code
     fragment:

          gsl_error (reason, __FILE__, __LINE__, gsl_errno);
          return gsl_errno;

     The macro definition in ‘gsl_errno.h’ actually wraps the code in a
     ‘do { ... } while (0)’ block to prevent possible parsing problems.

Here is an example of how the macro could be used to report that a
routine did not achieve a requested tolerance.  To report the error the
routine needs to return the error code ‘GSL_ETOL’:

     if (residual > tolerance)
       {
         GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
       }

 -- Macro: GSL_ERROR_VAL (reason, gsl_errno, value)

     This macro is the same as ‘GSL_ERROR’ but returns a user-defined
     value of ‘value’ instead of an error code.  It can be used for
     mathematical functions that return a floating point value.

The following example shows how to return a ‘NaN’ at a mathematical
singularity using the ‘GSL_ERROR_VAL’ macro:

     if (x == 0)
       {
         GSL_ERROR_VAL("argument lies on singularity", GSL_ERANGE, GSL_NAN);
       }


File: gsl-ref.info,  Node: Examples,  Prev: Using GSL error reporting in your own functions,  Up: Error Handling

3.5 Examples
============

Here is an example of some code which checks the return value of a
function where an error might be reported:

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>

     ...
       int status;
       size_t n = 37;

       gsl_set_error_handler_off();

       status = gsl_fft_complex_radix2_forward (data, stride, n);

       if (status) {
         if (status == GSL_EINVAL) {
            fprintf (stderr, "invalid argument, n=%d\n", n);
         } else {
            fprintf (stderr, "failed, gsl_errno=%d\n", status);
         }
         exit (-1);
       }
     ...

The function *note gsl_fft_complex_radix2_forward(): 35. only accepts
integer lengths which are a power of two.  If the variable ‘n’ is not a
power of two then the call to the library function will return
‘GSL_EINVAL’, indicating that the length argument is invalid.  The
function call to *note gsl_set_error_handler_off(): 30. stops the
default error handler from aborting the program.  The ‘else’ clause
catches any other possible errors.


File: gsl-ref.info,  Node: Mathematical Functions,  Next: Complex Numbers,  Prev: Error Handling,  Up: Top

4 Mathematical Functions
************************

This chapter describes basic mathematical functions.  Some of these
functions are present in system libraries, but the alternative versions
given here can be used as a substitute when the system functions are not
available.

The functions and macros described in this chapter are defined in the
header file ‘gsl_math.h’.

* Menu:

* Mathematical Constants::
* Infinities and Not-a-number::
* Elementary Functions::
* Small integer powers::
* Testing the Sign of Numbers::
* Testing for Odd and Even Numbers::
* Maximum and Minimum functions::
* Approximate Comparison of Floating Point Numbers::


File: gsl-ref.info,  Node: Mathematical Constants,  Next: Infinities and Not-a-number,  Up: Mathematical Functions

4.1 Mathematical Constants
==========================

The library ensures that the standard BSD mathematical constants are
defined.  For reference, here is a list of the constants:

‘M_E’                     The base of exponentials, e
                          
                          
‘M_LOG2E’                 The base-2 logarithm of e, \log_2 (e)
                          
                          
‘M_LOG10E’                The base-10 logarithm of e, \log_{10} (e)
                          
                          
‘M_SQRT2’                 The square root of two, \sqrt 2
                          
                          
‘M_SQRT1_2’               The square root of one-half, \sqrt{1/2}
                          
                          
‘M_SQRT3’                 The square root of three, \sqrt 3
                          
                          
‘M_PI’                    The constant pi, \pi
                          
                          
‘M_PI_2’                  Pi divided by two, \pi/2
                          
                          
‘M_PI_4’                  Pi divided by four, \pi/4
                          
                          
‘M_SQRTPI’                The square root of pi, \sqrt\pi
                          
                          
‘M_2_SQRTPI’              Two divided by the square root of pi, 2/\sqrt\pi
                          
                          
‘M_1_PI’                  The reciprocal of pi, 1/\pi
                          
                          
‘M_2_PI’                  Twice the reciprocal of pi, 2/\pi
                          
                          
‘M_LN10’                  The natural logarithm of ten, \ln(10)
                          
                          
‘M_LN2’                   The natural logarithm of two, \ln(2)
                          
                          
‘M_LNPI’                  The natural logarithm of pi, \ln(\pi)
                          
                          
‘M_EULER’                 Euler’s constant, \gamma
                          


File: gsl-ref.info,  Node: Infinities and Not-a-number,  Next: Elementary Functions,  Prev: Mathematical Constants,  Up: Mathematical Functions

4.2 Infinities and Not-a-number
===============================

 -- Macro: GSL_POSINF

     This macro contains the IEEE representation of positive infinity,
     +\infty.  It is computed from the expression ‘+1.0/0.0’.

 -- Macro: GSL_NEGINF

     This macro contains the IEEE representation of negative infinity,
     -\infty.  It is computed from the expression ‘-1.0/0.0’.

 -- Macro: GSL_NAN

     This macro contains the IEEE representation of the Not-a-Number
     symbol, ‘NaN’.  It is computed from the ratio ‘0.0/0.0’.

 -- Function: int gsl_isnan (const double x)

     This function returns 1 if *note x: 3d. is not-a-number.

 -- Function: int gsl_isinf (const double x)

     This function returns +1 if *note x: 3e. is positive infinity, -1
     if *note x: 3e. is negative infinity and 0 otherwise.  (1)

 -- Function: int gsl_finite (const double x)

     This function returns 1 if *note x: 3f. is a real number, and 0 if
     it is infinite or not-a-number.

   ---------- Footnotes ----------

   (1) (1) Note that the C99 standard only requires the system ‘isinf()’
function to return a non-zero value, without the sign of the infinity.
The implementation in some earlier versions of GSL used the system
‘isinf()’ function and may have this behavior on some platforms.
Therefore, it is advisable to test the sign of ‘x’ separately, if
needed, rather than relying the sign of the return value from *note
gsl_isinf(): 3e.


File: gsl-ref.info,  Node: Elementary Functions,  Next: Small integer powers,  Prev: Infinities and Not-a-number,  Up: Mathematical Functions

4.3 Elementary Functions
========================

The following routines provide portable implementations of functions
found in the BSD math library.  When native versions are not available
the functions described here can be used instead.  The substitution can
be made automatically if you use ‘autoconf’ to compile your application
(see *note Portability functions: 19.).

 -- Function: double gsl_log1p (const double x)

     This function computes the value of \log(1+x) in a way that is
     accurate for small *note x: 41.  It provides an alternative to the
     BSD math function ‘log1p(x)’.

 -- Function: double gsl_expm1 (const double x)

     This function computes the value of \exp(x)-1 in a way that is
     accurate for small *note x: 42.  It provides an alternative to the
     BSD math function ‘expm1(x)’.

 -- Function: double gsl_hypot (const double x, const double y)

     This function computes the value of \sqrt{x^2 + y^2} in a way that
     avoids overflow.  It provides an alternative to the BSD math
     function ‘hypot(x,y)’.

 -- Function: double gsl_hypot3 (const double x, const double y, const
          double z)

     This function computes the value of \sqrt{x^2 + y^2 + z^2} in a way
     that avoids overflow.

 -- Function: double gsl_acosh (const double x)

     This function computes the value of \arccosh{(x)}.  It provides an
     alternative to the standard math function ‘acosh(x)’.

 -- Function: double gsl_asinh (const double x)

     This function computes the value of \arcsinh{(x)}.  It provides an
     alternative to the standard math function ‘asinh(x)’.

 -- Function: double gsl_atanh (const double x)

     This function computes the value of \arctanh{(x)}.  It provides an
     alternative to the standard math function ‘atanh(x)’.

 -- Function: double gsl_ldexp (double x, int e)

     This function computes the value of x * 2^e.  It provides an
     alternative to the standard math function ‘ldexp(x,e)’.

 -- Function: double gsl_frexp (double x, int *e)

     This function splits the number *note x: 48. into its normalized
     fraction f and exponent e, such that x = f * 2^e and 0.5 <= f < 1.
     The function returns f and stores the exponent in e.  If x is zero,
     both f and e are set to zero.  This function provides an
     alternative to the standard math function ‘frexp(x, e)’.


File: gsl-ref.info,  Node: Small integer powers,  Next: Testing the Sign of Numbers,  Prev: Elementary Functions,  Up: Mathematical Functions

4.4 Small integer powers
========================

A common complaint about the standard C library is its lack of a
function for calculating (small) integer powers.  GSL provides some
simple functions to fill this gap.  For reasons of efficiency, these
functions do not check for overflow or underflow conditions.

 -- Function: double gsl_pow_int (double x, int n)
 -- Function: double gsl_pow_uint (double x, unsigned int n)

     These routines computes the power x^n for integer *note n: 4b.  The
     power is computed efficiently—for example, x^8 is computed as
     ((x^2)^2)^2, requiring only 3 multiplications.  A version of this
     function which also computes the numerical error in the result is
     available as *note gsl_sf_pow_int_e(): 4c.

 -- Function: double gsl_pow_2 (const double x)
 -- Function: double gsl_pow_3 (const double x)
 -- Function: double gsl_pow_4 (const double x)
 -- Function: double gsl_pow_5 (const double x)
 -- Function: double gsl_pow_6 (const double x)
 -- Function: double gsl_pow_7 (const double x)
 -- Function: double gsl_pow_8 (const double x)
 -- Function: double gsl_pow_9 (const double x)

     These functions can be used to compute small integer powers x^2,
     x^3, etc.  efficiently.  The functions will be inlined when
     ‘HAVE_INLINE’ is defined, so that use of these functions should be
     as efficient as explicitly writing the corresponding product
     expression:

          #include <gsl/gsl_math.h>
          double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */


File: gsl-ref.info,  Node: Testing the Sign of Numbers,  Next: Testing for Odd and Even Numbers,  Prev: Small integer powers,  Up: Mathematical Functions

4.5 Testing the Sign of Numbers
===============================

 -- Macro: GSL_SIGN (x)

     This macro returns the sign of ‘x’.  It is defined as ‘((x) >= 0 ?
     1 : -1)’.  Note that with this definition the sign of zero is
     positive (regardless of its IEEE sign bit).


File: gsl-ref.info,  Node: Testing for Odd and Even Numbers,  Next: Maximum and Minimum functions,  Prev: Testing the Sign of Numbers,  Up: Mathematical Functions

4.6 Testing for Odd and Even Numbers
====================================

 -- Macro: GSL_IS_ODD (n)

     This macro evaluates to 1 if ‘n’ is odd and 0 if ‘n’ is even.  The
     argument ‘n’ must be of integer type.

 -- Macro: GSL_IS_EVEN (n)

     This macro is the opposite of *note GSL_IS_ODD: 58.  It evaluates
     to 1 if ‘n’ is even and 0 if ‘n’ is odd.  The argument ‘n’ must be
     of integer type.


File: gsl-ref.info,  Node: Maximum and Minimum functions,  Next: Approximate Comparison of Floating Point Numbers,  Prev: Testing for Odd and Even Numbers,  Up: Mathematical Functions

4.7 Maximum and Minimum functions
=================================

Note that the following macros perform multiple evaluations of their
arguments, so they should not be used with arguments that have side
effects (such as a call to a random number generator).

 -- Macro: GSL_MAX (a, b)

     This macro returns the maximum of ‘a’ and ‘b’.  It is defined as
     ‘((a) > (b) ? (a):(b))’.

 -- Macro: GSL_MIN (a, b)

     This macro returns the minimum of ‘a’ and ‘b’.  It is defined as
     ‘((a) < (b) ? (a):(b))’.

 -- Function: extern inline double GSL_MAX_DBL (double a, double b)

     This function returns the maximum of the double precision numbers
     *note a: 5d. and *note b: 5d. using an inline function.  The use of
     a function allows for type checking of the arguments as an extra
     safety feature.  On platforms where inline functions are not
     available the macro *note GSL_MAX: 5b. will be automatically
     substituted.

 -- Function: extern inline double GSL_MIN_DBL (double a, double b)

     This function returns the minimum of the double precision numbers
     *note a: 5e. and *note b: 5e. using an inline function.  The use of
     a function allows for type checking of the arguments as an extra
     safety feature.  On platforms where inline functions are not
     available the macro *note GSL_MIN: 5c. will be automatically
     substituted.

 -- Function: extern inline int GSL_MAX_INT (int a, int b)
 -- Function: extern inline int GSL_MIN_INT (int a, int b)

     These functions return the maximum or minimum of the integers *note
     a: 60. and *note b: 60. using an inline function.  On platforms
     where inline functions are not available the macros *note GSL_MAX:
     5b. or *note GSL_MIN: 5c. will be automatically substituted.

 -- Function: extern inline long double GSL_MAX_LDBL (long double a,
          long double b)
 -- Function: extern inline long double GSL_MIN_LDBL (long double a,
          long double b)

     These functions return the maximum or minimum of the long doubles
     *note a: 62. and *note b: 62. using an inline function.  On
     platforms where inline functions are not available the macros *note
     GSL_MAX: 5b. or *note GSL_MIN: 5c. will be automatically
     substituted.


File: gsl-ref.info,  Node: Approximate Comparison of Floating Point Numbers,  Prev: Maximum and Minimum functions,  Up: Mathematical Functions

4.8 Approximate Comparison of Floating Point Numbers
====================================================

It is sometimes useful to be able to compare two floating point numbers
approximately, to allow for rounding and truncation errors.  The
following function implements the approximate floating-point comparison
algorithm proposed by D.E. Knuth in Section 4.2.2 of “Seminumerical
Algorithms” (3rd edition).

 -- Function: int gsl_fcmp (double x, double y, double epsilon)

     This function determines whether *note x: 64. and *note y: 64. are
     approximately equal to a relative accuracy *note epsilon: 64.

     The relative accuracy is measured using an interval of size 2
     \delta, where \delta = 2^k \epsilon and k is the maximum base-2
     exponent of x and y as computed by the function ‘frexp()’.

     If x and y lie within this interval, they are considered
     approximately equal and the function returns 0.  Otherwise if x <
     y, the function returns -1, or if x > y, the function returns +1.

     Note that x and y are compared to relative accuracy, so this
     function is not suitable for testing whether a value is
     approximately zero.

     The implementation is based on the package ‘fcmp’ by T.C. Belding.


File: gsl-ref.info,  Node: Complex Numbers,  Next: Polynomials,  Prev: Mathematical Functions,  Up: Top

5 Complex Numbers
*****************

The functions described in this chapter provide support for complex
numbers.  The algorithms take care to avoid unnecessary intermediate
underflows and overflows, allowing the functions to be evaluated over as
much of the complex plane as possible.

For multiple-valued functions the branch cuts have been chosen to follow
the conventions of Abramowitz and Stegun.  The functions return
principal values which are the same as those in GNU Calc, which in turn
are the same as those in “Common Lisp, The Language (Second Edition)”
(1) and the HP-28/48 series of calculators.

The complex types are defined in the header file ‘gsl_complex.h’, while
the corresponding complex functions and arithmetic operations are
defined in ‘gsl_complex_math.h’.

* Menu:

* Representation of complex numbers::
* Complex number macros::
* Assigning complex numbers::
* Properties of complex numbers::
* Complex arithmetic operators::
* Elementary Complex Functions::
* Complex Trigonometric Functions::
* Inverse Complex Trigonometric Functions::
* Complex Hyperbolic Functions::
* Inverse Complex Hyperbolic Functions::
* References and Further Reading::

   ---------- Footnotes ----------

   (1) (1) Note that the first edition uses different definitions.


File: gsl-ref.info,  Node: Representation of complex numbers,  Next: Complex number macros,  Up: Complex Numbers

5.1 Representation of complex numbers
=====================================

Complex numbers are represented using the type ‘gsl_complex’.  The
default interface defines ‘gsl_complex’ as:

     typedef struct
     {
       double dat[2];
     } gsl_complex;

The real and imaginary part are stored in contiguous elements of a two
element array.  This eliminates any padding between the real and
imaginary parts, ‘dat[0]’ and ‘dat[1]’, allowing the struct to be mapped
correctly onto packed complex arrays.

If a C compiler is available which supports the C11 standard, and the
‘<complex.h>’ header file is included `prior' to ‘gsl_complex.h’, then
‘gsl_complex’ will be defined to be the native C complex type:

     typedef double complex gsl_complex

This allows users to use ‘gsl_complex’ in ordinary operations such as:

     gsl_complex x = 2 + 5 * I;
     gsl_complex y = x + (3 - 4*I);

     Important: Native C support for complex numbers was introduced in
     the C99 standard, and additional functionality was added in C11.
     When ‘<complex.h>’ is included in a user’s program prior to
     ‘gsl_complex.h’, GSL uses the new C11 functionality to define the
     *note GSL_REAL: 68. and *note GSL_IMAG: 69. macros.  It does not
     appear possible to properly define these macros using the C99
     standard, and so using a C99 compiler will not define ‘gsl_complex’
     to the native complex type.

     Some compilers, such as the gcc 4.8 series implement only a portion
     of the C11 standard and so they may fail to correctly compile GSL
     code when a user tries to turn on native complex functionality.  A
     workaround for this issue is to either remove ‘<complex.h>’ from
     the include list, or add ‘-DGSL_COMPLEX_LEGACY’ to the compiler
     flags, which will use the older struct-based definition of
     ‘gsl_complex’.


File: gsl-ref.info,  Node: Complex number macros,  Next: Assigning complex numbers,  Prev: Representation of complex numbers,  Up: Complex Numbers

5.2 Complex number macros
=========================

The following C macros offer convenient ways to manipulate complex
numbers.

 -- Macro: GSL_REAL (z)
 -- Macro: GSL_IMAG (z)

     These macros return a memory location (lvalue) corresponding to the
     real and imaginary parts respectively of the complex number ‘z’.
     This allows users to perform operations like:

          gsl_complex x, y;

          GSL_REAL(x) = 4;
          GSL_IMAG(x) = 2;

          GSL_REAL(y) = GSL_REAL(x);
          GSL_IMAG(y) = GSL_REAL(x);

     In other words, these macros can both read and write to the real
     and imaginary parts of a complex variable.

 -- Macro: GSL_SET_COMPLEX (zp, x, y)

     This macro uses the Cartesian components (‘x’, ‘y’) to set the real
     and imaginary parts of the complex number pointed to by ‘zp’.  For
     example:

          GSL_SET_COMPLEX(&z, 3, 4)

     sets z to be 3 + 4i.


File: gsl-ref.info,  Node: Assigning complex numbers,  Next: Properties of complex numbers,  Prev: Complex number macros,  Up: Complex Numbers

5.3 Assigning complex numbers
=============================

 -- Function: gsl_complex gsl_complex_rect (double x, double y)

     This function uses the rectangular Cartesian components (x,y) to
     return the complex number z = x + i y.  An inline version of this
     function is used when ‘HAVE_INLINE’ is defined.

 -- Function: gsl_complex gsl_complex_polar (double r, double theta)

     This function returns the complex number z = r \exp(i \theta) = r
     (\cos(\theta) + i \sin(\theta)) from the polar representation
     (*note r: 6e, *note theta: 6e.).


File: gsl-ref.info,  Node: Properties of complex numbers,  Next: Complex arithmetic operators,  Prev: Assigning complex numbers,  Up: Complex Numbers

5.4 Properties of complex numbers
=================================

 -- Function: double gsl_complex_arg (gsl_complex z)

     This function returns the argument of the complex number *note z:
     70, \arg(z), where -\pi < \arg(z) <= \pi.

 -- Function: double gsl_complex_abs (gsl_complex z)

     This function returns the magnitude of the complex number *note z:
     71, |z|.

 -- Function: double gsl_complex_abs2 (gsl_complex z)

     This function returns the squared magnitude of the complex number
     *note z: 72, |z|^2.

 -- Function: double gsl_complex_logabs (gsl_complex z)

     This function returns the natural logarithm of the magnitude of the
     complex number *note z: 73, \log|z|.  It allows an accurate
     evaluation of \log|z| when |z| is close to one.  The direct
     evaluation of ‘log(gsl_complex_abs(z))’ would lead to a loss of
     precision in this case.


File: gsl-ref.info,  Node: Complex arithmetic operators,  Next: Elementary Complex Functions,  Prev: Properties of complex numbers,  Up: Complex Numbers

5.5 Complex arithmetic operators
================================

 -- Function: gsl_complex gsl_complex_add (gsl_complex a, gsl_complex b)

     This function returns the sum of the complex numbers *note a: 75.
     and *note b: 75, z=a+b.

 -- Function: gsl_complex gsl_complex_sub (gsl_complex a, gsl_complex b)

     This function returns the difference of the complex numbers *note
     a: 76. and *note b: 76, z=a-b.

 -- Function: gsl_complex gsl_complex_mul (gsl_complex a, gsl_complex b)

     This function returns the product of the complex numbers *note a:
     77. and *note b: 77, z=ab.

 -- Function: gsl_complex gsl_complex_div (gsl_complex a, gsl_complex b)

     This function returns the quotient of the complex numbers *note a:
     78. and *note b: 78, z=a/b.

 -- Function: gsl_complex gsl_complex_add_real (gsl_complex a, double x)

     This function returns the sum of the complex number *note a: 79.
     and the real number *note x: 79, z=a+x.

 -- Function: gsl_complex gsl_complex_sub_real (gsl_complex a, double x)

     This function returns the difference of the complex number *note a:
     7a. and the real number *note x: 7a, z=a-x.

 -- Function: gsl_complex gsl_complex_mul_real (gsl_complex a, double x)

     This function returns the product of the complex number *note a:
     7b. and the real number *note x: 7b, z=ax.

 -- Function: gsl_complex gsl_complex_div_real (gsl_complex a, double x)

     This function returns the quotient of the complex number *note a:
     7c. and the real number *note x: 7c, z=a/x.

 -- Function: gsl_complex gsl_complex_add_imag (gsl_complex a, double y)

     This function returns the sum of the complex number *note a: 7d.
     and the imaginary number iy, z=a+iy.

 -- Function: gsl_complex gsl_complex_sub_imag (gsl_complex a, double y)

     This function returns the difference of the complex number *note a:
     7e. and the imaginary number iy, z=a-iy.

 -- Function: gsl_complex gsl_complex_mul_imag (gsl_complex a, double y)

     This function returns the product of the complex number *note a:
     7f. and the imaginary number iy, z=a*(iy).

 -- Function: gsl_complex gsl_complex_div_imag (gsl_complex a, double y)

     This function returns the quotient of the complex number *note a:
     80. and the imaginary number iy, z=a/(iy).

 -- Function: gsl_complex gsl_complex_conjugate (gsl_complex z)

     This function returns the complex conjugate of the complex number
     *note z: 81, z^* = x - i y.

 -- Function: gsl_complex gsl_complex_inverse (gsl_complex z)

     This function returns the inverse, or reciprocal, of the complex
     number *note z: 82, 1/z = (x - i y)/(x^2 + y^2).

 -- Function: gsl_complex gsl_complex_negative (gsl_complex z)

     This function returns the negative of the complex number *note z:
     83, -z = (-x) + i(-y).


File: gsl-ref.info,  Node: Elementary Complex Functions,  Next: Complex Trigonometric Functions,  Prev: Complex arithmetic operators,  Up: Complex Numbers

5.6 Elementary Complex Functions
================================

 -- Function: gsl_complex gsl_complex_sqrt (gsl_complex z)

     This function returns the square root of the complex number *note
     z: 85, \sqrt z.  The branch cut is the negative real axis.  The
     result always lies in the right half of the complex plane.

 -- Function: gsl_complex gsl_complex_sqrt_real (double x)

     This function returns the complex square root of the real number
     *note x: 86, where *note x: 86. may be negative.

 -- Function: gsl_complex gsl_complex_pow (gsl_complex z, gsl_complex a)

     The function returns the complex number *note z: 87. raised to the
     complex power *note a: 87, z^a.  This is computed as
     \exp(\log(z)*a) using complex logarithms and complex exponentials.

 -- Function: gsl_complex gsl_complex_pow_real (gsl_complex z, double x)

     This function returns the complex number *note z: 88. raised to the
     real power *note x: 88, z^x.

 -- Function: gsl_complex gsl_complex_exp (gsl_complex z)

     This function returns the complex exponential of the complex number
     *note z: 89, \exp(z).

 -- Function: gsl_complex gsl_complex_log (gsl_complex z)

     This function returns the complex natural logarithm (base e) of the
     complex number *note z: 8a, \log(z).  The branch cut is the
     negative real axis.

 -- Function: gsl_complex gsl_complex_log10 (gsl_complex z)

     This function returns the complex base-10 logarithm of the complex
     number *note z: 8b, \log_{10} (z).

 -- Function: gsl_complex gsl_complex_log_b (gsl_complex z, gsl_complex
          b)

     This function returns the complex base-*note b: 8c. logarithm of
     the complex number *note z: 8c, \log_b(z).  This quantity is
     computed as the ratio \log(z)/\log(b).


File: gsl-ref.info,  Node: Complex Trigonometric Functions,  Next: Inverse Complex Trigonometric Functions,  Prev: Elementary Complex Functions,  Up: Complex Numbers

5.7 Complex Trigonometric Functions
===================================

 -- Function: gsl_complex gsl_complex_sin (gsl_complex z)

     This function returns the complex sine of the complex number *note
     z: 8e, \sin(z) = (\exp(iz) - \exp(-iz))/(2i).

 -- Function: gsl_complex gsl_complex_cos (gsl_complex z)

     This function returns the complex cosine of the complex number
     *note z: 8f, \cos(z) = (\exp(iz) + \exp(-iz))/2.

 -- Function: gsl_complex gsl_complex_tan (gsl_complex z)

     This function returns the complex tangent of the complex number
     *note z: 90, \tan(z) = \sin(z)/\cos(z).

 -- Function: gsl_complex gsl_complex_sec (gsl_complex z)

     This function returns the complex secant of the complex number
     *note z: 91, \sec(z) = 1/\cos(z).

 -- Function: gsl_complex gsl_complex_csc (gsl_complex z)

     This function returns the complex cosecant of the complex number
     *note z: 92, \csc(z) = 1/\sin(z).

 -- Function: gsl_complex gsl_complex_cot (gsl_complex z)

     This function returns the complex cotangent of the complex number
     *note z: 93, \cot(z) = 1/\tan(z).


File: gsl-ref.info,  Node: Inverse Complex Trigonometric Functions,  Next: Complex Hyperbolic Functions,  Prev: Complex Trigonometric Functions,  Up: Complex Numbers

5.8 Inverse Complex Trigonometric Functions
===========================================

 -- Function: gsl_complex gsl_complex_arcsin (gsl_complex z)

     This function returns the complex arcsine of the complex number
     *note z: 95, \arcsin(z).  The branch cuts are on the real axis,
     less than -1 and greater than 1.

 -- Function: gsl_complex gsl_complex_arcsin_real (double z)

     This function returns the complex arcsine of the real number *note
     z: 96, \arcsin(z).  For z between -1 and 1, the function returns a
     real value in the range [-\pi/2,\pi/2].  For z less than -1 the
     result has a real part of -\pi/2 and a positive imaginary part.
     For z greater than 1 the result has a real part of \pi/2 and a
     negative imaginary part.

 -- Function: gsl_complex gsl_complex_arccos (gsl_complex z)

     This function returns the complex arccosine of the complex number
     *note z: 97, \arccos(z).  The branch cuts are on the real axis,
     less than -1 and greater than 1.

 -- Function: gsl_complex gsl_complex_arccos_real (double z)

     This function returns the complex arccosine of the real number
     *note z: 98, \arccos(z).  For z between -1 and 1, the function
     returns a real value in the range [0,\pi].  For z less than -1 the
     result has a real part of \pi and a negative imaginary part.  For z
     greater than 1 the result is purely imaginary and positive.

 -- Function: gsl_complex gsl_complex_arctan (gsl_complex z)

     This function returns the complex arctangent of the complex number
     *note z: 99, \arctan(z).  The branch cuts are on the imaginary
     axis, below -i and above i.

 -- Function: gsl_complex gsl_complex_arcsec (gsl_complex z)

     This function returns the complex arcsecant of the complex number
     *note z: 9a, \arcsec(z) = \arccos(1/z).

 -- Function: gsl_complex gsl_complex_arcsec_real (double z)

     This function returns the complex arcsecant of the real number
     *note z: 9b, \arcsec(z) = \arccos(1/z).

 -- Function: gsl_complex gsl_complex_arccsc (gsl_complex z)

     This function returns the complex arccosecant of the complex number
     *note z: 9c, \arccsc(z) = \arcsin(1/z).

 -- Function: gsl_complex gsl_complex_arccsc_real (double z)

     This function returns the complex arccosecant of the real number
     *note z: 9d, \arccsc(z) = \arcsin(1/z).

 -- Function: gsl_complex gsl_complex_arccot (gsl_complex z)

     This function returns the complex arccotangent of the complex
     number *note z: 9e, \arccot(z) = \arctan(1/z).


File: gsl-ref.info,  Node: Complex Hyperbolic Functions,  Next: Inverse Complex Hyperbolic Functions,  Prev: Inverse Complex Trigonometric Functions,  Up: Complex Numbers

5.9 Complex Hyperbolic Functions
================================

 -- Function: gsl_complex gsl_complex_sinh (gsl_complex z)

     This function returns the complex hyperbolic sine of the complex
     number *note z: a0, \sinh(z) = (\exp(z) - \exp(-z))/2.

 -- Function: gsl_complex gsl_complex_cosh (gsl_complex z)

     This function returns the complex hyperbolic cosine of the complex
     number *note z: a1, \cosh(z) = (\exp(z) + \exp(-z))/2.

 -- Function: gsl_complex gsl_complex_tanh (gsl_complex z)

     This function returns the complex hyperbolic tangent of the complex
     number *note z: a2, \tanh(z) = \sinh(z)/\cosh(z).

 -- Function: gsl_complex gsl_complex_sech (gsl_complex z)

     This function returns the complex hyperbolic secant of the complex
     number *note z: a3, \sech(z) = 1/\cosh(z).

 -- Function: gsl_complex gsl_complex_csch (gsl_complex z)

     This function returns the complex hyperbolic cosecant of the
     complex number *note z: a4, \csch(z) = 1/\sinh(z).

 -- Function: gsl_complex gsl_complex_coth (gsl_complex z)

     This function returns the complex hyperbolic cotangent of the
     complex number *note z: a5, \coth(z) = 1/\tanh(z).


File: gsl-ref.info,  Node: Inverse Complex Hyperbolic Functions,  Next: References and Further Reading,  Prev: Complex Hyperbolic Functions,  Up: Complex Numbers

5.10 Inverse Complex Hyperbolic Functions
=========================================

 -- Function: gsl_complex gsl_complex_arcsinh (gsl_complex z)

     This function returns the complex hyperbolic arcsine of the complex
     number *note z: a7, \arcsinh(z).  The branch cuts are on the
     imaginary axis, below -i and above i.

 -- Function: gsl_complex gsl_complex_arccosh (gsl_complex z)

     This function returns the complex hyperbolic arccosine of the
     complex number *note z: a8, \arccosh(z).  The branch cut is on the
     real axis, less than 1.  Note that in this case we use the negative
     square root in formula 4.6.21 of Abramowitz & Stegun giving
     \arccosh(z)=\log(z-\sqrt{z^2-1}).

 -- Function: gsl_complex gsl_complex_arccosh_real (double z)

     This function returns the complex hyperbolic arccosine of the real
     number *note z: a9, \arccosh(z).

 -- Function: gsl_complex gsl_complex_arctanh (gsl_complex z)

     This function returns the complex hyperbolic arctangent of the
     complex number *note z: aa, \arctanh(z).  The branch cuts are on
     the real axis, less than -1 and greater than 1.

 -- Function: gsl_complex gsl_complex_arctanh_real (double z)

     This function returns the complex hyperbolic arctangent of the real
     number *note z: ab, \arctanh(z).

 -- Function: gsl_complex gsl_complex_arcsech (gsl_complex z)

     This function returns the complex hyperbolic arcsecant of the
     complex number *note z: ac, \arcsech(z) = \arccosh(1/z).

 -- Function: gsl_complex gsl_complex_arccsch (gsl_complex z)

     This function returns the complex hyperbolic arccosecant of the
     complex number *note z: ad, \arccsch(z) = \arcsinh(1/z).

 -- Function: gsl_complex gsl_complex_arccoth (gsl_complex z)

     This function returns the complex hyperbolic arccotangent of the
     complex number *note z: ae, \arccoth(z) = \arctanh(1/z).


File: gsl-ref.info,  Node: References and Further Reading,  Prev: Inverse Complex Hyperbolic Functions,  Up: Complex Numbers

5.11 References and Further Reading
===================================

The implementations of the elementary and trigonometric functions are
based on the following papers,

   * T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
     “Implementing Complex Elementary Functions Using Exception
     Handling”, ACM Transactions on Mathematical Software, Volume 20
     (1994), pp 215–244, Corrigenda, p553

   * T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
     “Implementing the complex arcsin and arccosine functions using
     exception handling”, ACM Transactions on Mathematical Software,
     Volume 23 (1997) pp 299–335

The general formulas and details of branch cuts can be found in the
following books,

   * Abramowitz and Stegun, Handbook of Mathematical Functions,
     “Circular Functions in Terms of Real and Imaginary Parts”, Formulas
     4.3.55–58, “Inverse Circular Functions in Terms of Real and
     Imaginary Parts”, Formulas 4.4.37–39, “Hyperbolic Functions in
     Terms of Real and Imaginary Parts”, Formulas 4.5.49–52, “Inverse
     Hyperbolic Functions—relation to Inverse Circular Functions”,
     Formulas 4.6.14–19.

   * Dave Gillespie, Calc Manual, Free Software Foundation, ISBN
     1-882114-18-3


File: gsl-ref.info,  Node: Polynomials,  Next: Special Functions,  Prev: Complex Numbers,  Up: Top

6 Polynomials
*************

This chapter describes functions for evaluating and solving polynomials.
There are routines for finding real and complex roots of quadratic and
cubic equations using analytic methods.  An iterative polynomial solver
is also available for finding the roots of general polynomials with real
coefficients (of any order).  The functions are declared in the header
file ‘gsl_poly.h’.

* Menu:

* Polynomial Evaluation::
* Divided Difference Representation of Polynomials::
* Quadratic Equations::
* Cubic Equations::
* General Polynomial Equations::
* Examples: Examples<2>.
* References and Further Reading: References and Further Reading<2>.


File: gsl-ref.info,  Node: Polynomial Evaluation,  Next: Divided Difference Representation of Polynomials,  Up: Polynomials

6.1 Polynomial Evaluation
=========================

The functions described here evaluate the polynomial

P(x) = c[0] + c[1] x + c[2] x^2 + … + c[len-1] x^{len-1}

using Horner’s method for stability.  Inline versions of these functions
are used when ‘HAVE_INLINE’ is defined.

 -- Function: double gsl_poly_eval (const double c[], const int len,
          const double x)

     This function evaluates a polynomial with real coefficients for the
     real variable *note x: b3.

 -- Function: gsl_complex gsl_poly_complex_eval (const double c[], const
          int len, const gsl_complex z)

     This function evaluates a polynomial with real coefficients for the
     complex variable *note z: b4.

 -- Function: gsl_complex gsl_complex_poly_complex_eval (const
          gsl_complex c[], const int len, const gsl_complex z)

     This function evaluates a polynomial with complex coefficients for
     the complex variable *note z: b5.

 -- Function: int gsl_poly_eval_derivs (const double c[], const size_t
          lenc, const double x, double res[], const size_t lenres)

     This function evaluates a polynomial and its derivatives storing
     the results in the array *note res: b6. of size *note lenres: b6.
     The output array contains the values of d^k P(x)/d x^k for the
     specified value of *note x: b6. starting with k = 0.


File: gsl-ref.info,  Node: Divided Difference Representation of Polynomials,  Next: Quadratic Equations,  Prev: Polynomial Evaluation,  Up: Polynomials

6.2 Divided Difference Representation of Polynomials
====================================================

The functions described here manipulate polynomials stored in Newton’s
divided-difference representation.  The use of divided-differences is
described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden
and Faires, chapter 3, and discussed briefly below.

Given a function f(x), an nth degree interpolating polynomial P_{n}(x)
can be constructed which agrees with f at n+1 distinct points
x_0,x_1,...,x_{n}.  This polynomial can be written in a form known as
Newton’s divided-difference representation

P_{n}(x) = f(x_0) + sum_{k=1}^n [x_0,x_1,…,x_k] (x-x_0)(x-x_1) …
(x-x_{k-1})

where the divided differences [x_0,x_1,...,x_k] are defined in section
25.1.4 of Abramowitz and Stegun.  Additionally, it is possible to
construct an interpolating polynomial of degree 2n+1 which also matches
the first derivatives of f at the points x_0,x_1,...,x_n.  This is
called the Hermite interpolating polynomial and is defined as

H_{2n+1}(x) = f(z_0) + sum_{k=1}^{2n+1} [z_0,z_1,…,z_k] (x-z_0)(x-z_1) …
(x-z_{k-1})

where the elements of z = \{x_0,x_0,x_1,x_1,...,x_n,x_n\} are defined by
z_{2k} = z_{2k+1} = x_k.  The divided-differences [z_0,z_1,...,z_k] are
discussed in Burden and Faires, section 3.4.

 -- Function: int gsl_poly_dd_init (double dd[], const double xa[],
          const double ya[], size_t size)

     This function computes a divided-difference representation of the
     interpolating polynomial for the points (x, y) stored in the arrays
     *note xa: b8. and *note ya: b8. of length *note size: b8.  On
     output the divided-differences of (*note xa: b8, *note ya: b8.) are
     stored in the array *note dd: b8, also of length *note size: b8.
     Using the notation above, dd[k] = [x_0,x_1,...,x_k].

 -- Function: double gsl_poly_dd_eval (const double dd[], const double
          xa[], const size_t size, const double x)

     This function evaluates the polynomial stored in divided-difference
     form in the arrays *note dd: b9. and *note xa: b9. of length *note
     size: b9. at the point *note x: b9.  An inline version of this
     function is used when ‘HAVE_INLINE’ is defined.

 -- Function: int gsl_poly_dd_taylor (double c[], double xp, const
          double dd[], const double xa[], size_t size, double w[])

     This function converts the divided-difference representation of a
     polynomial to a Taylor expansion.  The divided-difference
     representation is supplied in the arrays *note dd: ba. and *note
     xa: ba. of length *note size: ba.  On output the Taylor
     coefficients of the polynomial expanded about the point *note xp:
     ba. are stored in the array *note c: ba. also of length *note size:
     ba.  A workspace of length *note size: ba. must be provided in the
     array *note w: ba.

 -- Function: int gsl_poly_dd_hermite_init (double dd[], double za[],
          const double xa[], const double ya[], const double dya[],
          const size_t size)

     This function computes a divided-difference representation of the
     interpolating Hermite polynomial for the points (x,y) stored in the
     arrays *note xa: bb. and *note ya: bb. of length *note size: bb.
     Hermite interpolation constructs polynomials which also match first
     derivatives dy/dx which are provided in the array *note dya: bb.
     also of length *note size: bb.  The first derivatives can be
     incorported into the usual divided-difference algorithm by forming
     a new dataset z = \{x_0,x_0,x_1,x_1,...\}, which is stored in the
     array *note za: bb. of length 2**note size: bb. on output.  On
     output the divided-differences of the Hermite representation are
     stored in the array *note dd: bb, also of length 2**note size: bb.
     Using the notation above, dd[k] = [z_0,z_1,...,z_k].  The resulting
     Hermite polynomial can be evaluated by calling *note
     gsl_poly_dd_eval(): b9. and using *note za: bb. for the input
     argument *note xa: bb.


File: gsl-ref.info,  Node: Quadratic Equations,  Next: Cubic Equations,  Prev: Divided Difference Representation of Polynomials,  Up: Polynomials

6.3 Quadratic Equations
=======================

 -- Function: int gsl_poly_solve_quadratic (double a, double b, double
          c, double *x0, double *x1)

     This function finds the real roots of the quadratic equation,

          a x^2 + b x + c = 0

     The number of real roots (either zero, one or two) is returned, and
     their locations are stored in *note x0: bd. and *note x1: bd.  If
     no real roots are found then *note x0: bd. and *note x1: bd. are
     not modified.  If one real root is found (i.e.  if a=0) then it is
     stored in *note x0: bd.  When two real roots are found they are
     stored in *note x0: bd. and *note x1: bd. in ascending order.  The
     case of coincident roots is not considered special.  For example
     (x-1)^2=0 will have two roots, which happen to have exactly equal
     values.

     The number of roots found depends on the sign of the discriminant
     b^2 - 4 a c.  This will be subject to rounding and cancellation
     errors when computed in double precision, and will also be subject
     to errors if the coefficients of the polynomial are inexact.  These
     errors may cause a discrete change in the number of roots.
     However, for polynomials with small integer coefficients the
     discriminant can always be computed exactly.

 -- Function: int gsl_poly_complex_solve_quadratic (double a, double b,
          double c, gsl_complex *z0, gsl_complex *z1)

     This function finds the complex roots of the quadratic equation,

          a z^2 + b z + c = 0

     The number of complex roots is returned (either one or two) and the
     locations of the roots are stored in *note z0: be. and *note z1:
     be.  The roots are returned in ascending order, sorted first by
     their real components and then by their imaginary components.  If
     only one real root is found (i.e.  if a=0) then it is stored in
     *note z0: be.


File: gsl-ref.info,  Node: Cubic Equations,  Next: General Polynomial Equations,  Prev: Quadratic Equations,  Up: Polynomials

6.4 Cubic Equations
===================

 -- Function: int gsl_poly_solve_cubic (double a, double b, double c,
          double *x0, double *x1, double *x2)

     This function finds the real roots of the cubic equation,

          x^3 + a x^2 + b x + c = 0

     with a leading coefficient of unity.  The number of real roots
     (either one or three) is returned, and their locations are stored
     in *note x0: c0, *note x1: c0. and *note x2: c0.  If one real root
     is found then only *note x0: c0. is modified.  When three real
     roots are found they are stored in *note x0: c0, *note x1: c0. and
     *note x2: c0. in ascending order.  The case of coincident roots is
     not considered special.  For example, the equation (x-1)^3=0 will
     have three roots with exactly equal values.  As in the quadratic
     case, finite precision may cause equal or closely-spaced real roots
     to move off the real axis into the complex plane, leading to a
     discrete change in the number of real roots.

 -- Function: int gsl_poly_complex_solve_cubic (double a, double b,
          double c, gsl_complex *z0, gsl_complex *z1, gsl_complex *z2)

     This function finds the complex roots of the cubic equation,

          z^3 + a z^2 + b z + c = 0

     The number of complex roots is returned (always three) and the
     locations of the roots are stored in *note z0: c1, *note z1: c1.
     and *note z2: c1.  The roots are returned in ascending order,
     sorted first by their real components and then by their imaginary
     components.


File: gsl-ref.info,  Node: General Polynomial Equations,  Next: Examples<2>,  Prev: Cubic Equations,  Up: Polynomials

6.5 General Polynomial Equations
================================

The roots of polynomial equations cannot be found analytically beyond
the special cases of the quadratic, cubic and quartic equation.  The
algorithm described in this section uses an iterative method to find the
approximate locations of roots of higher order polynomials.

 -- Type: gsl_poly_complex_workspace

     This workspace contains parameters used for finding roots of
     general polynomials

 -- Function: *note gsl_poly_complex_workspace: c3.
          *gsl_poly_complex_workspace_alloc (size_t n)

     This function allocates space for a *note
     gsl_poly_complex_workspace: c3. struct and a workspace suitable for
     solving a polynomial with *note n: c4. coefficients using the
     routine *note gsl_poly_complex_solve(): c5.

     The function returns a pointer to the newly allocated *note
     gsl_poly_complex_workspace: c3. if no errors were detected, and a
     null pointer in the case of error.

 -- Function: void gsl_poly_complex_workspace_free
          (gsl_poly_complex_workspace *w)

     This function frees all the memory associated with the workspace
     *note w: c6.

 -- Function: int gsl_poly_complex_solve (const double *a, size_t n,
          gsl_poly_complex_workspace *w, gsl_complex_packed_ptr z)

     This function computes the roots of the general polynomial

     P(x) = a_0 + a_1 x + a_2 x^2 + … + a_{n-1} x^{n-1}

     using balanced-QR reduction of the companion matrix.  The parameter
     *note n: c5. specifies the length of the coefficient array.  The
     coefficient of the highest order term must be non-zero.  The
     function requires a workspace *note w: c5. of the appropriate size.
     The n-1 roots are returned in the packed complex array *note z: c5.
     of length 2(n-1), alternating real and imaginary parts.

     The function returns ‘GSL_SUCCESS’ if all the roots are found.  If
     the QR reduction does not converge, the error handler is invoked
     with an error code of ‘GSL_EFAILED’.  Note that due to finite
     precision, roots of higher multiplicity are returned as a cluster
     of simple roots with reduced accuracy.  The solution of polynomials
     with higher-order roots requires specialized algorithms that take
     the multiplicity structure into account (see e.g.  Z. Zeng,
     Algorithm 835, ACM Transactions on Mathematical Software, Volume
     30, Issue 2 (2004), pp 218–236).


File: gsl-ref.info,  Node: Examples<2>,  Next: References and Further Reading<2>,  Prev: General Polynomial Equations,  Up: Polynomials

6.6 Examples
============

To demonstrate the use of the general polynomial solver we will take the
polynomial P(x) = x^5 - 1 which has these roots:

1, e^{2*pi i / 5}, e^{4*pi i / 5}, e^{6*pi i / 5}, e^{8*pi i / 5}

The following program will find these roots.

     #include <stdio.h>
     #include <gsl/gsl_poly.h>

     int
     main (void)
     {
       int i;
       /* coefficients of P(x) =  -1 + x^5  */
       double a[6] = { -1, 0, 0, 0, 0, 1 };
       double z[10];

       gsl_poly_complex_workspace * w
           = gsl_poly_complex_workspace_alloc (6);

       gsl_poly_complex_solve (a, 6, w, z);

       gsl_poly_complex_workspace_free (w);

       for (i = 0; i < 5; i++)
         {
           printf ("z%d = %+.18f %+.18f\n",
                   i, z[2*i], z[2*i+1]);
         }

       return 0;
     }

The output of the program is

     z0 = -0.809016994374947673 +0.587785252292473359
     z1 = -0.809016994374947673 -0.587785252292473359
     z2 = +0.309016994374947507 +0.951056516295152976
     z3 = +0.309016994374947507 -0.951056516295152976
     z4 = +0.999999999999999889 +0.000000000000000000

which agrees with the analytic result, z_n = \exp(2 \pi n i/5).


File: gsl-ref.info,  Node: References and Further Reading<2>,  Prev: Examples<2>,  Up: Polynomials

6.7 References and Further Reading
==================================

The balanced-QR method and its error analysis are described in the
following papers,

   * R.S. Martin, G. Peters and J.H. Wilkinson, “The QR Algorithm for
     Real Hessenberg Matrices”, Numerische Mathematik, 14 (1970),
     219–231.

   * B.N. Parlett and C. Reinsch, “Balancing a Matrix for Calculation of
     Eigenvalues and Eigenvectors”, Numerische Mathematik, 13 (1969),
     293–304.

   * A. Edelman and H. Murakami, “Polynomial roots from companion matrix
     eigenvalues”, Mathematics of Computation, Vol.: 64, No.: 210
     (1995), 763–776.

The formulas for divided differences are given in the following texts,

   * Abramowitz and Stegun, Handbook of Mathematical Functions, Sections
     25.1.4 and 25.2.26.

   * R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition,
     ISBN 0-538-73351-9, 2011.


File: gsl-ref.info,  Node: Special Functions,  Next: Vectors and Matrices,  Prev: Polynomials,  Up: Top

7 Special Functions
*******************

This chapter describes the GSL special function library.  The library
includes routines for calculating the values of Airy functions, Bessel
functions, Clausen functions, Coulomb wave functions, Coupling
coefficients, the Dawson function, Debye functions, Dilogarithms,
Elliptic integrals, Jacobi elliptic functions, Error functions,
Exponential integrals, Fermi-Dirac functions, Gamma functions,
Gegenbauer functions, Hermite polynomials and functions, Hypergeometric
functions, Laguerre functions, Legendre functions and Spherical
Harmonics, the Psi (Digamma) Function, Synchrotron functions, Transport
functions, Trigonometric functions and Zeta functions.  Each routine
also computes an estimate of the numerical error in the calculated value
of the function.

The functions in this chapter are declared in individual header files,
such as ‘gsl_sf_airy.h’, ‘gsl_sf_bessel.h’, etc.  The complete set of
header files can be included using the file ‘gsl_sf.h’.

* Menu:

* Usage::
* The gsl_sf_result struct::
* Modes::
* Airy Functions and Derivatives::
* Bessel Functions::
* Clausen Functions::
* Coulomb Functions::
* Coupling Coefficients::
* Dawson Function::
* Debye Functions::
* Dilogarithm::
* Elementary Operations::
* Elliptic Integrals::
* Elliptic Functions (Jacobi): Elliptic Functions Jacobi.
* Error Functions::
* Exponential Functions::
* Exponential Integrals::
* Fermi-Dirac Function::
* Gamma and Beta Functions::
* Gegenbauer Functions::
* Hermite Polynomials and Functions::
* Hypergeometric Functions::
* Laguerre Functions::
* Lambert W Functions::
* Legendre Functions and Spherical Harmonics::
* Logarithm and Related Functions::
* Mathieu Functions::
* Power Function::
* Psi (Digamma) Function: Psi Digamma Function.
* Synchrotron Functions::
* Transport Functions::
* Trigonometric Functions::
* Zeta Functions::
* Examples: Examples<3>.
* References and Further Reading: References and Further Reading<3>.


File: gsl-ref.info,  Node: Usage,  Next: The gsl_sf_result struct,  Up: Special Functions

7.1 Usage
=========

The special functions are available in two calling conventions, a
`natural form' which returns the numerical value of the function and an
`error-handling form' which returns an error code.  The two types of
function provide alternative ways of accessing the same underlying code.

The `natural form' returns only the value of the function and can be
used directly in mathematical expressions.  For example, the following
function call will compute the value of the Bessel function J_0(x):

     double y = gsl_sf_bessel_J0 (x);

There is no way to access an error code or to estimate the error using
this method.  To allow access to this information the alternative
error-handling form stores the value and error in a modifiable argument:

     gsl_sf_result result;
     int status = gsl_sf_bessel_J0_e (x, &result);

The error-handling functions have the suffix ‘_e’.  The returned status
value indicates error conditions such as overflow, underflow or loss of
precision.  If there are no errors the error-handling functions return
‘GSL_SUCCESS’.


File: gsl-ref.info,  Node: The gsl_sf_result struct,  Next: Modes,  Prev: Usage,  Up: Special Functions

7.2 The gsl_sf_result struct
============================

The error handling form of the special functions always calculate an
error estimate along with the value of the result.  Therefore,
structures are provided for amalgamating a value and error estimate.
These structures are declared in the header file ‘gsl_sf_result.h’.

The following struct contains value and error fields.

 -- Type: gsl_sf_result

          typedef struct
          {
            double val;
            double err;
          } gsl_sf_result;

     The field ‘val’ contains the value and the field ‘err’ contains an
     estimate of the absolute error in the value.

In some cases, an overflow or underflow can be detected and handled by a
function.  In this case, it may be possible to return a scaling exponent
as well as an error/value pair in order to save the result from
exceeding the dynamic range of the built-in types.  The following struct
contains value and error fields as well as an exponent field such that
the actual result is obtained as ‘result * 10^(e10)’.

 -- Type: gsl_sf_result_e10

          typedef struct
          {
            double val;
            double err;
            int    e10;
          } gsl_sf_result_e10;


File: gsl-ref.info,  Node: Modes,  Next: Airy Functions and Derivatives,  Prev: The gsl_sf_result struct,  Up: Special Functions

7.3 Modes
=========

The goal of the library is to achieve double precision accuracy wherever
possible.  However the cost of evaluating some special functions to
double precision can be significant, particularly where very high order
terms are required.  In these cases a ‘mode’ argument, of type *note
gsl_mode_t: d0. allows the accuracy of the function to be reduced in
order to improve performance.  The following precision levels are
available for the mode argument,

 -- Type: gsl_mode_t

      -- Macro: GSL_PREC_DOUBLE

          Double-precision, a relative accuracy of approximately 2 *
          10^{-16}.

      -- Macro: GSL_PREC_SINGLE

          Single-precision, a relative accuracy of approximately
          10^{-7}.

      -- Macro: GSL_PREC_APPROX

          Approximate values, a relative accuracy of approximately 5 *
          10^{-4}.

The approximate mode provides the fastest evaluation at the lowest
accuracy.


File: gsl-ref.info,  Node: Airy Functions and Derivatives,  Next: Bessel Functions,  Prev: Modes,  Up: Special Functions

7.4 Airy Functions and Derivatives
==================================

The Airy functions Ai(x) and Bi(x) are defined by the integral
representations,

     Ai(x) = (1/pi) int_0^infty cos((1/3) t^3 + xt) dt 
     Bi(x) = (1/pi) int_0^infty (e^(-(1/3) t^3 + xt) + sin((1/3) t^3 + xt)) dt 

For further information see Abramowitz & Stegun, Section 10.4.  The Airy
functions are defined in the header file ‘gsl_sf_airy.h’.

* Menu:

* Airy Functions::
* Derivatives of Airy Functions::
* Zeros of Airy Functions::
* Zeros of Derivatives of Airy Functions::


File: gsl-ref.info,  Node: Airy Functions,  Next: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives

7.4.1 Airy Functions
--------------------

 -- Function: double gsl_sf_airy_Ai (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Ai_e (double x, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute the Airy function Ai(x) with an accuracy
     specified by *note mode: d7.

 -- Function: double gsl_sf_airy_Bi (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Bi_e (double x, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute the Airy function Bi(x) with an accuracy
     specified by *note mode: d9.

 -- Function: double gsl_sf_airy_Ai_scaled (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Ai_scaled_e (double x, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute a scaled version of the Airy function S_A(x)
     Ai(x).  For x > 0 the scaling factor S_A(x) is \exp(+(2/3)
     x^{3/2}), and is 1 for x < 0.

 -- Function: double gsl_sf_airy_Bi_scaled (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Bi_scaled_e (double x, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute a scaled version of the Airy function S_B(x)
     Bi(x).  For x > 0 the scaling factor S_B(x) is exp(-(2/3) x^{3/2}),
     and is 1 for x < 0.


File: gsl-ref.info,  Node: Derivatives of Airy Functions,  Next: Zeros of Airy Functions,  Prev: Airy Functions,  Up: Airy Functions and Derivatives

7.4.2 Derivatives of Airy Functions
-----------------------------------

 -- Function: double gsl_sf_airy_Ai_deriv (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Ai_deriv_e (double x, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute the Airy function derivative Ai'(x) with an
     accuracy specified by *note mode: e0.

 -- Function: double gsl_sf_airy_Bi_deriv (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Bi_deriv_e (double x, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute the Airy function derivative Bi'(x) with an
     accuracy specified by *note mode: e2.

 -- Function: double gsl_sf_airy_Ai_deriv_scaled (double x, gsl_mode_t
          mode)
 -- Function: int gsl_sf_airy_Ai_deriv_scaled_e (double x, gsl_mode_t
          mode, gsl_sf_result *result)

     These routines compute the scaled Airy function derivative S_A(x)
     Ai'(x).  For x > 0 the scaling factor S_A(x) is \exp(+(2/3)
     x^{3/2}), and is 1 for x < 0.

 -- Function: double gsl_sf_airy_Bi_deriv_scaled (double x, gsl_mode_t
          mode)
 -- Function: int gsl_sf_airy_Bi_deriv_scaled_e (double x, gsl_mode_t
          mode, gsl_sf_result *result)

     These routines compute the scaled Airy function derivative S_B(x)
     Bi'(x).  For x > 0 the scaling factor S_B(x) is exp(-(2/3)
     x^{3/2}), and is 1 for x < 0.


File: gsl-ref.info,  Node: Zeros of Airy Functions,  Next: Zeros of Derivatives of Airy Functions,  Prev: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives

7.4.3 Zeros of Airy Functions
-----------------------------

 -- Function: double gsl_sf_airy_zero_Ai (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl_sf_result
          *result)

     These routines compute the location of the *note s: e9.-th zero of
     the Airy function Ai(x).

 -- Function: double gsl_sf_airy_zero_Bi (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl_sf_result
          *result)

     These routines compute the location of the *note s: eb.-th zero of
     the Airy function Bi(x).


File: gsl-ref.info,  Node: Zeros of Derivatives of Airy Functions,  Prev: Zeros of Airy Functions,  Up: Airy Functions and Derivatives

7.4.4 Zeros of Derivatives of Airy Functions
--------------------------------------------

 -- Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s,
          gsl_sf_result *result)

     These routines compute the location of the *note s: ee.-th zero of
     the Airy function derivative Ai'(x).

 -- Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s,
          gsl_sf_result *result)

     These routines compute the location of the *note s: f0.-th zero of
     the Airy function derivative Bi'(x).


File: gsl-ref.info,  Node: Bessel Functions,  Next: Clausen Functions,  Prev: Airy Functions and Derivatives,  Up: Special Functions

7.5 Bessel Functions
====================

The routines described in this section compute the Cylindrical Bessel
functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x),
K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified
Spherical Bessel functions i_l(x), k_l(x).  For more information see
Abramowitz & Stegun, Chapters 9 and 10.  The Bessel functions are
defined in the header file ‘gsl_sf_bessel.h’.

* Menu:

* Regular Cylindrical Bessel Functions::
* Irregular Cylindrical Bessel Functions::
* Regular Modified Cylindrical Bessel Functions::
* Irregular Modified Cylindrical Bessel Functions::
* Regular Spherical Bessel Functions::
* Irregular Spherical Bessel Functions::
* Regular Modified Spherical Bessel Functions::
* Irregular Modified Spherical Bessel Functions::
* Regular Bessel Function—Fractional Order::
* Irregular Bessel Functions—Fractional Order::
* Regular Modified Bessel Functions—Fractional Order::
* Irregular Modified Bessel Functions—Fractional Order::
* Zeros of Regular Bessel Functions::


File: gsl-ref.info,  Node: Regular Cylindrical Bessel Functions,  Next: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.1 Regular Cylindrical Bessel Functions
------------------------------------------

 -- Function: double gsl_sf_bessel_J0 (double x)
 -- Function: int gsl_sf_bessel_J0_e (double x, gsl_sf_result *result)

     These routines compute the regular cylindrical Bessel function of
     zeroth order, J_0(x).

 -- Function: double gsl_sf_bessel_J1 (double x)
 -- Function: int gsl_sf_bessel_J1_e (double x, gsl_sf_result *result)

     These routines compute the regular cylindrical Bessel function of
     first order, J_1(x).

 -- Function: double gsl_sf_bessel_Jn (int n, double x)
 -- Function: int gsl_sf_bessel_Jn_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the regular cylindrical Bessel function of
     order *note n: f8, J_n(x).

 -- Function: int gsl_sf_bessel_Jn_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the regular cylindrical Bessel
     functions J_n(x) for n from *note nmin: f9. to *note nmax: f9.
     inclusive, storing the results in the array *note result_array: f9.
     The values are computed using recurrence relations for efficiency,
     and therefore may differ slightly from the exact values.


File: gsl-ref.info,  Node: Irregular Cylindrical Bessel Functions,  Next: Regular Modified Cylindrical Bessel Functions,  Prev: Regular Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.2 Irregular Cylindrical Bessel Functions
--------------------------------------------

 -- Function: double gsl_sf_bessel_Y0 (double x)
 -- Function: int gsl_sf_bessel_Y0_e (double x, gsl_sf_result *result)

     These routines compute the irregular cylindrical Bessel function of
     zeroth order, Y_0(x), for x>0.

 -- Function: double gsl_sf_bessel_Y1 (double x)
 -- Function: int gsl_sf_bessel_Y1_e (double x, gsl_sf_result *result)

     These routines compute the irregular cylindrical Bessel function of
     first order, Y_1(x), for x>0.

 -- Function: double gsl_sf_bessel_Yn (int n, double x)
 -- Function: int gsl_sf_bessel_Yn_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the irregular cylindrical Bessel function of
     order *note n: 100, Y_n(x), for x>0.

 -- Function: int gsl_sf_bessel_Yn_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the irregular cylindrical
     Bessel functions Y_n(x) for n from *note nmin: 101. to *note nmax:
     101. inclusive, storing the results in the array *note
     result_array: 101.  The domain of the function is x>0.  The values
     are computed using recurrence relations for efficiency, and
     therefore may differ slightly from the exact values.


File: gsl-ref.info,  Node: Regular Modified Cylindrical Bessel Functions,  Next: Irregular Modified Cylindrical Bessel Functions,  Prev: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.3 Regular Modified Cylindrical Bessel Functions
---------------------------------------------------

 -- Function: double gsl_sf_bessel_I0 (double x)
 -- Function: int gsl_sf_bessel_I0_e (double x, gsl_sf_result *result)

     These routines compute the regular modified cylindrical Bessel
     function of zeroth order, I_0(x).

 -- Function: double gsl_sf_bessel_I1 (double x)
 -- Function: int gsl_sf_bessel_I1_e (double x, gsl_sf_result *result)

     These routines compute the regular modified cylindrical Bessel
     function of first order, I_1(x).

 -- Function: double gsl_sf_bessel_In (int n, double x)
 -- Function: int gsl_sf_bessel_In_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the regular modified cylindrical Bessel
     function of order *note n: 108, I_n(x).

 -- Function: int gsl_sf_bessel_In_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the regular modified
     cylindrical Bessel functions I_n(x) for n from *note nmin: 109. to
     *note nmax: 109. inclusive, storing the results in the array *note
     result_array: 109.  The start of the range *note nmin: 109. must be
     positive or zero.  The values are computed using recurrence
     relations for efficiency, and therefore may differ slightly from
     the exact values.

 -- Function: double gsl_sf_bessel_I0_scaled (double x)
 -- Function: int gsl_sf_bessel_I0_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled regular modified cylindrical
     Bessel function of zeroth order \exp(-|x|) I_0(x).

 -- Function: double gsl_sf_bessel_I1_scaled (double x)
 -- Function: int gsl_sf_bessel_I1_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled regular modified cylindrical
     Bessel function of first order \exp(-|x|) I_1(x).

 -- Function: double gsl_sf_bessel_In_scaled (int n, double x)
 -- Function: int gsl_sf_bessel_In_scaled_e (int n, double x,
          gsl_sf_result *result)

     These routines compute the scaled regular modified cylindrical
     Bessel function of order *note n: 10f, \exp(-|x|) I_n(x)

 -- Function: int gsl_sf_bessel_In_scaled_array (int nmin, int nmax,
          double x, double result_array[])

     This routine computes the values of the scaled regular cylindrical
     Bessel functions \exp(-|x|) I_n(x) for n from *note nmin: 110. to
     *note nmax: 110. inclusive, storing the results in the array *note
     result_array: 110.  The start of the range *note nmin: 110. must be
     positive or zero.  The values are computed using recurrence
     relations for efficiency, and therefore may differ slightly from
     the exact values.


File: gsl-ref.info,  Node: Irregular Modified Cylindrical Bessel Functions,  Next: Regular Spherical Bessel Functions,  Prev: Regular Modified Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.4 Irregular Modified Cylindrical Bessel Functions
-----------------------------------------------------

 -- Function: double gsl_sf_bessel_K0 (double x)
 -- Function: int gsl_sf_bessel_K0_e (double x, gsl_sf_result *result)

     These routines compute the irregular modified cylindrical Bessel
     function of zeroth order, K_0(x), for x > 0.

 -- Function: double gsl_sf_bessel_K1 (double x)
 -- Function: int gsl_sf_bessel_K1_e (double x, gsl_sf_result *result)

     These routines compute the irregular modified cylindrical Bessel
     function of first order, K_1(x), for x > 0.

 -- Function: double gsl_sf_bessel_Kn (int n, double x)
 -- Function: int gsl_sf_bessel_Kn_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the irregular modified cylindrical Bessel
     function of order *note n: 117, K_n(x), for x > 0.

 -- Function: int gsl_sf_bessel_Kn_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the irregular modified
     cylindrical Bessel functions K_n(x) for n from *note nmin: 118. to
     *note nmax: 118. inclusive, storing the results in the array *note
     result_array: 118.  The start of the range *note nmin: 118. must be
     positive or zero.  The domain of the function is x>0.  The values
     are computed using recurrence relations for efficiency, and
     therefore may differ slightly from the exact values.

 -- Function: double gsl_sf_bessel_K0_scaled (double x)
 -- Function: int gsl_sf_bessel_K0_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled irregular modified cylindrical
     Bessel function of zeroth order \exp(x) K_0(x) for x>0.

 -- Function: double gsl_sf_bessel_K1_scaled (double x)
 -- Function: int gsl_sf_bessel_K1_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled irregular modified cylindrical
     Bessel function of first order \exp(x) K_1(x) for x>0.

 -- Function: double gsl_sf_bessel_Kn_scaled (int n, double x)
 -- Function: int gsl_sf_bessel_Kn_scaled_e (int n, double x,
          gsl_sf_result *result)

     These routines compute the scaled irregular modified cylindrical
     Bessel function of order *note n: 11e, \exp(x) K_n(x), for x>0.

 -- Function: int gsl_sf_bessel_Kn_scaled_array (int nmin, int nmax,
          double x, double result_array[])

     This routine computes the values of the scaled irregular
     cylindrical Bessel functions \exp(x) K_n(x) for n from *note nmin:
     11f. to *note nmax: 11f. inclusive, storing the results in the
     array *note result_array: 11f.  The start of the range *note nmin:
     11f. must be positive or zero.  The domain of the function is x>0.
     The values are computed using recurrence relations for efficiency,
     and therefore may differ slightly from the exact values.


File: gsl-ref.info,  Node: Regular Spherical Bessel Functions,  Next: Irregular Spherical Bessel Functions,  Prev: Irregular Modified Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.5 Regular Spherical Bessel Functions
----------------------------------------

 -- Function: double gsl_sf_bessel_j0 (double x)
 -- Function: int gsl_sf_bessel_j0_e (double x, gsl_sf_result *result)

     These routines compute the regular spherical Bessel function of
     zeroth order, j_0(x) = \sin(x)/x.

 -- Function: double gsl_sf_bessel_j1 (double x)
 -- Function: int gsl_sf_bessel_j1_e (double x, gsl_sf_result *result)

     These routines compute the regular spherical Bessel function of
     first order, j_1(x) = (\sin(x)/x - \cos(x))/x.

 -- Function: double gsl_sf_bessel_j2 (double x)
 -- Function: int gsl_sf_bessel_j2_e (double x, gsl_sf_result *result)

     These routines compute the regular spherical Bessel function of
     second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.

 -- Function: double gsl_sf_bessel_jl (int l, double x)
 -- Function: int gsl_sf_bessel_jl_e (int l, double x, gsl_sf_result
          *result)

     These routines compute the regular spherical Bessel function of
     order *note l: 128, j_l(x), for l \geq 0 and x \geq 0.

 -- Function: int gsl_sf_bessel_jl_array (int lmax, double x, double
          result_array[])

     This routine computes the values of the regular spherical Bessel
     functions j_l(x) for l from 0 to *note lmax: 129. inclusive for
     lmax \geq 0 and x \geq 0, storing the results in the array *note
     result_array: 129.  The values are computed using recurrence
     relations for efficiency, and therefore may differ slightly from
     the exact values.

 -- Function: int gsl_sf_bessel_jl_steed_array (int lmax, double x,
          double *result_array)

     This routine uses Steed’s method to compute the values of the
     regular spherical Bessel functions j_l(x) for l from 0 to *note
     lmax: 12a. inclusive for lmax \geq 0 and x \geq 0, storing the
     results in the array *note result_array: 12a.  The Steed/Barnett
     algorithm is described in Comp.  Phys.  Comm.  21, 297 (1981).
     Steed’s method is more stable than the recurrence used in the other
     functions but is also slower.


File: gsl-ref.info,  Node: Irregular Spherical Bessel Functions,  Next: Regular Modified Spherical Bessel Functions,  Prev: Regular Spherical Bessel Functions,  Up: Bessel Functions

7.5.6 Irregular Spherical Bessel Functions
------------------------------------------

 -- Function: double gsl_sf_bessel_y0 (double x)
 -- Function: int gsl_sf_bessel_y0_e (double x, gsl_sf_result *result)

     These routines compute the irregular spherical Bessel function of
     zeroth order, y_0(x) = -\cos(x)/x.

 -- Function: double gsl_sf_bessel_y1 (double x)
 -- Function: int gsl_sf_bessel_y1_e (double x, gsl_sf_result *result)

     These routines compute the irregular spherical Bessel function of
     first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.

 -- Function: double gsl_sf_bessel_y2 (double x)
 -- Function: int gsl_sf_bessel_y2_e (double x, gsl_sf_result *result)

     These routines compute the irregular spherical Bessel function of
     second order, y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x).

 -- Function: double gsl_sf_bessel_yl (int l, double x)
 -- Function: int gsl_sf_bessel_yl_e (int l, double x, gsl_sf_result
          *result)

     These routines compute the irregular spherical Bessel function of
     order *note l: 133, y_l(x), for l \geq 0.

 -- Function: int gsl_sf_bessel_yl_array (int lmax, double x, double
          result_array[])

     This routine computes the values of the irregular spherical Bessel
     functions y_l(x) for l from 0 to *note lmax: 134. inclusive for
     lmax \geq 0, storing the results in the array *note result_array:
     134.  The values are computed using recurrence relations for
     efficiency, and therefore may differ slightly from the exact
     values.


File: gsl-ref.info,  Node: Regular Modified Spherical Bessel Functions,  Next: Irregular Modified Spherical Bessel Functions,  Prev: Irregular Spherical Bessel Functions,  Up: Bessel Functions

7.5.7 Regular Modified Spherical Bessel Functions
-------------------------------------------------

The regular modified spherical Bessel functions i_l(x) are related to
the modified Bessel functions of fractional order, i_l(x) =
\sqrt{\pi/(2x)} I_{l+1/2}(x)

 -- Function: double gsl_sf_bessel_i0_scaled (double x)
 -- Function: int gsl_sf_bessel_i0_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled regular modified spherical Bessel
     function of zeroth order, \exp(-|x|) i_0(x).

 -- Function: double gsl_sf_bessel_i1_scaled (double x)
 -- Function: int gsl_sf_bessel_i1_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled regular modified spherical Bessel
     function of first order, \exp(-|x|) i_1(x).

 -- Function: double gsl_sf_bessel_i2_scaled (double x)
 -- Function: int gsl_sf_bessel_i2_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled regular modified spherical Bessel
     function of second order, \exp(-|x|) i_2(x)

 -- Function: double gsl_sf_bessel_il_scaled (int l, double x)
 -- Function: int gsl_sf_bessel_il_scaled_e (int l, double x,
          gsl_sf_result *result)

     These routines compute the scaled regular modified spherical Bessel
     function of order *note l: 13d, \exp(-|x|) i_l(x)

 -- Function: int gsl_sf_bessel_il_scaled_array (int lmax, double x,
          double result_array[])

     This routine computes the values of the scaled regular modified
     spherical Bessel functions \exp(-|x|) i_l(x) for l from 0 to *note
     lmax: 13e. inclusive for lmax \geq 0, storing the results in the
     array *note result_array: 13e.  The values are computed using
     recurrence relations for efficiency, and therefore may differ
     slightly from the exact values.


File: gsl-ref.info,  Node: Irregular Modified Spherical Bessel Functions,  Next: Regular Bessel Function—Fractional Order,  Prev: Regular Modified Spherical Bessel Functions,  Up: Bessel Functions

7.5.8 Irregular Modified Spherical Bessel Functions
---------------------------------------------------

The irregular modified spherical Bessel functions k_l(x) are related to
the irregular modified Bessel functions of fractional order, k_l(x) =
\sqrt{\pi/(2x)} K_{l+1/2}(x).

 -- Function: double gsl_sf_bessel_k0_scaled (double x)
 -- Function: int gsl_sf_bessel_k0_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled irregular modified spherical
     Bessel function of zeroth order, \exp(x) k_0(x), for x>0.

 -- Function: double gsl_sf_bessel_k1_scaled (double x)
 -- Function: int gsl_sf_bessel_k1_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled irregular modified spherical
     Bessel function of first order, \exp(x) k_1(x), for x>0.

 -- Function: double gsl_sf_bessel_k2_scaled (double x)
 -- Function: int gsl_sf_bessel_k2_scaled_e (double x, gsl_sf_result
          *result)

     These routines compute the scaled irregular modified spherical
     Bessel function of second order, \exp(x) k_2(x), for x>0.

 -- Function: double gsl_sf_bessel_kl_scaled (int l, double x)
 -- Function: int gsl_sf_bessel_kl_scaled_e (int l, double x,
          gsl_sf_result *result)

     These routines compute the scaled irregular modified spherical
     Bessel function of order *note l: 147, \exp(x) k_l(x), for x>0.

 -- Function: int gsl_sf_bessel_kl_scaled_array (int lmax, double x,
          double result_array[])

     This routine computes the values of the scaled irregular modified
     spherical Bessel functions \exp(x) k_l(x) for l from 0 to *note
     lmax: 148. inclusive for lmax \geq 0 and x>0, storing the results
     in the array *note result_array: 148.  The values are computed
     using recurrence relations for efficiency, and therefore may differ
     slightly from the exact values.


File: gsl-ref.info,  Node: Regular Bessel Function—Fractional Order,  Next: Irregular Bessel Functions—Fractional Order,  Prev: Irregular Modified Spherical Bessel Functions,  Up: Bessel Functions

7.5.9 Regular Bessel Function—Fractional Order
----------------------------------------------

 -- Function: double gsl_sf_bessel_Jnu (double nu, double x)
 -- Function: int gsl_sf_bessel_Jnu_e (double nu, double x,
          gsl_sf_result *result)

     These routines compute the regular cylindrical Bessel function of
     fractional order \nu, J_\nu(x).

 -- Function: int gsl_sf_bessel_sequence_Jnu_e (double nu, gsl_mode_t
          mode, size_t size, double v[])

     This function computes the regular cylindrical Bessel function of
     fractional order \nu, J_\nu(x), evaluated at a series of x values.
     The array *note v: 14c. of length *note size: 14c. contains the x
     values.  They are assumed to be strictly ordered and positive.  The
     array is over-written with the values of J_\nu(x_i).


File: gsl-ref.info,  Node: Irregular Bessel Functions—Fractional Order,  Next: Regular Modified Bessel Functions—Fractional Order,  Prev: Regular Bessel Function—Fractional Order,  Up: Bessel Functions

7.5.10 Irregular Bessel Functions—Fractional Order
--------------------------------------------------

 -- Function: double gsl_sf_bessel_Ynu (double nu, double x)
 -- Function: int gsl_sf_bessel_Ynu_e (double nu, double x,
          gsl_sf_result *result)

     These routines compute the irregular cylindrical Bessel function of
     fractional order \nu, Y_\nu(x).


File: gsl-ref.info,  Node: Regular Modified Bessel Functions—Fractional Order,  Next: Irregular Modified Bessel Functions—Fractional Order,  Prev: Irregular Bessel Functions—Fractional Order,  Up: Bessel Functions

7.5.11 Regular Modified Bessel Functions—Fractional Order
---------------------------------------------------------

 -- Function: double gsl_sf_bessel_Inu (double nu, double x)
 -- Function: int gsl_sf_bessel_Inu_e (double nu, double x,
          gsl_sf_result *result)

     These routines compute the regular modified Bessel function of
     fractional order \nu, I_\nu(x) for x>0, \nu>0.

 -- Function: double gsl_sf_bessel_Inu_scaled (double nu, double x)
 -- Function: int gsl_sf_bessel_Inu_scaled_e (double nu, double x,
          gsl_sf_result *result)

     These routines compute the scaled regular modified Bessel function
     of fractional order \nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.


File: gsl-ref.info,  Node: Irregular Modified Bessel Functions—Fractional Order,  Next: Zeros of Regular Bessel Functions,  Prev: Regular Modified Bessel Functions—Fractional Order,  Up: Bessel Functions

7.5.12 Irregular Modified Bessel Functions—Fractional Order
-----------------------------------------------------------

 -- Function: double gsl_sf_bessel_Knu (double nu, double x)
 -- Function: int gsl_sf_bessel_Knu_e (double nu, double x,
          gsl_sf_result *result)

     These routines compute the irregular modified Bessel function of
     fractional order \nu, K_\nu(x) for x>0, \nu>0.

 -- Function: double gsl_sf_bessel_lnKnu (double nu, double x)
 -- Function: int gsl_sf_bessel_lnKnu_e (double nu, double x,
          gsl_sf_result *result)

     These routines compute the logarithm of the irregular modified
     Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0,
     \nu>0.

 -- Function: double gsl_sf_bessel_Knu_scaled (double nu, double x)
 -- Function: int gsl_sf_bessel_Knu_scaled_e (double nu, double x,
          gsl_sf_result *result)

     These routines compute the scaled irregular modified Bessel
     function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0,
     \nu>0.


File: gsl-ref.info,  Node: Zeros of Regular Bessel Functions,  Prev: Irregular Modified Bessel Functions—Fractional Order,  Up: Bessel Functions

7.5.13 Zeros of Regular Bessel Functions
----------------------------------------

 -- Function: double gsl_sf_bessel_zero_J0 (unsigned int s)
 -- Function: int gsl_sf_bessel_zero_J0_e (unsigned int s, gsl_sf_result
          *result)

     These routines compute the location of the *note s: 15e.-th
     positive zero of the Bessel function J_0(x).

 -- Function: double gsl_sf_bessel_zero_J1 (unsigned int s)
 -- Function: int gsl_sf_bessel_zero_J1_e (unsigned int s, gsl_sf_result
          *result)

     These routines compute the location of the *note s: 160.-th
     positive zero of the Bessel function J_1(x).

 -- Function: double gsl_sf_bessel_zero_Jnu (double nu, unsigned int s)
 -- Function: int gsl_sf_bessel_zero_Jnu_e (double nu, unsigned int s,
          gsl_sf_result *result)

     These routines compute the location of the *note s: 162.-th
     positive zero of the Bessel function J_\nu(x).  The current
     implementation does not support negative values of *note nu: 162.


File: gsl-ref.info,  Node: Clausen Functions,  Next: Coulomb Functions,  Prev: Bessel Functions,  Up: Special Functions

7.6 Clausen Functions
=====================

The Clausen function is defined by the following integral,

Cl_2(x) = - int_0^x dt log( 2 sin(t/2) )

It is related to the *note dilogarithm: 164. by Cl_2(\theta) = \Im
Li_2(\exp(i\theta)).  The Clausen functions are declared in the header
file ‘gsl_sf_clausen.h’.

 -- Function: double gsl_sf_clausen (double x)
 -- Function: int gsl_sf_clausen_e (double x, gsl_sf_result *result)

     These routines compute the Clausen integral Cl_2(x).


File: gsl-ref.info,  Node: Coulomb Functions,  Next: Coupling Coefficients,  Prev: Clausen Functions,  Up: Special Functions

7.7 Coulomb Functions
=====================

The prototypes of the Coulomb functions are declared in the header file
‘gsl_sf_coulomb.h’.  Both bound state and scattering solutions are
available.

* Menu:

* Normalized Hydrogenic Bound States::
* Coulomb Wave Functions::
* Coulomb Wave Function Normalization Constant::


File: gsl-ref.info,  Node: Normalized Hydrogenic Bound States,  Next: Coulomb Wave Functions,  Up: Coulomb Functions

7.7.1 Normalized Hydrogenic Bound States
----------------------------------------

 -- Function: double gsl_sf_hydrogenicR_1 (double Z, double r)
 -- Function: int gsl_sf_hydrogenicR_1_e (double Z, double r,
          gsl_sf_result *result)

     These routines compute the lowest-order normalized hydrogenic bound
     state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).

 -- Function: double gsl_sf_hydrogenicR (int n, int l, double Z, double
          r)
 -- Function: int gsl_sf_hydrogenicR_e (int n, int l, double Z, double
          r, gsl_sf_result *result)

     These routines compute the *note n: 16c.-th normalized hydrogenic
     bound state radial wavefunction,

          R_n := 2 (Z^{3/2}/n^2) sqrt{(n-l-1)!/(n+l)!} exp(-Z r/n) (2Zr/n)^l 
               L^{2l+1}_{n-l-1}(2Zr/n). 

     where L^a_b(x) is the *note generalized Laguerre polynomial: 16d.
     The normalization is chosen such that the wavefunction \psi is
     given by \psi(n,l,r) = R_n Y_{lm}.


File: gsl-ref.info,  Node: Coulomb Wave Functions,  Next: Coulomb Wave Function Normalization Constant,  Prev: Normalized Hydrogenic Bound States,  Up: Coulomb Functions

7.7.2 Coulomb Wave Functions
----------------------------

The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are described in
Abramowitz & Stegun, Chapter 14.  Because there can be a large dynamic
range of values for these functions, overflows are handled gracefully.
If an overflow occurs, ‘GSL_EOVRFLW’ is signalled and exponent(s) are
returned through the modifiable parameters ‘exp_F’, ‘exp_G’.  The full
solution can be reconstructed from the following relations,

     F_L(eta,x) = fc[k_L] * exp(exp_F) 
     G_L(eta,x) = gc[k_L] * exp(exp_G) 
      
     F_L’(eta,x) = fcp[k_L] * exp(exp_F) 
     G_L’(eta,x) = gcp[k_L] * exp(exp_G) 

 -- Function: int gsl_sf_coulomb_wave_FG_e (double eta, double x, double
          L_F, int k, gsl_sf_result *F, gsl_sf_result *Fp, gsl_sf_result
          *G, gsl_sf_result *Gp, double *exp_F, double *exp_G)

     This function computes the Coulomb wave functions F_L(\eta,x),
     G_{L-k}(\eta,x) and their derivatives F'_L(\eta,x),
     G'_{L-k}(\eta,x) with respect to x.  The parameters are restricted
     to L, L-k > -1/2, x > 0 and integer k.  Note that L itself is not
     restricted to being an integer.  The results are stored in the
     parameters F, G for the function values and *note Fp: 16f, *note
     Gp: 16f. for the derivative values.  If an overflow occurs,
     ‘GSL_EOVRFLW’ is returned and scaling exponents are stored in the
     modifiable parameters *note exp_F: 16f, *note exp_G: 16f.

 -- Function: int gsl_sf_coulomb_wave_F_array (double L_min, int kmax,
          double eta, double x, double fc_array[], double *F_exponent)

     This function computes the Coulomb wave function F_L(\eta,x) for L
     = Lmin \dots Lmin + kmax, storing the results in *note fc_array:
     170.  In the case of overflow the exponent is stored in *note
     F_exponent: 170.

 -- Function: int gsl_sf_coulomb_wave_FG_array (double L_min, int kmax,
          double eta, double x, double fc_array[], double gc_array[],
          double *F_exponent, double *G_exponent)

     This function computes the functions F_L(\eta,x), G_L(\eta,x) for L
     = Lmin \dots Lmin + kmax storing the results in *note fc_array:
     171. and *note gc_array: 171.  In the case of overflow the
     exponents are stored in *note F_exponent: 171. and *note
     G_exponent: 171.

 -- Function: int gsl_sf_coulomb_wave_FGp_array (double L_min, int kmax,
          double eta, double x, double fc_array[], double fcp_array[],
          double gc_array[], double gcp_array[], double *F_exponent,
          double *G_exponent)

     This function computes the functions F_L(\eta,x), G_L(\eta,x) and
     their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots
     Lmin + kmax storing the results in *note fc_array: 172, *note
     gc_array: 172, *note fcp_array: 172. and *note gcp_array: 172.  In
     the case of overflow the exponents are stored in *note F_exponent:
     172. and *note G_exponent: 172.

 -- Function: int gsl_sf_coulomb_wave_sphF_array (double L_min, int
          kmax, double eta, double x, double fc_array[], double
          F_exponent[])

     This function computes the Coulomb wave function divided by the
     argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the
     results in *note fc_array: 173.  In the case of overflow the
     exponent is stored in *note F_exponent: 173.  This function reduces
     to spherical Bessel functions in the limit \eta \to 0.


File: gsl-ref.info,  Node: Coulomb Wave Function Normalization Constant,  Prev: Coulomb Wave Functions,  Up: Coulomb Functions

7.7.3 Coulomb Wave Function Normalization Constant
--------------------------------------------------

The Coulomb wave function normalization constant is defined in
Abramowitz 14.1.7.

 -- Function: int gsl_sf_coulomb_CL_e (double L, double eta,
          gsl_sf_result *result)

     This function computes the Coulomb wave function normalization
     constant C_L(\eta) for L > -1.

 -- Function: int gsl_sf_coulomb_CL_array (double Lmin, int kmax, double
          eta, double cl[])

     This function computes the Coulomb wave function normalization
     constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.


File: gsl-ref.info,  Node: Coupling Coefficients,  Next: Dawson Function,  Prev: Coulomb Functions,  Up: Special Functions

7.8 Coupling Coefficients
=========================

The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for
combined angular momentum vectors.  Since the arguments of the standard
coupling coefficient functions are integer or half-integer, the
arguments of the following functions are, by convention, integers equal
to twice the actual spin value.  For information on the 3-j coefficients
see Abramowitz & Stegun, Section 27.9.  The functions described in this
section are declared in the header file ‘gsl_sf_coupling.h’.

* Menu:

* 3-j Symbols::
* 6-j Symbols::
* 9-j Symbols::


File: gsl-ref.info,  Node: 3-j Symbols,  Next: 6-j Symbols,  Up: Coupling Coefficients

7.8.1 3-j Symbols
-----------------

 -- Function: double gsl_sf_coupling_3j (int two_ja, int two_jb, int
          two_jc, int two_ma, int two_mb, int two_mc)
 -- Function: int gsl_sf_coupling_3j_e (int two_ja, int two_jb, int
          two_jc, int two_ma, int two_mb, int two_mc, gsl_sf_result
          *result)

     These routines compute the Wigner 3-j coefficient,

          ( ja jb jc ) 
          ( ma mb mc ) 

     where the arguments are given in half-integer units, ja = *note
     two_ja: 17a./2, ma = *note two_ma: 17a./2, etc.


File: gsl-ref.info,  Node: 6-j Symbols,  Next: 9-j Symbols,  Prev: 3-j Symbols,  Up: Coupling Coefficients

7.8.2 6-j Symbols
-----------------

 -- Function: double gsl_sf_coupling_6j (int two_ja, int two_jb, int
          two_jc, int two_jd, int two_je, int two_jf)
 -- Function: int gsl_sf_coupling_6j_e (int two_ja, int two_jb, int
          two_jc, int two_jd, int two_je, int two_jf, gsl_sf_result
          *result)

     These routines compute the Wigner 6-j coefficient,

          { ja jb jc } 
          { jd je jf } 

     where the arguments are given in half-integer units, ja = *note
     two_ja: 17d./2, ma = ‘two_ma’/2, etc.


File: gsl-ref.info,  Node: 9-j Symbols,  Prev: 6-j Symbols,  Up: Coupling Coefficients

7.8.3 9-j Symbols
-----------------

 -- Function: double gsl_sf_coupling_9j (int two_ja, int two_jb, int
          two_jc, int two_jd, int two_je, int two_jf, int two_jg, int
          two_jh, int two_ji)
 -- Function: int gsl_sf_coupling_9j_e (int two_ja, int two_jb, int
          two_jc, int two_jd, int two_je, int two_jf, int two_jg, int
          two_jh, int two_ji, gsl_sf_result *result)

     These routines compute the Wigner 9-j coefficient,

          { ja jb jc } 
          { jd je jf } 
          { jg jh ji } 

     where the arguments are given in half-integer units, ja = *note
     two_ja: 180./2, ma = ‘two_ma’/2, etc.


File: gsl-ref.info,  Node: Dawson Function,  Next: Debye Functions,  Prev: Coupling Coefficients,  Up: Special Functions

7.9 Dawson Function
===================

The Dawson integral is defined by

     \exp(-x^2) \int_0^x dt \exp(t^2)

A table of Dawson’s integral can be found in Abramowitz & Stegun, Table
7.5.  The Dawson functions are declared in the header file
‘gsl_sf_dawson.h’.

 -- Function: double gsl_sf_dawson (double x)
 -- Function: int gsl_sf_dawson_e (double x, gsl_sf_result *result)

     These routines compute the value of Dawson’s integral for *note x:
     183.


File: gsl-ref.info,  Node: Debye Functions,  Next: Dilogarithm,  Prev: Dawson Function,  Up: Special Functions

7.10 Debye Functions
====================

The Debye functions D_n(x) are defined by the following integral,

     D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))

For further information see Abramowitz & Stegun, Section 27.1.  The
Debye functions are declared in the header file ‘gsl_sf_debye.h’.

 -- Function: double gsl_sf_debye_1 (double x)
 -- Function: int gsl_sf_debye_1_e (double x, gsl_sf_result *result)

     These routines compute the first-order Debye function D_1(x).

 -- Function: double gsl_sf_debye_2 (double x)
 -- Function: int gsl_sf_debye_2_e (double x, gsl_sf_result *result)

     These routines compute the second-order Debye function D_2(x).

 -- Function: double gsl_sf_debye_3 (double x)
 -- Function: int gsl_sf_debye_3_e (double x, gsl_sf_result *result)

     These routines compute the third-order Debye function D_3(x).

 -- Function: double gsl_sf_debye_4 (double x)
 -- Function: int gsl_sf_debye_4_e (double x, gsl_sf_result *result)

     These routines compute the fourth-order Debye function D_4(x).

 -- Function: double gsl_sf_debye_5 (double x)
 -- Function: int gsl_sf_debye_5_e (double x, gsl_sf_result *result)

     These routines compute the fifth-order Debye function D_5(x).

 -- Function: double gsl_sf_debye_6 (double x)
 -- Function: int gsl_sf_debye_6_e (double x, gsl_sf_result *result)

     These routines compute the sixth-order Debye function D_6(x).


File: gsl-ref.info,  Node: Dilogarithm,  Next: Elementary Operations,  Prev: Debye Functions,  Up: Special Functions

7.11 Dilogarithm
================

The dilogarithm is defined as

     Li_2(z) = - \int_0^z ds log(1-s) / s

The functions described in this section are declared in the header file
‘gsl_sf_dilog.h’.

* Menu:

* Real Argument::
* Complex Argument::


File: gsl-ref.info,  Node: Real Argument,  Next: Complex Argument,  Up: Dilogarithm

7.11.1 Real Argument
--------------------

 -- Function: double gsl_sf_dilog (double x)
 -- Function: int gsl_sf_dilog_e (double x, gsl_sf_result *result)

     These routines compute the dilogarithm for a real argument.  In
     Lewin’s notation this is Li_2(x), the real part of the dilogarithm
     of a real x.  It is defined by the integral representation

          Li_2(x) = - \Re \int_0^x ds \log(1-s) / s

     Note that \Im(Li_2(x)) = 0 for x \le 1, and -\pi\log(x) for x > 1.

     Note that Abramowitz & Stegun refer to the Spence integral S(x) =
     Li_2(1 - x) as the dilogarithm rather than Li_2(x).


File: gsl-ref.info,  Node: Complex Argument,  Prev: Real Argument,  Up: Dilogarithm

7.11.2 Complex Argument
-----------------------

 -- Function: int gsl_sf_complex_dilog_e (double r, double theta,
          gsl_sf_result *result_re, gsl_sf_result *result_im)

     This function computes the full complex-valued dilogarithm for the
     complex argument z = r \exp(i \theta).  The real and imaginary
     parts of the result are returned in *note result_re: 196, *note
     result_im: 196.


File: gsl-ref.info,  Node: Elementary Operations,  Next: Elliptic Integrals,  Prev: Dilogarithm,  Up: Special Functions

7.12 Elementary Operations
==========================

The following functions allow for the propagation of errors when
combining quantities by multiplication.  The functions are declared in
the header file ‘gsl_sf_elementary.h’.

 -- Function: double gsl_sf_multiply (double x, double y)
 -- Function: int gsl_sf_multiply_e (double x, double y, gsl_sf_result
          *result)

     This function multiplies *note x: 199. and *note y: 199. storing
     the product and its associated error in *note result: 199.

 -- Function: int gsl_sf_multiply_err_e (double x, double dx, double y,
          double dy, gsl_sf_result *result)

     This function multiplies *note x: 19a. and *note y: 19a. with
     associated absolute errors *note dx: 19a. and *note dy: 19a.  The
     product xy \pm xy \sqrt{(dx/x)^2 +(dy/y)^2} is stored in *note
     result: 19a.


File: gsl-ref.info,  Node: Elliptic Integrals,  Next: Elliptic Functions Jacobi,  Prev: Elementary Operations,  Up: Special Functions

7.13 Elliptic Integrals
=======================

The functions described in this section are declared in the header file
‘gsl_sf_ellint.h’.  Further information about the elliptic integrals can
be found in Abramowitz & Stegun, Chapter 17.

* Menu:

* Definition of Legendre Forms::
* Definition of Carlson Forms::
* Legendre Form of Complete Elliptic Integrals::
* Legendre Form of Incomplete Elliptic Integrals::
* Carlson Forms::


File: gsl-ref.info,  Node: Definition of Legendre Forms,  Next: Definition of Carlson Forms,  Up: Elliptic Integrals

7.13.1 Definition of Legendre Forms
-----------------------------------

The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and
\Pi(\phi,k,n) are defined by,

     F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
     E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))
     Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))

The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) =
E(\pi/2, k).

The notation used here is based on Carlson, “Numerische Mathematik” 33
(1979) 1 and differs slightly from that used by Abramowitz & Stegun,
where the functions are given in terms of the parameter m = k^2 and n is
replaced by -n.


File: gsl-ref.info,  Node: Definition of Carlson Forms,  Next: Legendre Form of Complete Elliptic Integrals,  Prev: Definition of Legendre Forms,  Up: Elliptic Integrals

7.13.2 Definition of Carlson Forms
----------------------------------

The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z),
RF(x,y,z) and RJ(x,y,z,p) are defined by,

     RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
     RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
     RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
     RJ(x,y,z,p) = 3/2 \int_0^\infty dt
                      (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)


File: gsl-ref.info,  Node: Legendre Form of Complete Elliptic Integrals,  Next: Legendre Form of Incomplete Elliptic Integrals,  Prev: Definition of Carlson Forms,  Up: Elliptic Integrals

7.13.3 Legendre Form of Complete Elliptic Integrals
---------------------------------------------------

 -- Function: double gsl_sf_ellint_Kcomp (double k, gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_Kcomp_e (double k, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute the complete elliptic integral K(k) to the
     accuracy specified by the mode variable *note mode: 1a0.  Note that
     Abramowitz & Stegun define this function in terms of the parameter
     m = k^2.

 -- Function: double gsl_sf_ellint_Ecomp (double k, gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_Ecomp_e (double k, gsl_mode_t mode,
          gsl_sf_result *result)

     These routines compute the complete elliptic integral E(k) to the
     accuracy specified by the mode variable *note mode: 1a2.  Note that
     Abramowitz & Stegun define this function in terms of the parameter
     m = k^2.

 -- Function: double gsl_sf_ellint_Pcomp (double k, double n, gsl_mode_t
          mode)
 -- Function: int gsl_sf_ellint_Pcomp_e (double k, double n, gsl_mode_t
          mode, gsl_sf_result *result)

     These routines compute the complete elliptic integral \Pi(k,n) to
     the accuracy specified by the mode variable *note mode: 1a4.  Note
     that Abramowitz & Stegun define this function in terms of the
     parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of
     sign n \to -n.


File: gsl-ref.info,  Node: Legendre Form of Incomplete Elliptic Integrals,  Next: Carlson Forms,  Prev: Legendre Form of Complete Elliptic Integrals,  Up: Elliptic Integrals

7.13.4 Legendre Form of Incomplete Elliptic Integrals
-----------------------------------------------------

 -- Function: double gsl_sf_ellint_F (double phi, double k, gsl_mode_t
          mode)
 -- Function: int gsl_sf_ellint_F_e (double phi, double k, gsl_mode_t
          mode, gsl_sf_result *result)

     These routines compute the incomplete elliptic integral F(\phi,k)
     to the accuracy specified by the mode variable *note mode: 1a7.
     Note that Abramowitz & Stegun define this function in terms of the
     parameter m = k^2.

 -- Function: double gsl_sf_ellint_E (double phi, double k, gsl_mode_t
          mode)
 -- Function: int gsl_sf_ellint_E_e (double phi, double k, gsl_mode_t
          mode, gsl_sf_result *result)

     These routines compute the incomplete elliptic integral E(\phi,k)
     to the accuracy specified by the mode variable *note mode: 1a9.
     Note that Abramowitz & Stegun define this function in terms of the
     parameter m = k^2.

 -- Function: double gsl_sf_ellint_P (double phi, double k, double n,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_P_e (double phi, double k, double n,
          gsl_mode_t mode, gsl_sf_result *result)

     These routines compute the incomplete elliptic integral
     \Pi(\phi,k,n) to the accuracy specified by the mode variable *note
     mode: 1ab.  Note that Abramowitz & Stegun define this function in
     terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the
     change of sign n \to -n.

 -- Function: double gsl_sf_ellint_D (double phi, double k, gsl_mode_t
          mode)
 -- Function: int gsl_sf_ellint_D_e (double phi, double k, gsl_mode_t
          mode, gsl_sf_result *result)

     These functions compute the incomplete elliptic integral D(\phi,k)
     which is defined through the Carlson form RD(x,y,z) by the
     following relation,

          D(\phi,k) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).


File: gsl-ref.info,  Node: Carlson Forms,  Prev: Legendre Form of Incomplete Elliptic Integrals,  Up: Elliptic Integrals

7.13.5 Carlson Forms
--------------------

 -- Function: double gsl_sf_ellint_RC (double x, double y, gsl_mode_t
          mode)
 -- Function: int gsl_sf_ellint_RC_e (double x, double y, gsl_mode_t
          mode, gsl_sf_result *result)

     These routines compute the incomplete elliptic integral RC(x,y) to
     the accuracy specified by the mode variable *note mode: 1b0.

 -- Function: double gsl_sf_ellint_RD (double x, double y, double z,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_RD_e (double x, double y, double z,
          gsl_mode_t mode, gsl_sf_result *result)

     These routines compute the incomplete elliptic integral RD(x,y,z)
     to the accuracy specified by the mode variable *note mode: 1b2.

 -- Function: double gsl_sf_ellint_RF (double x, double y, double z,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_RF_e (double x, double y, double z,
          gsl_mode_t mode, gsl_sf_result *result)

     These routines compute the incomplete elliptic integral RF(x,y,z)
     to the accuracy specified by the mode variable *note mode: 1b4.

 -- Function: double gsl_sf_ellint_RJ (double x, double y, double z,
          double p, gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_RJ_e (double x, double y, double z,
          double p, gsl_mode_t mode, gsl_sf_result *result)

     These routines compute the incomplete elliptic integral RJ(x,y,z,p)
     to the accuracy specified by the mode variable *note mode: 1b6.


File: gsl-ref.info,  Node: Elliptic Functions Jacobi,  Next: Error Functions,  Prev: Elliptic Integrals,  Up: Special Functions

7.14 Elliptic Functions (Jacobi)
================================

The Jacobian Elliptic functions are defined in Abramowitz & Stegun,
Chapter 16.  The functions are declared in the header file
‘gsl_sf_elljac.h’.

 -- Function: int gsl_sf_elljac_e (double u, double m, double *sn,
          double *cn, double *dn)

     This function computes the Jacobian elliptic functions sn(u|m),
     cn(u|m), dn(u|m) by descending Landen transformations.


File: gsl-ref.info,  Node: Error Functions,  Next: Exponential Functions,  Prev: Elliptic Functions Jacobi,  Up: Special Functions

7.15 Error Functions
====================

The error function is described in Abramowitz & Stegun, Chapter 7.  The
functions in this section are declared in the header file
‘gsl_sf_erf.h’.

* Menu:

* Error Function::
* Complementary Error Function::
* Log Complementary Error Function::
* Probability functions::


File: gsl-ref.info,  Node: Error Function,  Next: Complementary Error Function,  Up: Error Functions

7.15.1 Error Function
---------------------

 -- Function: double gsl_sf_erf (double x)
 -- Function: int gsl_sf_erf_e (double x, gsl_sf_result *result)

     These routines compute the error function \erf(x), where \erf(x) =
     (2/\sqrt{\pi}) \int_0^x dt \exp(-t^2).


File: gsl-ref.info,  Node: Complementary Error Function,  Next: Log Complementary Error Function,  Prev: Error Function,  Up: Error Functions

7.15.2 Complementary Error Function
-----------------------------------

 -- Function: double gsl_sf_erfc (double x)
 -- Function: int gsl_sf_erfc_e (double x, gsl_sf_result *result)

     These routines compute the complementary error function \erfc(x) =
     1 - \erf(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2)


File: gsl-ref.info,  Node: Log Complementary Error Function,  Next: Probability functions,  Prev: Complementary Error Function,  Up: Error Functions

7.15.3 Log Complementary Error Function
---------------------------------------

 -- Function: double gsl_sf_log_erfc (double x)
 -- Function: int gsl_sf_log_erfc_e (double x, gsl_sf_result *result)

     These routines compute the logarithm of the complementary error
     function \log(\erfc(x)).


File: gsl-ref.info,  Node: Probability functions,  Prev: Log Complementary Error Function,  Up: Error Functions

7.15.4 Probability functions
----------------------------

The probability functions for the Normal or Gaussian distribution are
described in Abramowitz & Stegun, Section 26.2.

 -- Function: double gsl_sf_erf_Z (double x)
 -- Function: int gsl_sf_erf_Z_e (double x, gsl_sf_result *result)

     These routines compute the Gaussian probability density function
     Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)

 -- Function: double gsl_sf_erf_Q (double x)
 -- Function: int gsl_sf_erf_Q_e (double x, gsl_sf_result *result)

     These routines compute the upper tail of the Gaussian probability
     function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)

The `hazard function' for the normal distribution, also known as the
inverse Mills’ ratio, is defined as,

     h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)

It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim
x as x approaches +\infty.

 -- Function: double gsl_sf_hazard (double x)
 -- Function: int gsl_sf_hazard_e (double x, gsl_sf_result *result)

     These routines compute the hazard function for the normal
     distribution.


File: gsl-ref.info,  Node: Exponential Functions,  Next: Exponential Integrals,  Prev: Error Functions,  Up: Special Functions

7.16 Exponential Functions
==========================

The functions described in this section are declared in the header file
‘gsl_sf_exp.h’.

* Menu:

* Exponential Function::
* Relative Exponential Functions::
* Exponentiation With Error Estimate::


File: gsl-ref.info,  Node: Exponential Function,  Next: Relative Exponential Functions,  Up: Exponential Functions

7.16.1 Exponential Function
---------------------------

 -- Function: double gsl_sf_exp (double x)
 -- Function: int gsl_sf_exp_e (double x, gsl_sf_result *result)

     These routines provide an exponential function \exp(x) using GSL
     semantics and error checking.

 -- Function: int gsl_sf_exp_e10_e (double x, gsl_sf_result_e10 *result)

     This function computes the exponential \exp(x) using the *note
     gsl_sf_result_e10: ce. type to return a result with extended range.
     This function may be useful if the value of \exp(x) would overflow
     the numeric range of ‘double’.

 -- Function: double gsl_sf_exp_mult (double x, double y)
 -- Function: int gsl_sf_exp_mult_e (double x, double y, gsl_sf_result
          *result)

     These routines exponentiate *note x: 1d0. and multiply by the
     factor *note y: 1d0. to return the product y \exp(x).

 -- Function: int gsl_sf_exp_mult_e10_e (const double x, const double y,
          gsl_sf_result_e10 *result)

     This function computes the product y \exp(x) using the *note
     gsl_sf_result_e10: ce. type to return a result with extended
     numeric range.


File: gsl-ref.info,  Node: Relative Exponential Functions,  Next: Exponentiation With Error Estimate,  Prev: Exponential Function,  Up: Exponential Functions

7.16.2 Relative Exponential Functions
-------------------------------------

 -- Function: double gsl_sf_expm1 (double x)
 -- Function: int gsl_sf_expm1_e (double x, gsl_sf_result *result)

     These routines compute the quantity \exp(x)-1 using an algorithm
     that is accurate for small x.

 -- Function: double gsl_sf_exprel (double x)
 -- Function: int gsl_sf_exprel_e (double x, gsl_sf_result *result)

     These routines compute the quantity (\exp(x)-1)/x using an
     algorithm that is accurate for small *note x: 1d6.  For small *note
     x: 1d6. the algorithm is based on the expansion (\exp(x)-1)/x = 1 +
     x/2 + x^2/(2*3) + x^3/(2*3*4) + \dots.

 -- Function: double gsl_sf_exprel_2 (double x)
 -- Function: int gsl_sf_exprel_2_e (double x, gsl_sf_result *result)

     These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an
     algorithm that is accurate for small *note x: 1d8.  For small *note
     x: 1d8. the algorithm is based on the expansion 2(\exp(x)-1-x)/x^2
     = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots.

 -- Function: double gsl_sf_exprel_n (int n, double x)
 -- Function: int gsl_sf_exprel_n_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the N-relative exponential, which is the
     *note n: 1da.-th generalization of the functions *note
     gsl_sf_exprel(): 1d5. and *note gsl_sf_exprel_2(): 1d7.  The
     N-relative exponential is given by,

          exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
                      = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
                      = 1F1 (1,1+N,x)


File: gsl-ref.info,  Node: Exponentiation With Error Estimate,  Prev: Relative Exponential Functions,  Up: Exponential Functions

7.16.3 Exponentiation With Error Estimate
-----------------------------------------

 -- Function: int gsl_sf_exp_err_e (double x, double dx, gsl_sf_result
          *result)

     This function exponentiates *note x: 1dc. with an associated
     absolute error *note dx: 1dc.

 -- Function: int gsl_sf_exp_err_e10_e (double x, double dx,
          gsl_sf_result_e10 *result)

     This function exponentiates a quantity *note x: 1dd. with an
     associated absolute error *note dx: 1dd. using the *note
     gsl_sf_result_e10: ce. type to return a result with extended range.

 -- Function: int gsl_sf_exp_mult_err_e (double x, double dx, double y,
          double dy, gsl_sf_result *result)

     This routine computes the product y \exp(x) for the quantities
     *note x: 1de, *note y: 1de. with associated absolute errors *note
     dx: 1de, *note dy: 1de.

 -- Function: int gsl_sf_exp_mult_err_e10_e (double x, double dx, double
          y, double dy, gsl_sf_result_e10 *result)

     This routine computes the product y \exp(x) for the quantities
     *note x: 1df, *note y: 1df. with associated absolute errors *note
     dx: 1df, *note dy: 1df. using the *note gsl_sf_result_e10: ce. type
     to return a result with extended range.


File: gsl-ref.info,  Node: Exponential Integrals,  Next: Fermi-Dirac Function,  Prev: Exponential Functions,  Up: Special Functions

7.17 Exponential Integrals
==========================

Information on the exponential integrals can be found in Abramowitz &
Stegun, Chapter 5.  These functions are declared in the header file
‘gsl_sf_expint.h’.

* Menu:

* Exponential Integral::
* Ei(x): Ei x.
* Hyperbolic Integrals::
* Ei_3(x): Ei_3 x.
* Trigonometric Integrals::
* Arctangent Integral::


File: gsl-ref.info,  Node: Exponential Integral,  Next: Ei x,  Up: Exponential Integrals

7.17.1 Exponential Integral
---------------------------

 -- Function: double gsl_sf_expint_E1 (double x)
 -- Function: int gsl_sf_expint_E1_e (double x, gsl_sf_result *result)

     These routines compute the exponential integral E_1(x),

          E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.

 -- Function: double gsl_sf_expint_E2 (double x)
 -- Function: int gsl_sf_expint_E2_e (double x, gsl_sf_result *result)

     These routines compute the second-order exponential integral
     E_2(x),

          E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2

 -- Function: double gsl_sf_expint_En (int n, double x)
 -- Function: int gsl_sf_expint_En_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the exponential integral E_n(x) of order
     *note n: 1e7,

          E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.


File: gsl-ref.info,  Node: Ei x,  Next: Hyperbolic Integrals,  Prev: Exponential Integral,  Up: Exponential Integrals

7.17.2 Ei(x)
------------

 -- Function: double gsl_sf_expint_Ei (double x)
 -- Function: int gsl_sf_expint_Ei_e (double x, gsl_sf_result *result)

     These routines compute the exponential integral Ei(x),

          Ei(x) = - PV(\int_{-x}^\infty dt \exp(-t)/t)

     where PV denotes the principal value of the integral.


File: gsl-ref.info,  Node: Hyperbolic Integrals,  Next: Ei_3 x,  Prev: Ei x,  Up: Exponential Integrals

7.17.3 Hyperbolic Integrals
---------------------------

 -- Function: double gsl_sf_Shi (double x)
 -- Function: int gsl_sf_Shi_e (double x, gsl_sf_result *result)

     These routines compute the integral

          Shi(x) = \int_0^x dt \sinh(t)/t

 -- Function: double gsl_sf_Chi (double x)
 -- Function: int gsl_sf_Chi_e (double x, gsl_sf_result *result)

     These routines compute the integral

          Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh(t)-1)/t ]

     where \gamma_E is the Euler constant (available as the macro
     ‘M_EULER’).


File: gsl-ref.info,  Node: Ei_3 x,  Next: Trigonometric Integrals,  Prev: Hyperbolic Integrals,  Up: Exponential Integrals

7.17.4 Ei_3(x)
--------------

 -- Function: double gsl_sf_expint_3 (double x)
 -- Function: int gsl_sf_expint_3_e (double x, gsl_sf_result *result)

     These routines compute the third-order exponential integral

          Ei_3(x) = \int_0^x dt \exp(-t^3)`

     for x \ge 0.


File: gsl-ref.info,  Node: Trigonometric Integrals,  Next: Arctangent Integral,  Prev: Ei_3 x,  Up: Exponential Integrals

7.17.5 Trigonometric Integrals
------------------------------

 -- Function: double gsl_sf_Si (const double x)
 -- Function: int gsl_sf_Si_e (double x, gsl_sf_result *result)

     These routines compute the Sine integral

          Si(x) = \int_0^x dt \sin(t)/t

 -- Function: double gsl_sf_Ci (const double x)
 -- Function: int gsl_sf_Ci_e (double x, gsl_sf_result *result)

     These routines compute the Cosine integral

          Ci(x) = -\int_x^\infty dt \cos(t)/t}

     for x > 0


File: gsl-ref.info,  Node: Arctangent Integral,  Prev: Trigonometric Integrals,  Up: Exponential Integrals

7.17.6 Arctangent Integral
--------------------------

 -- Function: double gsl_sf_atanint (double x)
 -- Function: int gsl_sf_atanint_e (double x, gsl_sf_result *result)

     These routines compute the Arctangent integral, which is defined as

          AtanInt(x) = \int_0^x dt \arctan(t)/t


File: gsl-ref.info,  Node: Fermi-Dirac Function,  Next: Gamma and Beta Functions,  Prev: Exponential Integrals,  Up: Special Functions

7.18 Fermi-Dirac Function
=========================

The functions described in this section are declared in the header file
‘gsl_sf_fermi_dirac.h’.

* Menu:

* Complete Fermi-Dirac Integrals::
* Incomplete Fermi-Dirac Integrals::


File: gsl-ref.info,  Node: Complete Fermi-Dirac Integrals,  Next: Incomplete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function

7.18.1 Complete Fermi-Dirac Integrals
-------------------------------------

The complete Fermi-Dirac integral F_j(x) is given by,

     F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

Note that the Fermi-Dirac integral is sometimes defined without the
normalisation factor in other texts.

 -- Function: double gsl_sf_fermi_dirac_m1 (double x)
 -- Function: int gsl_sf_fermi_dirac_m1_e (double x, gsl_sf_result
          *result)

     These routines compute the complete Fermi-Dirac integral with an
     index of -1.  This integral is given by F_{-1}(x) = e^x / (1 +
     e^x).

 -- Function: double gsl_sf_fermi_dirac_0 (double x)
 -- Function: int gsl_sf_fermi_dirac_0_e (double x, gsl_sf_result
          *result)

     These routines compute the complete Fermi-Dirac integral with an
     index of 0.  This integral is given by F_0(x) = \ln(1 + e^x).

 -- Function: double gsl_sf_fermi_dirac_1 (double x)
 -- Function: int gsl_sf_fermi_dirac_1_e (double x, gsl_sf_result
          *result)

     These routines compute the complete Fermi-Dirac integral with an
     index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).

 -- Function: double gsl_sf_fermi_dirac_2 (double x)
 -- Function: int gsl_sf_fermi_dirac_2_e (double x, gsl_sf_result
          *result)

     These routines compute the complete Fermi-Dirac integral with an
     index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).

 -- Function: double gsl_sf_fermi_dirac_int (int j, double x)
 -- Function: int gsl_sf_fermi_dirac_int_e (int j, double x,
          gsl_sf_result *result)

     These routines compute the complete Fermi-Dirac integral with an
     integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j
     /(\exp(t-x)+1)).

 -- Function: double gsl_sf_fermi_dirac_mhalf (double x)
 -- Function: int gsl_sf_fermi_dirac_mhalf_e (double x, gsl_sf_result
          *result)

     These routines compute the complete Fermi-Dirac integral
     F_{-1/2}(x).

 -- Function: double gsl_sf_fermi_dirac_half (double x)
 -- Function: int gsl_sf_fermi_dirac_half_e (double x, gsl_sf_result
          *result)

     These routines compute the complete Fermi-Dirac integral
     F_{1/2}(x).

 -- Function: double gsl_sf_fermi_dirac_3half (double x)
 -- Function: int gsl_sf_fermi_dirac_3half_e (double x, gsl_sf_result
          *result)

     These routines compute the complete Fermi-Dirac integral
     F_{3/2}(x).


File: gsl-ref.info,  Node: Incomplete Fermi-Dirac Integrals,  Prev: Complete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function

7.18.2 Incomplete Fermi-Dirac Integrals
---------------------------------------

The incomplete Fermi-Dirac integral F_j(x,b) is given by,

     F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))

 -- Function: double gsl_sf_fermi_dirac_inc_0 (double x, double b)
 -- Function: int gsl_sf_fermi_dirac_inc_0_e (double x, double b,
          gsl_sf_result *result)

     These routines compute the incomplete Fermi-Dirac integral with an
     index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x)


File: gsl-ref.info,  Node: Gamma and Beta Functions,  Next: Gegenbauer Functions,  Prev: Fermi-Dirac Function,  Up: Special Functions

7.19 Gamma and Beta Functions
=============================

The following routines compute the gamma and beta functions in their
full and incomplete forms, as well as various kinds of factorials.  The
functions described in this section are declared in the header file
‘gsl_sf_gamma.h’.

* Menu:

* Gamma Functions::
* Factorials::
* Pochhammer Symbol::
* Incomplete Gamma Functions::
* Beta Functions::
* Incomplete Beta Function::


File: gsl-ref.info,  Node: Gamma Functions,  Next: Factorials,  Up: Gamma and Beta Functions

7.19.1 Gamma Functions
----------------------

The Gamma function is defined by the following integral,

     \Gamma(x) = \int_0^{\infty} dt t^{x-1} \exp(-t)

It is related to the factorial function by \Gamma(n) = (n-1)! for
positive integer n.  Further information on the Gamma function can be
found in Abramowitz & Stegun, Chapter 6.

 -- Function: double gsl_sf_gamma (double x)
 -- Function: int gsl_sf_gamma_e (double x, gsl_sf_result *result)

     These routines compute the Gamma function \Gamma(x), subject to x
     not being a negative integer or zero.  The function is computed
     using the real Lanczos method.  The maximum value of x such that
     \Gamma(x) is not considered an overflow is given by the macro
     ‘GSL_SF_GAMMA_XMAX’ and is 171.0.

 -- Function: double gsl_sf_lngamma (double x)
 -- Function: int gsl_sf_lngamma_e (double x, gsl_sf_result *result)

     These routines compute the logarithm of the Gamma function,
     \log(\Gamma(x)), subject to x not being a negative integer or zero.
     For x < 0 the real part of \log(\Gamma(x)) is returned, which is
     equivalent to \log(|\Gamma(x)|).  The function is computed using
     the real Lanczos method.

 -- Function: int gsl_sf_lngamma_sgn_e (double x, gsl_sf_result
          *result_lg, double *sgn)

     This routine computes the sign of the gamma function and the
     logarithm of its magnitude, subject to x not being a negative
     integer or zero.  The function is computed using the real Lanczos
     method.  The value of the gamma function and its error can be
     reconstructed using the relation \Gamma(x) = sgn *
     \exp(result\_lg), taking into account the two components of *note
     result_lg: 216.

 -- Function: double gsl_sf_gammastar (double x)
 -- Function: int gsl_sf_gammastar_e (double x, gsl_sf_result *result)

     These routines compute the regulated Gamma Function \Gamma^*(x) for
     x > 0.  The regulated gamma function is given by,

          \Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
                      = (1 + (1/12x) + ...)  for x \to \infty

     and is a useful suggestion of Temme.

 -- Function: double gsl_sf_gammainv (double x)
 -- Function: int gsl_sf_gammainv_e (double x, gsl_sf_result *result)

     These routines compute the reciprocal of the gamma function,
     1/\Gamma(x) using the real Lanczos method.

 -- Function: int gsl_sf_lngamma_complex_e (double zr, double zi,
          gsl_sf_result *lnr, gsl_sf_result *arg)

     This routine computes \log(\Gamma(z)) for complex z = z_r + i z_i
     and z not a negative integer or zero, using the complex Lanczos
     method.  The returned parameters are lnr = \log|\Gamma(z)| and arg
     = \arg(\Gamma(z)) in (-\pi,\pi].  Note that the phase part (*note
     arg: 21b.) is not well-determined when |z| is very large, due to
     inevitable roundoff in restricting to (-\pi,\pi].  This will result
     in a ‘GSL_ELOSS’ error when it occurs.  The absolute value part
     (*note lnr: 21b.), however, never suffers from loss of precision.


File: gsl-ref.info,  Node: Factorials,  Next: Pochhammer Symbol,  Prev: Gamma Functions,  Up: Gamma and Beta Functions

7.19.2 Factorials
-----------------

Although factorials can be computed from the Gamma function, using the
relation n! = \Gamma(n+1) for non-negative integer n, it is usually more
efficient to call the functions in this section, particularly for small
values of n, whose factorial values are maintained in hardcoded tables.

 -- Function: double gsl_sf_fact (unsigned int n)
 -- Function: int gsl_sf_fact_e (unsigned int n, gsl_sf_result *result)

     These routines compute the factorial n!.  The factorial is related
     to the Gamma function by n! = \Gamma(n+1).  The maximum value of n
     such that n! is not considered an overflow is given by the macro
     ‘GSL_SF_FACT_NMAX’ and is 170.

 -- Function: double gsl_sf_doublefact (unsigned int n)
 -- Function: int gsl_sf_doublefact_e (unsigned int n, gsl_sf_result
          *result)

     These routines compute the double factorial n!! = n(n-2)(n-4)
     \dots.  The maximum value of n such that n!! is not considered an
     overflow is given by the macro ‘GSL_SF_DOUBLEFACT_NMAX’ and is 297.

 -- Function: double gsl_sf_lnfact (unsigned int n)
 -- Function: int gsl_sf_lnfact_e (unsigned int n, gsl_sf_result
          *result)

     These routines compute the logarithm of the factorial of *note n:
     222, \log(n!).  The algorithm is faster than computing
     \ln(\Gamma(n+1)) via *note gsl_sf_lngamma(): 214. for n < 170, but
     defers for larger *note n: 222.

 -- Function: double gsl_sf_lndoublefact (unsigned int n)
 -- Function: int gsl_sf_lndoublefact_e (unsigned int n, gsl_sf_result
          *result)

     These routines compute the logarithm of the double factorial of
     *note n: 224, \log(n!!).

 -- Function: double gsl_sf_choose (unsigned int n, unsigned int m)
 -- Function: int gsl_sf_choose_e (unsigned int n, unsigned int m,
          gsl_sf_result *result)

     These routines compute the combinatorial factor ‘n choose m’ =
     n!/(m!(n-m)!)

 -- Function: double gsl_sf_lnchoose (unsigned int n, unsigned int m)
 -- Function: int gsl_sf_lnchoose_e (unsigned int n, unsigned int m,
          gsl_sf_result *result)

     These routines compute the logarithm of ‘n choose m’.  This is
     equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).

 -- Function: double gsl_sf_taylorcoeff (int n, double x)
 -- Function: int gsl_sf_taylorcoeff_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the Taylor coefficient x^n / n! for x \ge 0,
     n \ge 0


File: gsl-ref.info,  Node: Pochhammer Symbol,  Next: Incomplete Gamma Functions,  Prev: Factorials,  Up: Gamma and Beta Functions

7.19.3 Pochhammer Symbol
------------------------

 -- Function: double gsl_sf_poch (double a, double x)
 -- Function: int gsl_sf_poch_e (double a, double x, gsl_sf_result
          *result)

     These routines compute the Pochhammer symbol (a)_x = \Gamma(a +
     x)/\Gamma(a).  The Pochhammer symbol is also known as the Apell
     symbol and sometimes written as (a,x).  When a and a + x are
     negative integers or zero, the limiting value of the ratio is
     returned.

 -- Function: double gsl_sf_lnpoch (double a, double x)
 -- Function: int gsl_sf_lnpoch_e (double a, double x, gsl_sf_result
          *result)

     These routines compute the logarithm of the Pochhammer symbol,
     \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)).

 -- Function: int gsl_sf_lnpoch_sgn_e (double a, double x, gsl_sf_result
          *result, double *sgn)

     These routines compute the sign of the Pochhammer symbol and the
     logarithm of its magnitude.  The computed parameters are result =
     \log(|(a)_x|) with a corresponding error term, and sgn =
     \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a).

 -- Function: double gsl_sf_pochrel (double a, double x)
 -- Function: int gsl_sf_pochrel_e (double a, double x, gsl_sf_result
          *result)

     These routines compute the relative Pochhammer symbol ((a)_x - 1)/x
     where (a)_x = \Gamma(a + x)/\Gamma(a).


File: gsl-ref.info,  Node: Incomplete Gamma Functions,  Next: Beta Functions,  Prev: Pochhammer Symbol,  Up: Gamma and Beta Functions

7.19.4 Incomplete Gamma Functions
---------------------------------

 -- Function: double gsl_sf_gamma_inc (double a, double x)
 -- Function: int gsl_sf_gamma_inc_e (double a, double x, gsl_sf_result
          *result)

     These functions compute the unnormalized incomplete Gamma Function
     \Gamma(a,x) = \int_x^\infty dt t^{(a-1)} \exp(-t) for a real and x
     \ge 0.

 -- Function: double gsl_sf_gamma_inc_Q (double a, double x)
 -- Function: int gsl_sf_gamma_inc_Q_e (double a, double x,
          gsl_sf_result *result)

     These routines compute the normalized incomplete Gamma Function
     Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^{(a-1)} \exp(-t) for a > 0,
     x \ge 0.

 -- Function: double gsl_sf_gamma_inc_P (double a, double x)
 -- Function: int gsl_sf_gamma_inc_P_e (double a, double x,
          gsl_sf_result *result)

     These routines compute the complementary normalized incomplete
     Gamma Function P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt
     t^{(a-1)} \exp(-t) for a > 0, x \ge 0.

     Note that Abramowitz & Stegun call P(a,x) the incomplete gamma
     function (section 6.5).


File: gsl-ref.info,  Node: Beta Functions,  Next: Incomplete Beta Function,  Prev: Incomplete Gamma Functions,  Up: Gamma and Beta Functions

7.19.5 Beta Functions
---------------------

 -- Function: double gsl_sf_beta (double a, double b)
 -- Function: int gsl_sf_beta_e (double a, double b, gsl_sf_result
          *result)

     These routines compute the Beta Function, B(a,b) =
     \Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being
     negative integers.

 -- Function: double gsl_sf_lnbeta (double a, double b)
 -- Function: int gsl_sf_lnbeta_e (double a, double b, gsl_sf_result
          *result)

     These routines compute the logarithm of the Beta Function,
     \log(B(a,b)) subject to a and b not being negative integers.


File: gsl-ref.info,  Node: Incomplete Beta Function,  Prev: Beta Functions,  Up: Gamma and Beta Functions

7.19.6 Incomplete Beta Function
-------------------------------

 -- Function: double gsl_sf_beta_inc (double a, double b, double x)
 -- Function: int gsl_sf_beta_inc_e (double a, double b, double x,
          gsl_sf_result *result)

     These routines compute the normalized incomplete Beta function
     I_x(a,b) = B_x(a,b) / B(a,b) where

          B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt

     for 0 \le x \le 1.  For a > 0, b > 0 the value is computed using a
     continued fraction expansion.  For all other values it is computed
     using the relation

          I_x(a,b,x) = (1/a) x^a 2F1(a,1-b,a+1,x) / B(a,b)


File: gsl-ref.info,  Node: Gegenbauer Functions,  Next: Hermite Polynomials and Functions,  Prev: Gamma and Beta Functions,  Up: Special Functions

7.20 Gegenbauer Functions
=========================

The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter
22, where they are known as Ultraspherical polynomials.  The functions
described in this section are declared in the header file
‘gsl_sf_gegenbauer.h’.

 -- Function: double gsl_sf_gegenpoly_1 (double lambda, double x)
 -- Function: double gsl_sf_gegenpoly_2 (double lambda, double x)
 -- Function: double gsl_sf_gegenpoly_3 (double lambda, double x)
 -- Function: int gsl_sf_gegenpoly_1_e (double lambda, double x,
          gsl_sf_result *result)
 -- Function: int gsl_sf_gegenpoly_2_e (double lambda, double x,
          gsl_sf_result *result)
 -- Function: int gsl_sf_gegenpoly_3_e (double lambda, double x,
          gsl_sf_result *result)

     These functions evaluate the Gegenbauer polynomials
     C^{(\lambda)}_n(x) using explicit representations for n = 1, 2, 3.

 -- Function: double gsl_sf_gegenpoly_n (int n, double lambda, double x)
 -- Function: int gsl_sf_gegenpoly_n_e (int n, double lambda, double x,
          gsl_sf_result *result)

     These functions evaluate the Gegenbauer polynomial
     C^{(\lambda)}_n(x) for a specific value of *note n: 24b, *note
     lambda: 24b, *note x: 24b. subject to \lambda > -1/2, n \ge 0.

 -- Function: int gsl_sf_gegenpoly_array (int nmax, double lambda,
          double x, double result_array[])

     This function computes an array of Gegenbauer polynomials
     C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda
     > -1/2, nmax \ge 0.


File: gsl-ref.info,  Node: Hermite Polynomials and Functions,  Next: Hypergeometric Functions,  Prev: Gegenbauer Functions,  Up: Special Functions

7.21 Hermite Polynomials and Functions
======================================

Hermite polynomials and functions are discussed in Abramowitz & Stegun,
Chapter 22 and Szego, Gabor (1939, 1958, 1967), Orthogonal Polynomials,
American Mathematical Society.  The Hermite polynomials and functions
are defined in the header file ‘gsl_sf_hermite.h’.

* Menu:

* Hermite Polynomials::
* Derivatives of Hermite Polynomials::
* Hermite Functions::
* Derivatives of Hermite Functions::
* Zeros of Hermite Polynomials and Hermite Functions::


File: gsl-ref.info,  Node: Hermite Polynomials,  Next: Derivatives of Hermite Polynomials,  Up: Hermite Polynomials and Functions

7.21.1 Hermite Polynomials
--------------------------

The Hermite polynomials exist in two variants: the physicist version
H_n(x) and the probabilist version He_n(x).  They are defined by the
derivatives

     H_n(x) = (-1)^n e^{x^2} (d / dx)^n e^{-x^2}
     He_n(x) = (-1)^n e^{x^2/2} (d / dx)^n e^{-x^2/2}

They are connected via

     H_n(x) = 2^{n/2} He_n(\sqrt{2} x)
     He_n(x) = 2^{-n/2} H_n(x / \sqrt{2})

and satisfy the ordinary differential equations

     H_n^{''}(x) - 2x H_n^{'}(x) + 2n H_n(x) = 0
     He_n^{''}(x) - x He_n^{'}(x) + n He_n(x) = 0

 -- Function: double gsl_sf_hermite (const int n, const double x)
 -- Function: int gsl_sf_hermite_e (const int n, const double x,
          gsl_sf_result *result)

     These routines evaluate the physicist Hermite polynomial H_n(x) of
     order *note n: 250. at position *note x: 250.  If an overflow is
     detected, ‘GSL_EOVRFLW’ is returned without calling the error
     handler.

 -- Function: int gsl_sf_hermite_array (const int nmax, const double x,
          double *result_array)

     This routine evaluates all physicist Hermite polynomials H_n up to
     order *note nmax: 251. at position *note x: 251.  The results are
     stored in *note result_array: 251.

 -- Function: double gsl_sf_hermite_series (const int n, const double x,
          const double *a)
 -- Function: int gsl_sf_hermite_series_e (const int n, const double x,
          const double *a, gsl_sf_result *result)

     These routines evaluate the series \sum_{j=0}^n a_j H_j(x) with H_j
     being the j-th physicist Hermite polynomial using the Clenshaw
     algorithm.

 -- Function: double gsl_sf_hermite_prob (const int n, const double x)
 -- Function: int gsl_sf_hermite_prob_e (const int n, const double x,
          gsl_sf_result *result)

     These routines evaluate the probabilist Hermite polynomial He_n(x)
     of order *note n: 255. at position *note x: 255.  If an overflow is
     detected, ‘GSL_EOVRFLW’ is returned without calling the error
     handler.

 -- Function: int gsl_sf_hermite_prob_array (const int nmax, const
          double x, double *result_array)

     This routine evaluates all probabilist Hermite polynomials He_n(x)
     up to order *note nmax: 256. at position *note x: 256.  The results
     are stored in *note result_array: 256.

 -- Function: double gsl_sf_hermite_prob_series (const int n, const
          double x, const double *a)
 -- Function: int gsl_sf_hermite_prob_series_e (const int n, const
          double x, const double *a, gsl_sf_result *result)

     These routines evaluate the series \sum_{j=0}^n a_j He_j(x) with
     He_j being the j-th probabilist Hermite polynomial using the
     Clenshaw algorithm.


File: gsl-ref.info,  Node: Derivatives of Hermite Polynomials,  Next: Hermite Functions,  Prev: Hermite Polynomials,  Up: Hermite Polynomials and Functions

7.21.2 Derivatives of Hermite Polynomials
-----------------------------------------

 -- Function: double gsl_sf_hermite_deriv (const int m, const int n,
          const double x)
 -- Function: int gsl_sf_hermite_deriv_e (const int m, const int n,
          const double x, gsl_sf_result *result)

     These routines evaluate the *note m: 25b.-th derivative of the
     physicist Hermite polynomial H_n(x) of order *note n: 25b. at
     position *note x: 25b.

 -- Function: int gsl_sf_hermite_array_deriv (const int m, const int
          nmax, const double x, double *result_array)

     This routine evaluates the *note m: 25c.-th derivative of all
     physicist Hermite polynomials H_n(x) from orders 0, \dots,
     \text{nmax} at position *note x: 25c.  The result d^m/dx^m H_n(x)
     is stored in ‘result_array[n]’.  The output *note result_array:
     25c. must have length at least ‘nmax + 1’.

 -- Function: int gsl_sf_hermite_deriv_array (const int mmax, const int
          n, const double x, double *result_array)

     This routine evaluates all derivative orders from 0, \dots,
     \text{mmax} of the physicist Hermite polynomial of order *note n:
     25d, H_n, at position *note x: 25d.  The result d^m/dx^m H_n(x) is
     stored in ‘result_array[m]’.  The output *note result_array: 25d.
     must have length at least ‘mmax + 1’.

 -- Function: double gsl_sf_hermite_prob_deriv (const int m, const int
          n, const double x)
 -- Function: int gsl_sf_hermite_prob_deriv_e (const int m, const int n,
          const double x, gsl_sf_result *result)

     These routines evaluate the *note m: 25f.-th derivative of the
     probabilist Hermite polynomial He_n(x) of order *note n: 25f. at
     position *note x: 25f.

 -- Function: int gsl_sf_hermite_prob_array_deriv (const int m, const
          int nmax, const double x, double *result_array)

     This routine evaluates the *note m: 260.-th derivative of all
     probabilist Hermite polynomials He_n(x) from orders 0, \dots,
     \text{nmax} at position *note x: 260.  The result d^m/dx^m He_n(x)
     is stored in ‘result_array[n]’.  The output *note result_array:
     260. must have length at least ‘nmax + 1’.

 -- Function: int gsl_sf_hermite_prob_deriv_array (const int mmax, const
          int n, const double x, double *result_array)

     This routine evaluates all derivative orders from 0, \dots,
     \text{mmax} of the probabilist Hermite polynomial of order *note n:
     261, He_n, at position *note x: 261.  The result d^m/dx^m He_n(x)
     is stored in ‘result_array[m]’.  The output *note result_array:
     261. must have length at least ‘mmax + 1’.


File: gsl-ref.info,  Node: Hermite Functions,  Next: Derivatives of Hermite Functions,  Prev: Derivatives of Hermite Polynomials,  Up: Hermite Polynomials and Functions

7.21.3 Hermite Functions
------------------------

The Hermite functions are defined by

     \psi_n(x) = ( 2^n n! \sqrt{\pi} )^{-1/2} e^{-x^2/2} H_n(x)

and satisfy the Schrödinger equation for a quantum mechanical harmonic
oscillator

     psi''_n(x) + (2n + 1 - x^2) psi_n(x) = 0

They are orthonormal,

     \int_{-\infty}^{\infty} \psi_m(x) \psi_n(x) dx = \delta_{mn}

and form an orthonormal basis of L^2(\mathbb{R}).  The Hermite functions
are also eigenfunctions of the continuous Fourier transform.  GSL offers
two methods for evaluating the Hermite functions.  The first uses the
standard three-term recurrence relation which has O(n) complexity and is
the most accurate.  The second uses a Cauchy integral approach due to
Bunck (2009) which has O(\sqrt{n}) complexity which represents a
significant speed improvement for large n, although it is slightly less
accurate.

 -- Function: double gsl_sf_hermite_func (const int n, const double x)
 -- Function: int gsl_sf_hermite_func_e (const int n, const double x,
          gsl_sf_result *result)

     These routines evaluate the Hermite function \psi_n(x) of order
     *note n: 264. at position *note x: 264. using a three term
     recurrence relation.  The algorithm complexity is O(n).

 -- Function: double gsl_sf_hermite_func_fast (const int n, const double
          x)
 -- Function: int gsl_sf_hermite_func_fast_e (const int n, const double
          x, gsl_sf_result *result)

     These routines evaluate the Hermite function \psi_n(x) of order
     *note n: 266. at position *note x: 266. using a the Cauchy integral
     algorithm due to Bunck, 2009.  The algorithm complexity is
     O(\sqrt{n}).

 -- Function: int gsl_sf_hermite_func_array (const int nmax, const
          double x, double *result_array)

     This routine evaluates all Hermite functions \psi_n(x) for orders n
     = 0, \dots, \textrm{nmax} at position *note x: 267, using the
     recurrence relation algorithm.  The results are stored in *note
     result_array: 267. which has length at least ‘nmax + 1’.

 -- Function: double gsl_sf_hermite_func_series (const int n, const
          double x, const double *a)
 -- Function: int gsl_sf_hermite_func_series_e (const int n, const
          double x, const double *a, gsl_sf_result *result)

     These routines evaluate the series \sum_{j=0}^n a_j \psi_j(x) with
     \psi_j being the j-th Hermite function using the Clenshaw
     algorithm.


File: gsl-ref.info,  Node: Derivatives of Hermite Functions,  Next: Zeros of Hermite Polynomials and Hermite Functions,  Prev: Hermite Functions,  Up: Hermite Polynomials and Functions

7.21.4 Derivatives of Hermite Functions
---------------------------------------

 -- Function: double gsl_sf_hermite_func_der (const int m, const int n,
          const double x)
 -- Function: int gsl_sf_hermite_func_der_e (const int m, const int n,
          const double x, gsl_sf_result *result)

     These routines evaluate the *note m: 26c.-th derivative of the
     Hermite function \psi_n(x) of order *note n: 26c. at position *note
     x: 26c.


File: gsl-ref.info,  Node: Zeros of Hermite Polynomials and Hermite Functions,  Prev: Derivatives of Hermite Functions,  Up: Hermite Polynomials and Functions

7.21.5 Zeros of Hermite Polynomials and Hermite Functions
---------------------------------------------------------

These routines calculate the s-th zero of the Hermite
polynomial/function of order n.  Since the zeros are symmetrical around
zero, only positive zeros are calculated, ordered from smallest to
largest, starting from index 1.  Only for odd polynomial orders a zeroth
zero exists, its value always being zero.

 -- Function: double gsl_sf_hermite_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_zero_e (const int n, const int s,
          gsl_sf_result *result)

     These routines evaluate the *note s: 26f.-th zero of the physicist
     Hermite polynomial H_n(x) of order *note n: 26f.

 -- Function: double gsl_sf_hermite_prob_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_prob_zero_e (const int n, const int s,
          gsl_sf_result *result)

     These routines evaluate the *note s: 271.-th zero of the
     probabilist Hermite polynomial He_n(x) of order *note n: 271.

 -- Function: double gsl_sf_hermite_func_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_func_zero_e (const int n, const int s,
          gsl_sf_result *result)

     These routines evaluate the *note s: 273.-th zero of the Hermite
     function \psi_n(x) of order *note n: 273.


File: gsl-ref.info,  Node: Hypergeometric Functions,  Next: Laguerre Functions,  Prev: Hermite Polynomials and Functions,  Up: Special Functions

7.22 Hypergeometric Functions
=============================

Hypergeometric functions are described in Abramowitz & Stegun, Chapters
13 and 15.  These functions are declared in the header file
‘gsl_sf_hyperg.h’.

 -- Function: double gsl_sf_hyperg_0F1 (double c, double x)
 -- Function: int gsl_sf_hyperg_0F1_e (double c, double x, gsl_sf_result
          *result)

     These routines compute the hypergeometric function

          0F1(c,x)

 -- Function: double gsl_sf_hyperg_1F1_int (int m, int n, double x)
 -- Function: int gsl_sf_hyperg_1F1_int_e (int m, int n, double x,
          gsl_sf_result *result)

     These routines compute the confluent hypergeometric function

          1F1(m,n,x) = M(m,n,x)

     for integer parameters *note m: 278, *note n: 278.

 -- Function: double gsl_sf_hyperg_1F1 (double a, double b, double x)
 -- Function: int gsl_sf_hyperg_1F1_e (double a, double b, double x,
          gsl_sf_result *result)

     These routines compute the confluent hypergeometric function

          1F1(a,b,x) = M(a,b,x)

     for general parameters *note a: 27a, *note b: 27a.

 -- Function: double gsl_sf_hyperg_U_int (int m, int n, double x)
 -- Function: int gsl_sf_hyperg_U_int_e (int m, int n, double x,
          gsl_sf_result *result)

     These routines compute the confluent hypergeometric function
     U(m,n,x) for integer parameters *note m: 27c, *note n: 27c.

 -- Function: int gsl_sf_hyperg_U_int_e10_e (int m, int n, double x,
          gsl_sf_result_e10 *result)

     This routine computes the confluent hypergeometric function
     U(m,n,x) for integer parameters *note m: 27d, *note n: 27d. using
     the *note gsl_sf_result_e10: ce. type to return a result with
     extended range.

 -- Function: double gsl_sf_hyperg_U (double a, double b, double x)
 -- Function: int gsl_sf_hyperg_U_e (double a, double b, double x,
          gsl_sf_result *result)

     These routines compute the confluent hypergeometric function
     U(a,b,x).

 -- Function: int gsl_sf_hyperg_U_e10_e (double a, double b, double x,
          gsl_sf_result_e10 *result)

     This routine computes the confluent hypergeometric function
     U(a,b,x) using the *note gsl_sf_result_e10: ce. type to return a
     result with extended range.

 -- Function: double gsl_sf_hyperg_2F1 (double a, double b, double c,
          double x)
 -- Function: int gsl_sf_hyperg_2F1_e (double a, double b, double c,
          double x, gsl_sf_result *result)

     These routines compute the Gauss hypergeometric function

          2F1(a,b,c,x) = F(a,b,c,x)

     for |x| < 1.  If the arguments (a,b,c,x) are too close to a
     singularity then the function can return the error code
     ‘GSL_EMAXITER’ when the series approximation converges too slowly.
     This occurs in the region of x = 1, c - a - b = m for integer m.

 -- Function: double gsl_sf_hyperg_2F1_conj (double aR, double aI,
          double c, double x)
 -- Function: int gsl_sf_hyperg_2F1_conj_e (double aR, double aI, double
          c, double x, gsl_sf_result *result)

     These routines compute the Gauss hypergeometric function

          2F1(a_R + i a_I, aR - i aI, c, x)

     with complex parameters for |x| < 1.

 -- Function: double gsl_sf_hyperg_2F1_renorm (double a, double b,
          double c, double x)
 -- Function: int gsl_sf_hyperg_2F1_renorm_e (double a, double b, double
          c, double x, gsl_sf_result *result)

     These routines compute the renormalized Gauss hypergeometric
     function

          2F1(a,b,c,x) / \Gamma(c)

     for |x| < 1.

 -- Function: double gsl_sf_hyperg_2F1_conj_renorm (double aR, double
          aI, double c, double x)
 -- Function: int gsl_sf_hyperg_2F1_conj_renorm_e (double aR, double aI,
          double c, double x, gsl_sf_result *result)

     These routines compute the renormalized Gauss hypergeometric
     function

          2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)

     for |x| < 1.

 -- Function: double gsl_sf_hyperg_2F0 (double a, double b, double x)
 -- Function: int gsl_sf_hyperg_2F0_e (double a, double b, double x,
          gsl_sf_result *result)

     These routines compute the hypergeometric function

          2F0(a,b,x)

     The series representation is a divergent hypergeometric series.
     However, for x < 0 we have

          2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)


File: gsl-ref.info,  Node: Laguerre Functions,  Next: Lambert W Functions,  Prev: Hypergeometric Functions,  Up: Special Functions

7.23 Laguerre Functions
=======================

The generalized Laguerre polynomials, sometimes referred to as
associated Laguerre polynomials, are defined in terms of confluent
hypergeometric functions as

L^a_n(x) = ((a+1)_n / n!)  1F1(-n,a+1,x)

where (a)_n is the *note Pochhammer symbol: 22c. (rising factorial).
They are related to the plain Laguerre polynomials L_n(x) by L^0_n(x) =
L_n(x) and L^k_n(x) = (-1)^k (d^k/dx^k) L_{(n+k)}(x) For more
information see Abramowitz & Stegun, Chapter 22.

The functions described in this section are declared in the header file
‘gsl_sf_laguerre.h’.

 -- Function: double gsl_sf_laguerre_1 (double a, double x)
 -- Function: double gsl_sf_laguerre_2 (double a, double x)
 -- Function: double gsl_sf_laguerre_3 (double a, double x)
 -- Function: int gsl_sf_laguerre_1_e (double a, double x, gsl_sf_result
          *result)
 -- Function: int gsl_sf_laguerre_2_e (double a, double x, gsl_sf_result
          *result)
 -- Function: int gsl_sf_laguerre_3_e (double a, double x, gsl_sf_result
          *result)

     These routines evaluate the generalized Laguerre polynomials
     L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.

 -- Function: double gsl_sf_laguerre_n (const int n, const double a,
          const double x)
 -- Function: int gsl_sf_laguerre_n_e (int n, double a, double x,
          gsl_sf_result *result)

     These routines evaluate the generalized Laguerre polynomials
     L^a_n(x) for a > -1, n \ge 0.


File: gsl-ref.info,  Node: Lambert W Functions,  Next: Legendre Functions and Spherical Harmonics,  Prev: Laguerre Functions,  Up: Special Functions

7.24 Lambert W Functions
========================

Lambert’s W functions, W(x), are defined to be solutions of the equation
W(x) \exp(W(x)) = x.  This function has multiple branches for x < 0;
however, it has only two real-valued branches.  We define W_0(x) to be
the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the
other real branch, where W < -1 for x < 0.  The Lambert functions are
declared in the header file ‘gsl_sf_lambert.h’.

 -- Function: double gsl_sf_lambert_W0 (double x)
 -- Function: int gsl_sf_lambert_W0_e (double x, gsl_sf_result *result)

     These compute the principal branch of the Lambert W function,
     W_0(x).

 -- Function: double gsl_sf_lambert_Wm1 (double x)
 -- Function: int gsl_sf_lambert_Wm1_e (double x, gsl_sf_result *result)

     These compute the secondary real-valued branch of the Lambert W
     function, W_{-1}(x).


File: gsl-ref.info,  Node: Legendre Functions and Spherical Harmonics,  Next: Logarithm and Related Functions,  Prev: Lambert W Functions,  Up: Special Functions

7.25 Legendre Functions and Spherical Harmonics
===============================================

The Legendre Functions and Legendre Polynomials are described in
Abramowitz & Stegun, Chapter 8.  These functions are declared in the
header file ‘gsl_sf_legendre.h’.

* Menu:

* Legendre Polynomials::
* Associated Legendre Polynomials and Spherical Harmonics::
* Conical Functions::
* Radial Functions for Hyperbolic Space::


File: gsl-ref.info,  Node: Legendre Polynomials,  Next: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics

7.25.1 Legendre Polynomials
---------------------------

 -- Function: double gsl_sf_legendre_P1 (double x)
 -- Function: double gsl_sf_legendre_P2 (double x)
 -- Function: double gsl_sf_legendre_P3 (double x)
 -- Function: int gsl_sf_legendre_P1_e (double x, gsl_sf_result *result)
 -- Function: int gsl_sf_legendre_P2_e (double x, gsl_sf_result *result)
 -- Function: int gsl_sf_legendre_P3_e (double x, gsl_sf_result *result)

     These functions evaluate the Legendre polynomials P_l(x) using
     explicit representations for l = 1, 2, 3.

 -- Function: double gsl_sf_legendre_Pl (int l, double x)
 -- Function: int gsl_sf_legendre_Pl_e (int l, double x, gsl_sf_result
          *result)

     These functions evaluate the Legendre polynomial P_l(x) for a
     specific value of *note l: 2a2, *note x: 2a2. subject to l \ge 0
     and |x| \le 1.

 -- Function: int gsl_sf_legendre_Pl_array (int lmax, double x, double
          result_array[])
 -- Function: int gsl_sf_legendre_Pl_deriv_array (int lmax, double x,
          double result_array[], double result_deriv_array[])

     These functions compute arrays of Legendre polynomials P_l(x) and
     derivatives dP_l(x)/dx for l = 0, \dots, lmax and |x| \le 1.

 -- Function: double gsl_sf_legendre_Q0 (double x)
 -- Function: int gsl_sf_legendre_Q0_e (double x, gsl_sf_result *result)

     These routines compute the Legendre function Q_0(x) for x > -1 and
     x \ne 1.

 -- Function: double gsl_sf_legendre_Q1 (double x)
 -- Function: int gsl_sf_legendre_Q1_e (double x, gsl_sf_result *result)

     These routines compute the Legendre function Q_1(x) for x > -1 and
     x \ne 1.

 -- Function: double gsl_sf_legendre_Ql (int l, double x)
 -- Function: int gsl_sf_legendre_Ql_e (int l, double x, gsl_sf_result
          *result)

     These routines compute the Legendre function Q_l(x) for x > -1, x
     \ne 1 and l \ge 0.


File: gsl-ref.info,  Node: Associated Legendre Polynomials and Spherical Harmonics,  Next: Conical Functions,  Prev: Legendre Polynomials,  Up: Legendre Functions and Spherical Harmonics

7.25.2 Associated Legendre Polynomials and Spherical Harmonics
--------------------------------------------------------------

The following functions compute the associated Legendre polynomials
P_l^m(x) which are solutions of the differential equation

     (1 - x^2) d^2 P_l^m(x) / dx^2 P_l^m(x) - 2x d/dx P_l^m(x) +
     ( l(l+1) - m^2 / (1 - x^2) ) P_l^m(x) = 0

where the degree l and order m satisfy 0 \le l and 0 \le m \le l.  The
functions P_l^m(x) grow combinatorially with l and can overflow for l
larger than about 150.  Alternatively, one may calculate normalized
associated Legendre polynomials.  There are a number of different
normalization conventions, and these functions can be stably computed up
to degree and order 2700.  The following normalizations are provided:

   * Schmidt semi-normalization

     Schmidt semi-normalized associated Legendre polynomials are often
     used in the magnetics community and are defined as

          S_l^0(x) = P_l^0(x)
          S_l^m(x) = (-1)^m \sqrt((2(l-m)! / (l+m)!)) P_l^m(x), m > 0

     The factor of (-1)^m is called the Condon-Shortley phase factor and
     can be excluded if desired by setting the parameter ‘csphase = 1’
     in the functions below.

   * Spherical Harmonic Normalization

     The associated Legendre polynomials suitable for calculating
     spherical harmonics are defined as

          Y_l^m(x) = (-1)^m \sqrt((2l + 1) * (l-m)! / (4 \pi) / (l+m)!) P_l^m(x)

     where again the phase factor (-1)^m can be included or excluded if
     desired.

   * Full Normalization

     The fully normalized associated Legendre polynomials are defined as

          N_l^m(x) = (-1)^m \sqrt((l + 1/2) (l-m)! / (l+m)!) P_l^m(x)

     and have the property

          \int_{-1}^1 N_l^m(x)^2 dx = 1

The normalized associated Legendre routines below use a recurrence
relation which is stable up to a degree and order of about 2700.  Beyond
this, the computed functions could suffer from underflow leading to
incorrect results.  Routines are provided to compute first and second
derivatives dP_l^m(x)/dx and d^2 P_l^m(x)/dx^2 as well as their
alternate versions d P_l^m(\cos{\theta})/d\theta and d^2
P_l^m(\cos{\theta})/d\theta^2.  While there is a simple scaling
relationship between the two forms, the derivatives involving \theta are
heavily used in spherical harmonic expansions and so these routines are
also provided.

In the functions below, a parameter of type *note gsl_sf_legendre_t:
2ac. specifies the type of normalization to use.  The possible values
are

 -- Type: gsl_sf_legendre_t

     Value                                  Description
                                            
     ---------------------------------------------------------------------------------------------------------------------------
                                            
     ‘GSL_SF_LEGENDRE_NONE’                 The unnormalized associated Legendre polynomials P_l^m(x)
                                            
                                            
     ‘GSL_SF_LEGENDRE_SCHMIDT’              The Schmidt semi-normalized associated Legendre polynomials S_l^m(x)
                                            
                                            
     ‘GSL_SF_LEGENDRE_SPHARM’               The spherical harmonic associated Legendre polynomials Y_l^m(x)
                                            
                                            
     ‘GSL_SF_LEGENDRE_FULL’                 The fully normalized associated Legendre polynomials N_l^m(x)
                                            

 -- Function: int gsl_sf_legendre_array (const gsl_sf_legendre_t norm,
          const size_t lmax, const double x, double result_array[])
 -- Function: int gsl_sf_legendre_array_e (const gsl_sf_legendre_t norm,
          const size_t lmax, const double x, const double csphase,
          double result_array[])

     These functions calculate all normalized associated Legendre
     polynomials for 0 \le l \le lmax and 0 \le m \le l for |x| \le 1.
     The *note norm: 2ae. parameter specifies which normalization is
     used.  The normalized P_l^m(x) values are stored in *note
     result_array: 2ae, whose minimum size can be obtained from calling
     *note gsl_sf_legendre_array_n(): 2af.  The array index of P_l^m(x)
     is obtained from calling ‘gsl_sf_legendre_array_index(l, m)’.  To
     include or exclude the Condon-Shortley phase factor of (-1)^m, set
     the parameter *note csphase: 2ae. to either -1 or 1 respectively in
     the ‘_e’ function.  This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv_array (const gsl_sf_legendre_t
          norm, const size_t lmax, const double x, double
          result_array[], double result_deriv_array[])
 -- Function: int gsl_sf_legendre_deriv_array_e (const gsl_sf_legendre_t
          norm, const size_t lmax, const double x, const double csphase,
          double result_array[], double result_deriv_array[])

     These functions calculate all normalized associated Legendre
     functions and their first derivatives up to degree *note lmax: 2b1.
     for |x| < 1.  The parameter *note norm: 2b1. specifies the
     normalization used.  The normalized P_l^m(x) values and their
     derivatives dP_l^m(x)/dx are stored in *note result_array: 2b1. and
     *note result_deriv_array: 2b1. respectively.  To include or exclude
     the Condon-Shortley phase factor of (-1)^m, set the parameter *note
     csphase: 2b1. to either -1 or 1 respectively in the ‘_e’ function.
     This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv_alt_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[])
 -- Function: int gsl_sf_legendre_deriv_alt_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[], double
          result_deriv_array[])

     These functions calculate all normalized associated Legendre
     functions and their (alternate) first derivatives up to degree
     *note lmax: 2b3. for |x| < 1.  The normalized P_l^m(x) values and
     their derivatives dP_l^m(\cos{\theta})/d\theta are stored in *note
     result_array: 2b3. and *note result_deriv_array: 2b3. respectively.
     To include or exclude the Condon-Shortley phase factor of (-1)^m,
     set the parameter *note csphase: 2b3. to either -1 or 1
     respectively in the ‘_e’ function.  This factor is excluded by
     default.

 -- Function: int gsl_sf_legendre_deriv2_array (const gsl_sf_legendre_t
          norm, const size_t lmax, const double x, double
          result_array[], double result_deriv_array[], double
          result_deriv2_array[])
 -- Function: int gsl_sf_legendre_deriv2_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[], double
          result_deriv_array[], double result_deriv2_array[])

     These functions calculate all normalized associated Legendre
     functions and their first and second derivatives up to degree *note
     lmax: 2b5. for |x| < 1.  The parameter *note norm: 2b5. specifies
     the normalization used.  The normalized P_l^m(x), their first
     derivatives dP_l^m(x)/dx, and their second derivatives d^2
     P_l^m(x)/dx^2 are stored in *note result_array: 2b5, *note
     result_deriv_array: 2b5, and *note result_deriv2_array: 2b5.
     respectively.  To include or exclude the Condon-Shortley phase
     factor of (-1)^m, set the parameter *note csphase: 2b5. to either
     -1 or 1 respectively in the ‘_e’ function.  This factor is excluded
     by default.

 -- Function: int gsl_sf_legendre_deriv2_alt_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[], double
          result_deriv2_array[])
 -- Function: int gsl_sf_legendre_deriv2_alt_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[], double
          result_deriv_array[], double result_deriv2_array[])

     These functions calculate all normalized associated Legendre
     functions and their (alternate) first and second derivatives up to
     degree *note lmax: 2b7. for |x| < 1.  The parameter *note norm:
     2b7. specifies the normalization used.  The normalized P_l^m(x),
     their first derivatives dP_l^m(\cos{\theta})/d\theta, and their
     second derivatives d^2 P_l^m(\cos{\theta})/d\theta^2 are stored in
     *note result_array: 2b7, *note result_deriv_array: 2b7, and *note
     result_deriv2_array: 2b7. respectively.  To include or exclude the
     Condon-Shortley phase factor of (-1)^m, set the parameter *note
     csphase: 2b7. to either -1 or 1 respectively in the ‘_e’ function.
     This factor is excluded by default.

 -- Function: size_t gsl_sf_legendre_nlm (const size_t lmax)

     This function returns the total number of associated Legendre
     functions P_l^m(x) for a given *note lmax: 2b8.  The number is
     ‘(lmax+1) * (lmax+2) / 2’.

 -- Function: size_t gsl_sf_legendre_array_n (const size_t lmax)

     This function returns the minimum array size for maximum degree
     *note lmax: 2af. needed for the array versions of the associated
     Legendre functions.  Size is calculated as the total number of
     P_l^m(x) functions (see *note gsl_sf_legendre_nlm(): 2b8.), plus
     extra space for precomputing multiplicative factors used in the
     recurrence relations.

 -- Function: size_t gsl_sf_legendre_array_index (const size_t l, const
          size_t m)

     This function returns the index into ‘result_array’,
     ‘result_deriv_array’, or ‘result_deriv2_array’ corresponding to
     P_l^m(x), P_l^{'m}(x), or P_l^{''m}(x).  The index is given by
     l(l+1)/2 + m.

     An inline version of this function is used if ‘HAVE_INLINE’ is
     defined.

 -- Function: double gsl_sf_legendre_Plm (int l, int m, double x)
 -- Function: int gsl_sf_legendre_Plm_e (int l, int m, double x,
          gsl_sf_result *result)

     These routines compute the associated Legendre polynomial P_l^m(x)
     for m \ge 0, l \ge m, and |x| \le 1.

 -- Function: double gsl_sf_legendre_sphPlm (int l, int m, double x)
 -- Function: int gsl_sf_legendre_sphPlm_e (int l, int m, double x,
          gsl_sf_result *result)

     These routines compute the normalized associated Legendre
     polynomial \sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)
     suitable for use in spherical harmonics.  The parameters must
     satisfy m \ge 0, l \ge m, and |x| \le 1.  These routines avoid the
     overflows that occur for the standard normalization of P_l^m(x).

 -- Function: int gsl_sf_legendre_Plm_array (int lmax, int m, double x,
          double result_array[])
 -- Function: int gsl_sf_legendre_Plm_deriv_array (int lmax, int m,
          double x, double result_array[], double result_deriv_array[])

     These functions are now deprecated and will be removed in a future
     release; see *note gsl_sf_legendre_array(): 2ad. and *note
     gsl_sf_legendre_deriv_array(): 2b0.

 -- Function: int gsl_sf_legendre_sphPlm_array (int lmax, int m, double
          x, double result_array[])
 -- Function: int gsl_sf_legendre_sphPlm_deriv_array (int lmax, int m,
          double x, double result_array[], double result_deriv_array[])

     These functions are now deprecated and will be removed in a future
     release; see *note gsl_sf_legendre_array(): 2ad. and *note
     gsl_sf_legendre_deriv_array(): 2b0.

 -- Function: int gsl_sf_legendre_array_size (const int lmax, const int
          m)

     This function is now deprecated and will be removed in a future
     release.


File: gsl-ref.info,  Node: Conical Functions,  Next: Radial Functions for Hyperbolic Space,  Prev: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics

7.25.3 Conical Functions
------------------------

The Conical Functions P^\mu_{-(1/2)+i\lambda}(x) and
Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section
8.12.

 -- Function: double gsl_sf_conicalP_half (double lambda, double x)
 -- Function: int gsl_sf_conicalP_half_e (double lambda, double x,
          gsl_sf_result *result)

     These routines compute the irregular Spherical Conical Function
     P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_mhalf (double lambda, double x)
 -- Function: int gsl_sf_conicalP_mhalf_e (double lambda, double x,
          gsl_sf_result *result)

     These routines compute the regular Spherical Conical Function
     P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_0 (double lambda, double x)
 -- Function: int gsl_sf_conicalP_0_e (double lambda, double x,
          gsl_sf_result *result)

     These routines compute the conical function P^0_{-1/2 + i
     \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_1 (double lambda, double x)
 -- Function: int gsl_sf_conicalP_1_e (double lambda, double x,
          gsl_sf_result *result)

     These routines compute the conical function P^1_{-1/2 + i
     \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_sph_reg (int l, double lambda,
          double x)
 -- Function: int gsl_sf_conicalP_sph_reg_e (int l, double lambda,
          double x, gsl_sf_result *result)

     These routines compute the Regular Spherical Conical Function
     P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1 and l \ge -1.

 -- Function: double gsl_sf_conicalP_cyl_reg (int m, double lambda,
          double x)
 -- Function: int gsl_sf_conicalP_cyl_reg_e (int m, double lambda,
          double x, gsl_sf_result *result)

     These routines compute the Regular Cylindrical Conical Function
     P^{-m}_{-1/2 + i \lambda}(x) for x > -1 and m \ge -1.


File: gsl-ref.info,  Node: Radial Functions for Hyperbolic Space,  Prev: Conical Functions,  Up: Legendre Functions and Spherical Harmonics

7.25.4 Radial Functions for Hyperbolic Space
--------------------------------------------

The following spherical functions are specializations of Legendre
functions which give the regular eigenfunctions of the Laplacian on a
3-dimensional hyperbolic space H^3.  Of particular interest is the flat
limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.

 -- Function: double gsl_sf_legendre_H3d_0 (double lambda, double eta)
 -- Function: int gsl_sf_legendre_H3d_0_e (double lambda, double eta,
          gsl_sf_result *result)

     These routines compute the zeroth radial eigenfunction of the
     Laplacian on the 3-dimensional hyperbolic space,

          L^{H3d}_0(\lambda,\eta) := {\sin(\lambda\eta) \over \lambda\sinh(\eta)}

     for \eta \ge 0.  In the flat limit this takes the form
     L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta).

 -- Function: double gsl_sf_legendre_H3d_1 (double lambda, double eta)
 -- Function: int gsl_sf_legendre_H3d_1_e (double lambda, double eta,
          gsl_sf_result *result)

     These routines compute the first radial eigenfunction of the
     Laplacian on the 3-dimensional hyperbolic space,

          L^{H3d}_1(\lambda,\eta) := {1\over\sqrt{\lambda^2 + 1}} {\left(\sin(\lambda \eta)\over \lambda \sinh(\eta)\right)} \left(\coth(\eta) - \lambda \cot(\lambda\eta)\right)

     for \eta \ge 0 In the flat limit this takes the form
     L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).

 -- Function: double gsl_sf_legendre_H3d (int l, double lambda, double
          eta)
 -- Function: int gsl_sf_legendre_H3d_e (int l, double lambda, double
          eta, gsl_sf_result *result)

     These routines compute the *note l: 2d6.-th radial eigenfunction of
     the Laplacian on the 3-dimensional hyperbolic space \eta \ge 0 and
     l \ge 0.  In the flat limit this takes the form
     L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta).

 -- Function: int gsl_sf_legendre_H3d_array (int lmax, double lambda,
          double eta, double result_array[])

     This function computes an array of radial eigenfunctions L^{H3d}_l(
     \lambda, \eta) for 0 \le l \le lmax.


File: gsl-ref.info,  Node: Logarithm and Related Functions,  Next: Mathieu Functions,  Prev: Legendre Functions and Spherical Harmonics,  Up: Special Functions

7.26 Logarithm and Related Functions
====================================

Information on the properties of the Logarithm function can be found in
Abramowitz & Stegun, Chapter 4.  The functions described in this section
are declared in the header file ‘gsl_sf_log.h’.

 -- Function: double gsl_sf_log (double x)
 -- Function: int gsl_sf_log_e (double x, gsl_sf_result *result)

     These routines compute the logarithm of *note x: 2da, \log(x), for
     x > 0.

 -- Function: double gsl_sf_log_abs (double x)
 -- Function: int gsl_sf_log_abs_e (double x, gsl_sf_result *result)

     These routines compute the logarithm of the magnitude of *note x:
     2dc, \log(|x|), for x \ne 0.

 -- Function: int gsl_sf_complex_log_e (double zr, double zi,
          gsl_sf_result *lnr, gsl_sf_result *theta)

     This routine computes the complex logarithm of z = z_r + i z_i.
     The results are returned as *note lnr: 2dd, *note theta: 2dd. such
     that \exp(lnr + i \theta) = z_r + i z_i, where \theta lies in the
     range [-\pi,\pi].

 -- Function: double gsl_sf_log_1plusx (double x)
 -- Function: int gsl_sf_log_1plusx_e (double x, gsl_sf_result *result)

     These routines compute \log(1 + x) for x > -1 using an algorithm
     that is accurate for small *note x: 2df.

 -- Function: double gsl_sf_log_1plusx_mx (double x)
 -- Function: int gsl_sf_log_1plusx_mx_e (double x, gsl_sf_result
          *result)

     These routines compute \log(1 + x) - x for x > -1 using an
     algorithm that is accurate for small *note x: 2e1.


File: gsl-ref.info,  Node: Mathieu Functions,  Next: Power Function,  Prev: Logarithm and Related Functions,  Up: Special Functions

7.27 Mathieu Functions
======================

The routines described in this section compute the angular and radial
Mathieu functions, and their characteristic values.  Mathieu functions
are the solutions of the following two differential equations:

     d^2y/dv^2 + (a - 2q\cos 2v)y = 0
     d^2f/du^2 - (a - 2q\cosh 2u)f = 0

The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd
periodic solutions of the first equation, which is known as Mathieu’s
equation.  These exist only for the discrete sequence of characteristic
values a = a_r(q) (even-periodic) and a = b_r(q) (odd-periodic).

The radial Mathieu functions Mc^{(j)}_{r}(z,q) and Ms^{(j)}_{r}(z,q) are
the solutions of the second equation, which is referred to as Mathieu’s
modified equation.  The radial Mathieu functions of the first, second,
third and fourth kind are denoted by the parameter j, which takes the
value 1, 2, 3 or 4.

For more information on the Mathieu functions, see Abramowitz and
Stegun, Chapter 20.  These functions are defined in the header file
‘gsl_sf_mathieu.h’.

* Menu:

* Mathieu Function Workspace::
* Mathieu Function Characteristic Values::
* Angular Mathieu Functions::
* Radial Mathieu Functions::


File: gsl-ref.info,  Node: Mathieu Function Workspace,  Next: Mathieu Function Characteristic Values,  Up: Mathieu Functions

7.27.1 Mathieu Function Workspace
---------------------------------

The Mathieu functions can be computed for a single order or for multiple
orders, using array-based routines.  The array-based routines require a
preallocated workspace.

 -- Type: gsl_sf_mathieu_workspace

     Workspace required for array-based routines

 -- Function: *note gsl_sf_mathieu_workspace: 2e4. *gsl_sf_mathieu_alloc
          (size_t n, double qmax)

     This function returns a workspace for the array versions of the
     Mathieu routines.  The arguments n and *note qmax: 2e5. specify the
     maximum order and q-value of Mathieu functions which can be
     computed with this workspace.

 -- Function: void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace *work)

     This function frees the workspace *note work: 2e6.


File: gsl-ref.info,  Node: Mathieu Function Characteristic Values,  Next: Angular Mathieu Functions,  Prev: Mathieu Function Workspace,  Up: Mathieu Functions

7.27.2 Mathieu Function Characteristic Values
---------------------------------------------

 -- Function: int gsl_sf_mathieu_a (int n, double q)
 -- Function: int gsl_sf_mathieu_a_e (int n, double q, gsl_sf_result
          *result)
 -- Function: int gsl_sf_mathieu_b (int n, double q)
 -- Function: int gsl_sf_mathieu_b_e (int n, double q, gsl_sf_result
          *result)

     These routines compute the characteristic values a_n(q), b_n(q) of
     the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.

 -- Function: int gsl_sf_mathieu_a_array (int order_min, int order_max,
          double q, gsl_sf_mathieu_workspace *work, double
          result_array[])
 -- Function: int gsl_sf_mathieu_b_array (int order_min, int order_max,
          double q, gsl_sf_mathieu_workspace *work, double
          result_array[])

     These routines compute a series of Mathieu characteristic values
     a_n(q), b_n(q) for n from *note order_min: 2ed. to *note order_max:
     2ed. inclusive, storing the results in the array *note
     result_array: 2ed.


File: gsl-ref.info,  Node: Angular Mathieu Functions,  Next: Radial Mathieu Functions,  Prev: Mathieu Function Characteristic Values,  Up: Mathieu Functions

7.27.3 Angular Mathieu Functions
--------------------------------

 -- Function: int gsl_sf_mathieu_ce (int n, double q, double x)
 -- Function: int gsl_sf_mathieu_ce_e (int n, double q, double x,
          gsl_sf_result *result)
 -- Function: int gsl_sf_mathieu_se (int n, double q, double x)
 -- Function: int gsl_sf_mathieu_se_e (int n, double q, double x,
          gsl_sf_result *result)

     These routines compute the angular Mathieu functions ce_n(q,x) and
     se_n(q,x), respectively.

 -- Function: int gsl_sf_mathieu_ce_array (int nmin, int nmax, double q,
          double x, gsl_sf_mathieu_workspace *work, double
          result_array[])
 -- Function: int gsl_sf_mathieu_se_array (int nmin, int nmax, double q,
          double x, gsl_sf_mathieu_workspace *work, double
          result_array[])

     These routines compute a series of the angular Mathieu functions
     ce_n(q,x) and se_n(q,x) of order n from *note nmin: 2f4. to *note
     nmax: 2f4. inclusive, storing the results in the array *note
     result_array: 2f4.


File: gsl-ref.info,  Node: Radial Mathieu Functions,  Prev: Angular Mathieu Functions,  Up: Mathieu Functions

7.27.4 Radial Mathieu Functions
-------------------------------

 -- Function: int gsl_sf_mathieu_Mc (int j, int n, double q, double x)
 -- Function: int gsl_sf_mathieu_Mc_e (int j, int n, double q, double x,
          gsl_sf_result *result)
 -- Function: int gsl_sf_mathieu_Ms (int j, int n, double q, double x)
 -- Function: int gsl_sf_mathieu_Ms_e (int j, int n, double q, double x,
          gsl_sf_result *result)

     These routines compute the radial *note j: 2f9.-th kind Mathieu
     functions Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order *note n:
     2f9.

     The allowed values of *note j: 2f9. are 1 and 2.  The functions for
     j = 3,4 can be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and
     M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or
     Ms_n^{(j)}.

 -- Function: int gsl_sf_mathieu_Mc_array (int j, int nmin, int nmax,
          double q, double x, gsl_sf_mathieu_workspace *work, double
          result_array[])
 -- Function: int gsl_sf_mathieu_Ms_array (int j, int nmin, int nmax,
          double q, double x, gsl_sf_mathieu_workspace *work, double
          result_array[])

     These routines compute a series of the radial Mathieu functions of
     kind *note j: 2fb, with order from *note nmin: 2fb. to *note nmax:
     2fb. inclusive, storing the results in the array *note
     result_array: 2fb.


File: gsl-ref.info,  Node: Power Function,  Next: Psi Digamma Function,  Prev: Mathieu Functions,  Up: Special Functions

7.28 Power Function
===================

The following functions are equivalent to the function *note
gsl_pow_int(): 4a. with an error estimate.  These functions are declared
in the header file ‘gsl_sf_pow_int.h’.

 -- Function: double gsl_sf_pow_int (double x, int n)
 -- Function: int gsl_sf_pow_int_e (double x, int n, gsl_sf_result
          *result)

     These routines compute the power x^n for integer *note n: 4c.  The
     power is computed using the minimum number of multiplications.  For
     example, x^8 is computed as ((x^2)^2)^2, requiring only 3
     multiplications.  For reasons of efficiency, these functions do not
     check for overflow or underflow conditions.  The following is a
     simple example:

          #include <gsl/gsl_sf_pow_int.h>
          /* compute 3.0**12 */
          double y = gsl_sf_pow_int(3.0, 12);


File: gsl-ref.info,  Node: Psi Digamma Function,  Next: Synchrotron Functions,  Prev: Power Function,  Up: Special Functions

7.29 Psi (Digamma) Function
===========================

The polygamma functions of order n are defined by

     \psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x))

where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function.
These functions are declared in the header file ‘gsl_sf_psi.h’.

* Menu:

* Digamma Function::
* Trigamma Function::
* Polygamma Function::


File: gsl-ref.info,  Node: Digamma Function,  Next: Trigamma Function,  Up: Psi Digamma Function

7.29.1 Digamma Function
-----------------------

 -- Function: double gsl_sf_psi_int (int n)
 -- Function: int gsl_sf_psi_int_e (int n, gsl_sf_result *result)

     These routines compute the digamma function \psi(n) for positive
     integer *note n: 301.  The digamma function is also called the Psi
     function.

 -- Function: double gsl_sf_psi (double x)
 -- Function: int gsl_sf_psi_e (double x, gsl_sf_result *result)

     These routines compute the digamma function \psi(x) for general
     *note x: 303, x \ne 0.

 -- Function: double gsl_sf_psi_1piy (double y)
 -- Function: int gsl_sf_psi_1piy_e (double y, gsl_sf_result *result)

     These routines compute the real part of the digamma function on the
     line 1 + i y, \Re[\psi(1 + i y)].


File: gsl-ref.info,  Node: Trigamma Function,  Next: Polygamma Function,  Prev: Digamma Function,  Up: Psi Digamma Function

7.29.2 Trigamma Function
------------------------

 -- Function: double gsl_sf_psi_1_int (int n)
 -- Function: int gsl_sf_psi_1_int_e (int n, gsl_sf_result *result)

     These routines compute the Trigamma function \psi'(n) for positive
     integer n.

 -- Function: double gsl_sf_psi_1 (double x)
 -- Function: int gsl_sf_psi_1_e (double x, gsl_sf_result *result)

     These routines compute the Trigamma function \psi'(x) for general
     *note x: 30a.


File: gsl-ref.info,  Node: Polygamma Function,  Prev: Trigamma Function,  Up: Psi Digamma Function

7.29.3 Polygamma Function
-------------------------

 -- Function: double gsl_sf_psi_n (int n, double x)
 -- Function: int gsl_sf_psi_n_e (int n, double x, gsl_sf_result
          *result)

     These routines compute the polygamma function \psi^{(n)}(x) for n
     \ge 0, x > 0.


File: gsl-ref.info,  Node: Synchrotron Functions,  Next: Transport Functions,  Prev: Psi Digamma Function,  Up: Special Functions

7.30 Synchrotron Functions
==========================

The functions described in this section are declared in the header file
‘gsl_sf_synchrotron.h’.

 -- Function: double gsl_sf_synchrotron_1 (double x)
 -- Function: int gsl_sf_synchrotron_1_e (double x, gsl_sf_result
          *result)

     These routines compute the first synchrotron function x
     \int_x^\infty dt K_{5/3}(t) for x \ge 0.

 -- Function: double gsl_sf_synchrotron_2 (double x)
 -- Function: int gsl_sf_synchrotron_2_e (double x, gsl_sf_result
          *result)

     These routines compute the second synchrotron function x K_{2/3}(x)
     for x \ge 0.


File: gsl-ref.info,  Node: Transport Functions,  Next: Trigonometric Functions,  Prev: Synchrotron Functions,  Up: Special Functions

7.31 Transport Functions
========================

The transport functions J(n,x) are defined by the integral
representations

     J(n,x) = \int_0^x t^n e^t /(e^t - 1)^2 dt

They are declared in the header file ‘gsl_sf_transport.h’.

 -- Function: double gsl_sf_transport_2 (double x)
 -- Function: int gsl_sf_transport_2_e (double x, gsl_sf_result *result)

     These routines compute the transport function J(2,x).

 -- Function: double gsl_sf_transport_3 (double x)
 -- Function: int gsl_sf_transport_3_e (double x, gsl_sf_result *result)

     These routines compute the transport function J(3,x).

 -- Function: double gsl_sf_transport_4 (double x)
 -- Function: int gsl_sf_transport_4_e (double x, gsl_sf_result *result)

     These routines compute the transport function J(4,x).

 -- Function: double gsl_sf_transport_5 (double x)
 -- Function: int gsl_sf_transport_5_e (double x, gsl_sf_result *result)

     These routines compute the transport function J(5,x).


File: gsl-ref.info,  Node: Trigonometric Functions,  Next: Zeta Functions,  Prev: Transport Functions,  Up: Special Functions

7.32 Trigonometric Functions
============================

The library includes its own trigonometric functions in order to provide
consistency across platforms and reliable error estimates.  These
functions are declared in the header file ‘gsl_sf_trig.h’.

* Menu:

* Circular Trigonometric Functions::
* Trigonometric Functions for Complex Arguments::
* Hyperbolic Trigonometric Functions::
* Conversion Functions::
* Restriction Functions::
* Trigonometric Functions With Error Estimates::


File: gsl-ref.info,  Node: Circular Trigonometric Functions,  Next: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions

7.32.1 Circular Trigonometric Functions
---------------------------------------

 -- Function: double gsl_sf_sin (double x)
 -- Function: int gsl_sf_sin_e (double x, gsl_sf_result *result)

     These routines compute the sine function \sin(x).

 -- Function: double gsl_sf_cos (double x)
 -- Function: int gsl_sf_cos_e (double x, gsl_sf_result *result)

     These routines compute the cosine function \cos(x).

 -- Function: double gsl_sf_hypot (double x, double y)
 -- Function: int gsl_sf_hypot_e (double x, double y, gsl_sf_result
          *result)

     These routines compute the hypotenuse function \sqrt{x^2 + y^2}
     avoiding overflow and underflow.

 -- Function: double gsl_sf_sinc (double x)
 -- Function: int gsl_sf_sinc_e (double x, gsl_sf_result *result)

     These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any
     value of *note x: 325.


File: gsl-ref.info,  Node: Trigonometric Functions for Complex Arguments,  Next: Hyperbolic Trigonometric Functions,  Prev: Circular Trigonometric Functions,  Up: Trigonometric Functions

7.32.2 Trigonometric Functions for Complex Arguments
----------------------------------------------------

 -- Function: int gsl_sf_complex_sin_e (double zr, double zi,
          gsl_sf_result *szr, gsl_sf_result *szi)

     This function computes the complex sine, \sin(z_r + i z_i) storing
     the real and imaginary parts in *note szr: 327, *note szi: 327.

 -- Function: int gsl_sf_complex_cos_e (double zr, double zi,
          gsl_sf_result *czr, gsl_sf_result *czi)

     This function computes the complex cosine, \cos(z_r + i z_i)
     storing the real and imaginary parts in *note czr: 328, *note czi:
     328.

 -- Function: int gsl_sf_complex_logsin_e (double zr, double zi,
          gsl_sf_result *lszr, gsl_sf_result *lszi)

     This function computes the logarithm of the complex sine,
     \log(\sin(z_r + i z_i)) storing the real and imaginary parts in
     *note lszr: 329, *note lszi: 329.


File: gsl-ref.info,  Node: Hyperbolic Trigonometric Functions,  Next: Conversion Functions,  Prev: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions

7.32.3 Hyperbolic Trigonometric Functions
-----------------------------------------

 -- Function: double gsl_sf_lnsinh (double x)
 -- Function: int gsl_sf_lnsinh_e (double x, gsl_sf_result *result)

     These routines compute \log(\sinh(x)) for x > 0.

 -- Function: double gsl_sf_lncosh (double x)
 -- Function: int gsl_sf_lncosh_e (double x, gsl_sf_result *result)

     These routines compute \log(\cosh(x)) for any *note x: 32e.


File: gsl-ref.info,  Node: Conversion Functions,  Next: Restriction Functions,  Prev: Hyperbolic Trigonometric Functions,  Up: Trigonometric Functions

7.32.4 Conversion Functions
---------------------------

 -- Function: int gsl_sf_polar_to_rect (double r, double theta,
          gsl_sf_result *x, gsl_sf_result *y)

     This function converts the polar coordinates (*note r: 330, *note
     theta: 330.) to rectilinear coordinates (*note x: 330, *note y:
     330.), x = r\cos(\theta), y = r\sin(\theta).

 -- Function: int gsl_sf_rect_to_polar (double x, double y,
          gsl_sf_result *r, gsl_sf_result *theta)

     This function converts the rectilinear coordinates (*note x: 331,
     *note y: 331.) to polar coordinates (*note r: 331, *note theta:
     331.), such that x = r\cos(\theta), y = r\sin(\theta).  The
     argument *note theta: 331. lies in the range [-\pi, \pi].


File: gsl-ref.info,  Node: Restriction Functions,  Next: Trigonometric Functions With Error Estimates,  Prev: Conversion Functions,  Up: Trigonometric Functions

7.32.5 Restriction Functions
----------------------------

 -- Function: double gsl_sf_angle_restrict_symm (double theta)
 -- Function: int gsl_sf_angle_restrict_symm_e (double *theta)

     These routines force the angle *note theta: 334. to lie in the
     range (-\pi,\pi].

     Note that the mathematical value of \pi is slightly greater than
     ‘M_PI’, so the machine numbers ‘M_PI’ and ‘-M_PI’ are included in
     the range.

 -- Function: double gsl_sf_angle_restrict_pos (double theta)
 -- Function: int gsl_sf_angle_restrict_pos_e (double *theta)

     These routines force the angle *note theta: 336. to lie in the
     range [0, 2\pi).

     Note that the mathematical value of 2\pi is slightly greater than
     ‘2*M_PI’, so the machine number ‘2*M_PI’ is included in the range.


File: gsl-ref.info,  Node: Trigonometric Functions With Error Estimates,  Prev: Restriction Functions,  Up: Trigonometric Functions

7.32.6 Trigonometric Functions With Error Estimates
---------------------------------------------------

 -- Function: int gsl_sf_sin_err_e (double x, double dx, gsl_sf_result
          *result)

     This routine computes the sine of an angle *note x: 338. with an
     associated absolute error *note dx: 338, \sin(x \pm dx).  Note that
     this function is provided in the error-handling form only since its
     purpose is to compute the propagated error.

 -- Function: int gsl_sf_cos_err_e (double x, double dx, gsl_sf_result
          *result)

     This routine computes the cosine of an angle *note x: 339. with an
     associated absolute error *note dx: 339, \cos(x \pm dx).  Note that
     this function is provided in the error-handling form only since its
     purpose is to compute the propagated error.


File: gsl-ref.info,  Node: Zeta Functions,  Next: Examples<3>,  Prev: Trigonometric Functions,  Up: Special Functions

7.33 Zeta Functions
===================

The Riemann zeta function is defined in Abramowitz & Stegun, Section
23.2.  The functions described in this section are declared in the
header file ‘gsl_sf_zeta.h’.

* Menu:

* Riemann Zeta Function::
* Riemann Zeta Function Minus One::
* Hurwitz Zeta Function::
* Eta Function::


File: gsl-ref.info,  Node: Riemann Zeta Function,  Next: Riemann Zeta Function Minus One,  Up: Zeta Functions

7.33.1 Riemann Zeta Function
----------------------------

The Riemann zeta function is defined by the infinite sum

     \zeta(s) = \sum_{k=1}^\infty k^{-s}

 -- Function: double gsl_sf_zeta_int (int n)
 -- Function: int gsl_sf_zeta_int_e (int n, gsl_sf_result *result)

     These routines compute the Riemann zeta function \zeta(n) for
     integer *note n: 33d, n \ne 1.

 -- Function: double gsl_sf_zeta (double s)
 -- Function: int gsl_sf_zeta_e (double s, gsl_sf_result *result)

     These routines compute the Riemann zeta function \zeta(s) for
     arbitrary *note s: 33f, s \ne 1.


File: gsl-ref.info,  Node: Riemann Zeta Function Minus One,  Next: Hurwitz Zeta Function,  Prev: Riemann Zeta Function,  Up: Zeta Functions

7.33.2 Riemann Zeta Function Minus One
--------------------------------------

For large positive argument, the Riemann zeta function approaches one.
In this region the fractional part is interesting, and therefore we need
a function to evaluate it explicitly.

 -- Function: double gsl_sf_zetam1_int (int n)
 -- Function: int gsl_sf_zetam1_int_e (int n, gsl_sf_result *result)

     These routines compute \zeta(n) - 1 for integer *note n: 342, n \ne
     1.

 -- Function: double gsl_sf_zetam1 (double s)
 -- Function: int gsl_sf_zetam1_e (double s, gsl_sf_result *result)

     These routines compute \zeta(s) - 1 for arbitrary *note s: 344, s
     \ne 1.


File: gsl-ref.info,  Node: Hurwitz Zeta Function,  Next: Eta Function,  Prev: Riemann Zeta Function Minus One,  Up: Zeta Functions

7.33.3 Hurwitz Zeta Function
----------------------------

The Hurwitz zeta function is defined by

     \zeta(s,q) = \sum_0^\infty (k+q)^{-s}

 -- Function: double gsl_sf_hzeta (double s, double q)
 -- Function: int gsl_sf_hzeta_e (double s, double q, gsl_sf_result
          *result)

     These routines compute the Hurwitz zeta function \zeta(s,q) for s >
     1, q > 0.


File: gsl-ref.info,  Node: Eta Function,  Prev: Hurwitz Zeta Function,  Up: Zeta Functions

7.33.4 Eta Function
-------------------

The eta function is defined by

     \eta(s) = (1-2^{1-s}) \zeta(s)

 -- Function: double gsl_sf_eta_int (int n)
 -- Function: int gsl_sf_eta_int_e (int n, gsl_sf_result *result)

     These routines compute the eta function \eta(n) for integer *note
     n: 34a.

 -- Function: double gsl_sf_eta (double s)
 -- Function: int gsl_sf_eta_e (double s, gsl_sf_result *result)

     These routines compute the eta function \eta(s) for arbitrary *note
     s: 34c.


File: gsl-ref.info,  Node: Examples<3>,  Next: References and Further Reading<3>,  Prev: Zeta Functions,  Up: Special Functions

7.34 Examples
=============

The following example demonstrates the use of the error handling form of
the special functions, in this case to compute the Bessel function
J_0(5.0),

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_sf_bessel.h>

     int
     main (void)
     {
       double x = 5.0;
       gsl_sf_result result;

       double expected = -0.17759677131433830434739701;

       int status = gsl_sf_bessel_J0_e (x, &result);

       printf ("status  = %s\n", gsl_strerror(status));
       printf ("J0(5.0) = %.18f\n"
               "      +/- % .18f\n",
               result.val, result.err);
       printf ("exact   = %.18f\n", expected);
       return status;
     }

Here are the results of running the program,

     status  = success
     J0(5.0) = -0.177596771314338264
           +/-  0.000000000000000193
     exact   = -0.177596771314338292

The next program computes the same quantity using the natural form of
the function.  In this case the error term ‘result.err’ and return
status are not accessible.

     #include <stdio.h>
     #include <gsl/gsl_sf_bessel.h>

     int
     main (void)
     {
       double x = 5.0;
       double expected = -0.17759677131433830434739701;

       double y = gsl_sf_bessel_J0 (x);

       printf ("J0(5.0) = %.18f\n", y);
       printf ("exact   = %.18f\n", expected);
       return 0;
     }

The results of the function are the same,

     J0(5.0) = -0.177596771314338264
     exact   = -0.177596771314338292


File: gsl-ref.info,  Node: References and Further Reading<3>,  Prev: Examples<3>,  Up: Special Functions

7.35 References and Further Reading
===================================

The library follows the conventions of the following book where
possible,

   * Handbook of Mathematical Functions, edited by Abramowitz & Stegun,
     Dover, ISBN 0486612724.

The following papers contain information on the algorithms used to
compute the special functions,

   * Allan J. MacLeod, MISCFUN: A software package to compute uncommon
     special functions.  ACM Trans.  Math.  Soft., vol.: 22, 1996,
     288–301

   * Bunck, B. F., A fast algorithm for evaluation of normalized Hermite
     functions, BIT Numer.  Math, 49: 281-295, 2009.

   * G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd
     Edition (Cambridge University Press, 1944).

   * G. Nemeth, Mathematical Approximations of Special Functions, Nova
     Science Publishers, ISBN 1-56072-052-2

   * B.C. Carlson, Special Functions of Applied Mathematics (1977)

   * N. M. Temme, Special Functions: An Introduction to the Classical
     Functions of Mathematical Physics (1996), ISBN 978-0471113133.

   * W.J. Thompson, Atlas for Computing Mathematical Functions, John
     Wiley & Sons, New York (1997).

   * Y.Y. Luke, Algorithms for the Computation of Mathematical
     Functions, Academic Press, New York (1977).

   * S. A. Holmes and W. E. Featherstone, A unified approach to the
     Clenshaw summation and the recursive computation of very high
     degree and order normalised associated Legendre functions, Journal
     of Geodesy, 76, pg.  279-299, 2002.


File: gsl-ref.info,  Node: Vectors and Matrices,  Next: Permutations,  Prev: Special Functions,  Up: Top

8 Vectors and Matrices
**********************

The functions described in this chapter provide a simple vector and
matrix interface to ordinary C arrays.  The memory management of these
arrays is implemented using a single underlying type, known as a block.
By writing your functions in terms of vectors and matrices you can pass
a single structure containing both data and dimensions as an argument
without needing additional function parameters.  The structures are
compatible with the vector and matrix formats used by BLAS routines.

* Menu:

* Data types::
* Blocks::
* Vectors::
* Matrices::


File: gsl-ref.info,  Node: Data types,  Next: Blocks,  Up: Vectors and Matrices

8.1 Data types
==============

All the functions are available for each of the standard data-types.
The versions for ‘double’ have the prefix ‘gsl_block’, ‘gsl_vector’ and
‘gsl_matrix’.  Similarly the versions for single-precision ‘float’
arrays have the prefix ‘gsl_block_float’, ‘gsl_vector_float’ and
‘gsl_matrix_float’.  The full list of available types is given below,

Prefix                              Type
                                    
-------------------------------------------------------------
                                    
gsl_block                           double
                                    
                                    
gsl_block_float                     float
                                    
                                    
gsl_block_long_double               long double
                                    
                                    
gsl_block_int                       int
                                    
                                    
gsl_block_uint                      unsigned int
                                    
                                    
gsl_block_long                      long
                                    
                                    
gsl_block_ulong                     unsigned long
                                    
                                    
gsl_block_short                     short
                                    
                                    
gsl_block_ushort                    unsigned short
                                    
                                    
gsl_block_char                      char
                                    
                                    
gsl_block_uchar                     unsigned char
                                    
                                    
gsl_block_complex                   complex double
                                    
                                    
gsl_block_complex_float             complex float
                                    
                                    
gsl_block_complex_long_double       complex long double
                                    

Corresponding types exist for the ‘gsl_vector’ and ‘gsl_matrix’
functions.


File: gsl-ref.info,  Node: Blocks,  Next: Vectors,  Prev: Data types,  Up: Vectors and Matrices

8.2 Blocks
==========

For consistency all memory is allocated through a ‘gsl_block’ structure.
The structure contains two components, the size of an area of memory and
a pointer to the memory.  The ‘gsl_block’ structure looks like this,

 -- Type: gsl_block

          typedef struct
          {
            size_t size;
            double * data;
          } gsl_block;

Vectors and matrices are made by `slicing' an underlying block.  A slice
is a set of elements formed from an initial offset and a combination of
indices and step-sizes.  In the case of a matrix the step-size for the
column index represents the row-length.  The step-size for a vector is
known as the `stride'.

The functions for allocating and deallocating blocks are defined in
‘gsl_block.h’.

* Menu:

* Block allocation::
* Reading and writing blocks::
* Example programs for blocks::


File: gsl-ref.info,  Node: Block allocation,  Next: Reading and writing blocks,  Up: Blocks

8.2.1 Block allocation
----------------------

The functions for allocating memory to a block follow the style of
‘malloc’ and ‘free’.  In addition they also perform their own error
checking.  If there is insufficient memory available to allocate a block
then the functions call the GSL error handler (with an error number of
*note GSL_ENOMEM: 2a.) in addition to returning a null pointer.  Thus if
you use the library error handler to abort your program then it isn’t
necessary to check every ‘alloc’.

 -- Function: *note gsl_block: 353. *gsl_block_alloc (size_t n)

     This function allocates memory for a block of *note n: 355.
     double-precision elements, returning a pointer to the block struct.
     The block is not initialized and so the values of its elements are
     undefined.  Use the function *note gsl_block_calloc(): 356. if you
     want to ensure that all the elements are initialized to zero.

     Zero-sized requests are valid and return a non-null result.  A null
     pointer is returned if insufficient memory is available to create
     the block.

 -- Function: *note gsl_block: 353. *gsl_block_calloc (size_t n)

     This function allocates memory for a block and initializes all the
     elements of the block to zero.

 -- Function: void gsl_block_free (gsl_block *b)

     This function frees the memory used by a block *note b: 357.
     previously allocated with *note gsl_block_alloc(): 355. or *note
     gsl_block_calloc(): 356.


File: gsl-ref.info,  Node: Reading and writing blocks,  Next: Example programs for blocks,  Prev: Block allocation,  Up: Blocks

8.2.2 Reading and writing blocks
--------------------------------

The library provides functions for reading and writing blocks to a file
as binary data or formatted text.

 -- Function: int gsl_block_fwrite (FILE *stream, const gsl_block *b)

     This function writes the elements of the block *note b: 359. to the
     stream *note stream: 359. in binary format.  The return value is 0
     for success and ‘GSL_EFAILED’ if there was a problem writing to the
     file.  Since the data is written in the native binary format it may
     not be portable between different architectures.

 -- Function: int gsl_block_fread (FILE *stream, gsl_block *b)

     This function reads into the block *note b: 35a. from the open
     stream *note stream: 35a. in binary format.  The block *note b:
     35a. must be preallocated with the correct length since the
     function uses the size of *note b: 35a. to determine how many bytes
     to read.  The return value is 0 for success and ‘GSL_EFAILED’ if
     there was a problem reading from the file.  The data is assumed to
     have been written in the native binary format on the same
     architecture.

 -- Function: int gsl_block_fprintf (FILE *stream, const gsl_block *b,
          const char *format)

     This function writes the elements of the block *note b: 35b.
     line-by-line to the stream *note stream: 35b. using the format
     specifier *note format: 35b, which should be one of the ‘%g’, ‘%e’
     or ‘%f’ formats for floating point numbers and ‘%d’ for integers.
     The function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem writing to the file.

 -- Function: int gsl_block_fscanf (FILE *stream, gsl_block *b)

     This function reads formatted data from the stream *note stream:
     35c. into the block *note b: 35c.  The block *note b: 35c. must be
     preallocated with the correct length since the function uses the
     size of *note b: 35c. to determine how many numbers to read.  The
     function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem reading from the file.


File: gsl-ref.info,  Node: Example programs for blocks,  Prev: Reading and writing blocks,  Up: Blocks

8.2.3 Example programs for blocks
---------------------------------

The following program shows how to allocate a block,

     #include <stdio.h>
     #include <gsl/gsl_block.h>

     int
     main (void)
     {
       gsl_block * b = gsl_block_alloc (100);

       printf ("length of block = %zu\n", b->size);
       printf ("block data address = %p\n", b->data);

       gsl_block_free (b);
       return 0;
     }

Here is the output from the program,

     length of block = 100
     block data address = 0x804b0d8


File: gsl-ref.info,  Node: Vectors,  Next: Matrices,  Prev: Blocks,  Up: Vectors and Matrices

8.3 Vectors
===========

Vectors are defined by a *note gsl_vector: 35f. structure which
describes a slice of a block.  Different vectors can be created which
point to the same block.  A vector slice is a set of equally-spaced
elements of an area of memory.

The *note gsl_vector: 35f. structure contains five components, the
`size', the `stride', a pointer to the memory where the elements are
stored, ‘data’, a pointer to the block owned by the vector, ‘block’, if
any, and an ownership flag, ‘owner’.  The structure is very simple and
looks like this,

 -- Type: gsl_vector

          typedef struct
          {
            size_t size;
            size_t stride;
            double * data;
            gsl_block * block;
            int owner;
          } gsl_vector;

The ‘size’ is simply the number of vector elements.  The range of valid
indices runs from 0 to ‘size-1’.  The ‘stride’ is the step-size from one
element to the next in physical memory, measured in units of the
appropriate datatype.  The pointer ‘data’ gives the location of the
first element of the vector in memory.  The pointer ‘block’ stores the
location of the memory block in which the vector elements are located
(if any).  If the vector owns this block then the ‘owner’ field is set
to one and the block will be deallocated when the vector is freed.  If
the vector points to a block owned by another object then the ‘owner’
field is zero and any underlying block will not be deallocated with the
vector.

The functions for allocating and accessing vectors are defined in
‘gsl_vector.h’.

* Menu:

* Vector allocation::
* Accessing vector elements::
* Initializing vector elements::
* Reading and writing vectors::
* Vector views::
* Copying vectors::
* Exchanging elements::
* Vector operations::
* Finding maximum and minimum elements of vectors::
* Vector properties::
* Example programs for vectors::


File: gsl-ref.info,  Node: Vector allocation,  Next: Accessing vector elements,  Up: Vectors

8.3.1 Vector allocation
-----------------------

The functions for allocating memory to a vector follow the style of
‘malloc’ and ‘free’.  In addition they also perform their own error
checking.  If there is insufficient memory available to allocate a
vector then the functions call the GSL error handler (with an error
number of *note GSL_ENOMEM: 2a.) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your
program then it isn’t necessary to check every ‘alloc’.

 -- Function: *note gsl_vector: 35f. *gsl_vector_alloc (size_t n)

     This function creates a vector of length n, returning a pointer to
     a newly initialized vector struct.  A new block is allocated for
     the elements of the vector, and stored in the ‘block’ component of
     the vector struct.  The block is “owned” by the vector, and will be
     deallocated when the vector is deallocated.  Zero-sized requests
     are valid and return a non-null result.

 -- Function: *note gsl_vector: 35f. *gsl_vector_calloc (size_t n)

     This function allocates memory for a vector of length *note n: 362.
     and initializes all the elements of the vector to zero.

 -- Function: void gsl_vector_free (gsl_vector *v)

     This function frees a previously allocated vector *note v: 363.  If
     the vector was created using *note gsl_vector_alloc(): 361. then
     the block underlying the vector will also be deallocated.  If the
     vector has been created from another object then the memory is
     still owned by that object and will not be deallocated.


File: gsl-ref.info,  Node: Accessing vector elements,  Next: Initializing vector elements,  Prev: Vector allocation,  Up: Vectors

8.3.2 Accessing vector elements
-------------------------------

Unlike Fortran compilers, C compilers do not usually provide support for
range checking of vectors and matrices.  (1) The functions *note
gsl_vector_get(): 365. and *note gsl_vector_set(): 366. can perform
portable range checking for you and report an error if you attempt to
access elements outside the allowed range.

The functions for accessing the elements of a vector or matrix are
defined in ‘gsl_vector.h’ and declared ‘extern inline’ to eliminate
function-call overhead.  You must compile your program with the
preprocessor macro ‘HAVE_INLINE’ defined to use these functions.

 -- Macro: GSL_RANGE_CHECK_OFF

     If necessary you can turn off range checking completely without
     modifying any source files by recompiling your program with the
     preprocessor definition *note GSL_RANGE_CHECK_OFF: 367.  Provided
     your compiler supports inline functions the effect of turning off
     range checking is to replace calls to ‘gsl_vector_get(v,i)’ by
     ‘v->data[i*v->stride]’ and calls to ‘gsl_vector_set(v,i,x)’ by
     ‘v->data[i*v->stride]=x’.  Thus there should be no performance
     penalty for using the range checking functions when range checking
     is turned off.

 -- Macro: GSL_C99_INLINE

     If you use a C99 compiler which requires inline functions in header
     files to be declared ‘inline’ instead of ‘extern inline’, define
     the macro *note GSL_C99_INLINE: 368. (see *note Inline functions:
     15.).  With GCC this is selected automatically when compiling in
     C99 mode (‘-std=c99’).

 -- Variable: int gsl_check_range

     If inline functions are not used, calls to the functions *note
     gsl_vector_get(): 365. and *note gsl_vector_set(): 366. will link
     to the compiled versions of these functions in the library itself.
     The range checking in these functions is controlled by the global
     integer variable ‘gsl_check_range’.  It is enabled by default—to
     disable it, set ‘gsl_check_range’ to zero.  Due to function-call
     overhead, there is less benefit in disabling range checking here
     than for inline functions.

 -- Function: double gsl_vector_get (const gsl_vector *v, const size_t
          i)

     This function returns the *note i: 365.-th element of a vector
     *note v: 365.  If *note i: 365. lies outside the allowed range of 0
     to ‘size - 1’ then the error handler is invoked and 0 is returned.
     An inline version of this function is used when ‘HAVE_INLINE’ is
     defined.

 -- Function: void gsl_vector_set (gsl_vector *v, const size_t i, double
          x)

     This function sets the value of the *note i: 366.-th element of a
     vector *note v: 366. to *note x: 366.  If *note i: 366. lies
     outside the allowed range of 0 to ‘size - 1’ then the error handler
     is invoked.  An inline version of this function is used when
     ‘HAVE_INLINE’ is defined.

 -- Function: double *gsl_vector_ptr (gsl_vector *v, size_t i)
 -- Function: const double *gsl_vector_const_ptr (const gsl_vector *v,
          size_t i)

     These functions return a pointer to the *note i: 36b.-th element of
     a vector *note v: 36b.  If *note i: 36b. lies outside the allowed
     range of 0 to ‘size - 1’ then the error handler is invoked and a
     null pointer is returned.  Inline versions of these functions are
     used when ‘HAVE_INLINE’ is defined.

   ---------- Footnotes ----------

   (1) (1) Range checking is available in the GNU C Compiler
bounds-checking extension, but it is not part of the default
installation of GCC. Memory accesses can also be checked with Valgrind
or the ‘gcc -fmudflap’ memory protection option.


File: gsl-ref.info,  Node: Initializing vector elements,  Next: Reading and writing vectors,  Prev: Accessing vector elements,  Up: Vectors

8.3.3 Initializing vector elements
----------------------------------

 -- Function: void gsl_vector_set_all (gsl_vector *v, double x)

     This function sets all the elements of the vector *note v: 36d. to
     the value *note x: 36d.

 -- Function: void gsl_vector_set_zero (gsl_vector *v)

     This function sets all the elements of the vector *note v: 36e. to
     zero.

 -- Function: int gsl_vector_set_basis (gsl_vector *v, size_t i)

     This function makes a basis vector by setting all the elements of
     the vector *note v: 36f. to zero except for the *note i: 36f.-th
     element which is set to one.


File: gsl-ref.info,  Node: Reading and writing vectors,  Next: Vector views,  Prev: Initializing vector elements,  Up: Vectors

8.3.4 Reading and writing vectors
---------------------------------

The library provides functions for reading and writing vectors to a file
as binary data or formatted text.

 -- Function: int gsl_vector_fwrite (FILE *stream, const gsl_vector *v)

     This function writes the elements of the vector *note v: 371. to
     the stream *note stream: 371. in binary format.  The return value
     is 0 for success and ‘GSL_EFAILED’ if there was a problem writing
     to the file.  Since the data is written in the native binary format
     it may not be portable between different architectures.

 -- Function: int gsl_vector_fread (FILE *stream, gsl_vector *v)

     This function reads into the vector *note v: 372. from the open
     stream *note stream: 372. in binary format.  The vector *note v:
     372. must be preallocated with the correct length since the
     function uses the size of *note v: 372. to determine how many bytes
     to read.  The return value is 0 for success and ‘GSL_EFAILED’ if
     there was a problem reading from the file.  The data is assumed to
     have been written in the native binary format on the same
     architecture.

 -- Function: int gsl_vector_fprintf (FILE *stream, const gsl_vector *v,
          const char *format)

     This function writes the elements of the vector *note v: 373.
     line-by-line to the stream *note stream: 373. using the format
     specifier *note format: 373, which should be one of the ‘%g’, ‘%e’
     or ‘%f’ formats for floating point numbers and ‘%d’ for integers.
     The function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem writing to the file.

 -- Function: int gsl_vector_fscanf (FILE *stream, gsl_vector *v)

     This function reads formatted data from the stream *note stream:
     374. into the vector *note v: 374.  The vector *note v: 374. must
     be preallocated with the correct length since the function uses the
     size of *note v: 374. to determine how many numbers to read.  The
     function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem reading from the file.


File: gsl-ref.info,  Node: Vector views,  Next: Copying vectors,  Prev: Reading and writing vectors,  Up: Vectors

8.3.5 Vector views
------------------

In addition to creating vectors from slices of blocks it is also
possible to slice vectors and create vector views.  For example, a
subvector of another vector can be described with a view, or two views
can be made which provide access to the even and odd elements of a
vector.

 -- Type: gsl_vector_view
 -- Type: gsl_vector_const_view

     A vector view is a temporary object, stored on the stack, which can
     be used to operate on a subset of vector elements.  Vector views
     can be defined for both constant and non-constant vectors, using
     separate types that preserve constness.  A vector view has the type
     *note gsl_vector_view: 376. and a constant vector view has the type
     *note gsl_vector_const_view: 377.  In both cases the elements of
     the view can be accessed as a *note gsl_vector: 35f. using the
     ‘vector’ component of the view object.  A pointer to a vector of
     type ‘gsl_vector *’ or ‘const gsl_vector *’ can be obtained by
     taking the address of this component with the ‘&’ operator.

     When using this pointer it is important to ensure that the view
     itself remains in scope—the simplest way to do so is by always
     writing the pointer as ‘&view.vector’, and never storing this value
     in another variable.

 -- Function: *note gsl_vector_view: 376. gsl_vector_subvector
          (gsl_vector *v, size_t offset, size_t n)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_vector_const_subvector (const gsl_vector *v, size_t
          offset, size_t n)

     These functions return a vector view of a subvector of another
     vector *note v: 379.  The start of the new vector is offset by
     *note offset: 379. elements from the start of the original vector.
     The new vector has *note n: 379. elements.  Mathematically, the
     ‘i’-th element of the new vector ‘v'’ is given by:

          v'(i) = v->data[(offset + i)*v->stride]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     The ‘data’ pointer of the returned vector struct is set to null if
     the combined parameters (*note offset: 379, *note n: 379.) overrun
     the end of the original vector.

     The new vector is only a view of the block underlying the original
     vector, *note v: 379.  The block containing the elements of *note
     v: 379. is not owned by the new vector.  When the view goes out of
     scope the original vector *note v: 379. and its block will continue
     to exist.  The original memory can only be deallocated by freeing
     the original vector.  Of course, the original vector should not be
     deallocated while the view is still in use.

     The function *note gsl_vector_const_subvector(): 379. is equivalent
     to *note gsl_vector_subvector(): 378. but can be used for vectors
     which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376.
          gsl_vector_subvector_with_stride (gsl_vector *v, size_t
          offset, size_t stride, size_t n)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_vector_const_subvector_with_stride (const gsl_vector *v,
          size_t offset, size_t stride, size_t n)

     These functions return a vector view of a subvector of another
     vector *note v: 37b. with an additional stride argument.  The
     subvector is formed in the same way as for *note
     gsl_vector_subvector(): 378. but the new vector has *note n: 37b.
     elements with a step-size of *note stride: 37b. from one element to
     the next in the original vector.  Mathematically, the ‘i’-th
     element of the new vector ‘v'’ is given by:

          v'(i) = v->data[(offset + i*stride)*v->stride]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     Note that subvector views give direct access to the underlying
     elements of the original vector.  For example, the following code
     will zero the even elements of the vector *note v: 37b. of length
     ‘n’, while leaving the odd elements untouched:

          gsl_vector_view v_even = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
          gsl_vector_set_zero (&v_even.vector);

     A vector view can be passed to any subroutine which takes a vector
     argument just as a directly allocated vector would be, using
     ‘&view.vector’.  For example, the following code computes the norm
     of the odd elements of *note v: 37b. using the BLAS routine
     ‘dnrm2’:

          gsl_vector_view v_odd = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
          double r = gsl_blas_dnrm2 (&v_odd.vector);

     The function *note gsl_vector_const_subvector_with_stride(): 37b.
     is equivalent to *note gsl_vector_subvector_with_stride(): 37a. but
     can be used for vectors which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_vector_complex_real
          (gsl_vector_complex *v)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_vector_complex_const_real (const gsl_vector_complex *v)

     These functions return a vector view of the real parts of the
     complex vector *note v: 37d.

     The function *note gsl_vector_complex_const_real(): 37d. is
     equivalent to *note gsl_vector_complex_real(): 37c. but can be used
     for vectors which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_vector_complex_imag
          (gsl_vector_complex *v)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_vector_complex_const_imag (const gsl_vector_complex *v)

     These functions return a vector view of the imaginary parts of the
     complex vector *note v: 37f.

     The function *note gsl_vector_complex_const_imag(): 37f. is
     equivalent to *note gsl_vector_complex_imag(): 37e. but can be used
     for vectors which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_vector_view_array (double
          *base, size_t n)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_vector_const_view_array (const double *base, size_t n)

     These functions return a vector view of an array.  The start of the
     new vector is given by *note base: 381. and has *note n: 381.
     elements.  Mathematically, the ‘i’-th element of the new vector
     ‘v'’ is given by:

          v'(i) = base[i]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     The array containing the elements of ‘v’ is not owned by the new
     vector view.  When the view goes out of scope the original array
     will continue to exist.  The original memory can only be
     deallocated by freeing the original pointer *note base: 381.  Of
     course, the original array should not be deallocated while the view
     is still in use.

     The function *note gsl_vector_const_view_array(): 381. is
     equivalent to *note gsl_vector_view_array(): 380. but can be used
     for arrays which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376.
          gsl_vector_view_array_with_stride (double *base, size_t
          stride, size_t n)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_vector_const_view_array_with_stride (const double *base,
          size_t stride, size_t n)

     These functions return a vector view of an array *note base: 383.
     with an additional stride argument.  The subvector is formed in the
     same way as for *note gsl_vector_view_array(): 380. but the new
     vector has *note n: 383. elements with a step-size of *note stride:
     383. from one element to the next in the original array.
     Mathematically, the ‘i’-th element of the new vector ‘v'’ is given
     by:

          v'(i) = base[i*stride]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     Note that the view gives direct access to the underlying elements
     of the original array.  A vector view can be passed to any
     subroutine which takes a vector argument just as a directly
     allocated vector would be, using ‘&view.vector’.

     The function *note gsl_vector_const_view_array_with_stride(): 383.
     is equivalent to *note gsl_vector_view_array_with_stride(): 382.
     but can be used for arrays which are declared ‘const’.


File: gsl-ref.info,  Node: Copying vectors,  Next: Exchanging elements,  Prev: Vector views,  Up: Vectors

8.3.6 Copying vectors
---------------------

Common operations on vectors such as addition and multiplication are
available in the BLAS part of the library (see *note BLAS Support: 11.).
However, it is useful to have a small number of utility functions which
do not require the full BLAS code.  The following functions fall into
this category.

 -- Function: int gsl_vector_memcpy (gsl_vector *dest, const gsl_vector
          *src)

     This function copies the elements of the vector *note src: 385.
     into the vector *note dest: 385.  The two vectors must have the
     same length.

 -- Function: int gsl_vector_swap (gsl_vector *v, gsl_vector *w)

     This function exchanges the elements of the vectors *note v: 386.
     and *note w: 386. by copying.  The two vectors must have the same
     length.


File: gsl-ref.info,  Node: Exchanging elements,  Next: Vector operations,  Prev: Copying vectors,  Up: Vectors

8.3.7 Exchanging elements
-------------------------

The following functions can be used to exchange, or permute, the
elements of a vector.

 -- Function: int gsl_vector_swap_elements (gsl_vector *v, size_t i,
          size_t j)

     This function exchanges the *note i: 388.-th and *note j: 388.-th
     elements of the vector *note v: 388. in-place.

 -- Function: int gsl_vector_reverse (gsl_vector *v)

     This function reverses the order of the elements of the vector
     *note v: 389.


File: gsl-ref.info,  Node: Vector operations,  Next: Finding maximum and minimum elements of vectors,  Prev: Exchanging elements,  Up: Vectors

8.3.8 Vector operations
-----------------------

 -- Function: int gsl_vector_add (gsl_vector *a, const gsl_vector *b)

     This function adds the elements of vector *note b: 38b. to the
     elements of vector *note a: 38b.  The result a_i \leftarrow a_i +
     b_i is stored in *note a: 38b. and *note b: 38b. remains unchanged.
     The two vectors must have the same length.

 -- Function: int gsl_vector_sub (gsl_vector *a, const gsl_vector *b)

     This function subtracts the elements of vector *note b: 38c. from
     the elements of vector *note a: 38c.  The result a_i \leftarrow a_i
     - b_i is stored in *note a: 38c. and *note b: 38c. remains
     unchanged.  The two vectors must have the same length.

 -- Function: int gsl_vector_mul (gsl_vector *a, const gsl_vector *b)

     This function multiplies the elements of vector *note a: 38d. by
     the elements of vector *note b: 38d.  The result a_i \leftarrow a_i
     * b_i is stored in *note a: 38d. and *note b: 38d. remains
     unchanged.  The two vectors must have the same length.

 -- Function: int gsl_vector_div (gsl_vector *a, const gsl_vector *b)

     This function divides the elements of vector *note a: 38e. by the
     elements of vector *note b: 38e.  The result a_i \leftarrow a_i /
     b_i is stored in *note a: 38e. and *note b: 38e. remains unchanged.
     The two vectors must have the same length.

 -- Function: int gsl_vector_scale (gsl_vector *a, const double x)

     This function multiplies the elements of vector *note a: 38f. by
     the constant factor *note x: 38f.  The result a_i \leftarrow x a_i
     is stored in *note a: 38f.

 -- Function: int gsl_vector_add_constant (gsl_vector *a, const double
          x)

     This function adds the constant value *note x: 390. to the elements
     of the vector *note a: 390.  The result a_i \leftarrow a_i + x is
     stored in *note a: 390.

 -- Function: double gsl_vector_sum (const gsl_vector *a)

     This function returns the sum of the elements of *note a: 391,
     defined as \sum_{i=1}^n a_i

 -- Function: int gsl_vector_axpby (const double alpha, const gsl_vector
          *x, const double beta, gsl_vector *y)

     This function performs the operation y \leftarrow \alpha x + \beta
     y.  The vectors *note x: 392. and *note y: 392. must have the same
     length.


File: gsl-ref.info,  Node: Finding maximum and minimum elements of vectors,  Next: Vector properties,  Prev: Vector operations,  Up: Vectors

8.3.9 Finding maximum and minimum elements of vectors
-----------------------------------------------------

The following operations are only defined for real vectors.

 -- Function: double gsl_vector_max (const gsl_vector *v)

     This function returns the maximum value in the vector *note v: 394.

 -- Function: double gsl_vector_min (const gsl_vector *v)

     This function returns the minimum value in the vector *note v: 395.

 -- Function: void gsl_vector_minmax (const gsl_vector *v, double
          *min_out, double *max_out)

     This function returns the minimum and maximum values in the vector
     *note v: 396, storing them in *note min_out: 396. and *note
     max_out: 396.

 -- Function: size_t gsl_vector_max_index (const gsl_vector *v)

     This function returns the index of the maximum value in the vector
     *note v: 397.  When there are several equal maximum elements then
     the lowest index is returned.

 -- Function: size_t gsl_vector_min_index (const gsl_vector *v)

     This function returns the index of the minimum value in the vector
     *note v: 398.  When there are several equal minimum elements then
     the lowest index is returned.

 -- Function: void gsl_vector_minmax_index (const gsl_vector *v, size_t
          *imin, size_t *imax)

     This function returns the indices of the minimum and maximum values
     in the vector *note v: 399, storing them in *note imin: 399. and
     *note imax: 399.  When there are several equal minimum or maximum
     elements then the lowest indices are returned.


File: gsl-ref.info,  Node: Vector properties,  Next: Example programs for vectors,  Prev: Finding maximum and minimum elements of vectors,  Up: Vectors

8.3.10 Vector properties
------------------------

The following functions are defined for real and complex vectors.  For
complex vectors both the real and imaginary parts must satisfy the
conditions.

 -- Function: int gsl_vector_isnull (const gsl_vector *v)
 -- Function: int gsl_vector_ispos (const gsl_vector *v)
 -- Function: int gsl_vector_isneg (const gsl_vector *v)
 -- Function: int gsl_vector_isnonneg (const gsl_vector *v)

     These functions return 1 if all the elements of the vector *note v:
     39e. are zero, strictly positive, strictly negative, or
     non-negative respectively, and 0 otherwise.

 -- Function: int gsl_vector_equal (const gsl_vector *u, const
          gsl_vector *v)

     This function returns 1 if the vectors *note u: 39f. and *note v:
     39f. are equal (by comparison of element values) and 0 otherwise.


File: gsl-ref.info,  Node: Example programs for vectors,  Prev: Vector properties,  Up: Vectors

8.3.11 Example programs for vectors
-----------------------------------

This program shows how to allocate, initialize and read from a vector
using the functions *note gsl_vector_alloc(): 361, *note
gsl_vector_set(): 366. and *note gsl_vector_get(): 365.

     #include <stdio.h>
     #include <gsl/gsl_vector.h>

     int
     main (void)
     {
       int i;
       gsl_vector * v = gsl_vector_alloc (3);

       for (i = 0; i < 3; i++)
         {
           gsl_vector_set (v, i, 1.23 + i);
         }

       for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
         {
           printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
         }

       gsl_vector_free (v);
       return 0;
     }

Here is the output from the program.  The final loop attempts to read
outside the range of the vector ‘v’, and the error is trapped by the
range-checking code in *note gsl_vector_get(): 365.

     $ ./a.out
     v_0 = 1.23
     v_1 = 2.23
     v_2 = 3.23
     gsl: vector_source.c:12: ERROR: index out of range
     Default GSL error handler invoked.
     Aborted (core dumped)

The next program shows how to write a vector to a file.

     #include <stdio.h>
     #include <gsl/gsl_vector.h>

     int
     main (void)
     {
       int i;
       gsl_vector * v = gsl_vector_alloc (100);

       for (i = 0; i < 100; i++)
         {
           gsl_vector_set (v, i, 1.23 + i);
         }

       {
          FILE * f = fopen ("test.dat", "w");
          gsl_vector_fprintf (f, v, "%.5g");
          fclose (f);
       }

       gsl_vector_free (v);
       return 0;
     }

After running this program the file ‘test.dat’ should contain the
elements of ‘v’, written using the format specifier ‘%.5g’.  The vector
could then be read back in using the function ‘gsl_vector_fscanf (f, v)’
as follows:

     #include <stdio.h>
     #include <gsl/gsl_vector.h>

     int
     main (void)
     {
       int i;
       gsl_vector * v = gsl_vector_alloc (10);

       {
          FILE * f = fopen ("test.dat", "r");
          gsl_vector_fscanf (f, v);
          fclose (f);
       }

       for (i = 0; i < 10; i++)
         {
           printf ("%g\n", gsl_vector_get(v, i));
         }

       gsl_vector_free (v);
       return 0;
     }


File: gsl-ref.info,  Node: Matrices,  Prev: Vectors,  Up: Vectors and Matrices

8.4 Matrices
============

Matrices are defined by a *note gsl_matrix: 3a2. structure which
describes a generalized slice of a block.  Like a vector it represents a
set of elements in an area of memory, but uses two indices instead of
one.

 -- Type: gsl_matrix

     The *note gsl_matrix: 3a2. structure contains six components, the
     two dimensions of the matrix, a physical dimension, a pointer to
     the memory where the elements of the matrix are stored, ‘data’, a
     pointer to the block owned by the matrix ‘block’, if any, and an
     ownership flag, ‘owner’.  The physical dimension determines the
     memory layout and can differ from the matrix dimension to allow the
     use of submatrices.  The *note gsl_matrix: 3a2. structure is very
     simple and looks like this:

          typedef struct
          {
            size_t size1;
            size_t size2;
            size_t tda;
            double * data;
            gsl_block * block;
            int owner;
          } gsl_matrix;

Matrices are stored in row-major order, meaning that each row of
elements forms a contiguous block in memory.  This is the standard
“C-language ordering” of two-dimensional arrays.  Note that Fortran
stores arrays in column-major order.  The number of rows is ‘size1’.
The range of valid row indices runs from 0 to ‘size1 - 1’.  Similarly
‘size2’ is the number of columns.  The range of valid column indices
runs from 0 to ‘size2 - 1’.  The physical row dimension ‘tda’, or
`trailing dimension', specifies the size of a row of the matrix as laid
out in memory.

For example, in the following matrix ‘size1’ is 3, ‘size2’ is 4, and
‘tda’ is 8.  The physical memory layout of the matrix begins in the top
left hand-corner and proceeds from left to right along each row in turn.

     00 01 02 03 XX XX XX XX
     10 11 12 13 XX XX XX XX
     20 21 22 23 XX XX XX XX

Each unused memory location is represented by “‘XX’”.  The pointer
‘data’ gives the location of the first element of the matrix in memory.
The pointer ‘block’ stores the location of the memory block in which the
elements of the matrix are located (if any).  If the matrix owns this
block then the ‘owner’ field is set to one and the block will be
deallocated when the matrix is freed.  If the matrix is only a slice of
a block owned by another object then the ‘owner’ field is zero and any
underlying block will not be freed.

The functions for allocating and accessing matrices are defined in
‘gsl_matrix.h’.

* Menu:

* Matrix allocation::
* Accessing matrix elements::
* Initializing matrix elements::
* Reading and writing matrices::
* Matrix views::
* Creating row and column views::
* Copying matrices::
* Copying rows and columns::
* Exchanging rows and columns::
* Matrix operations::
* Finding maximum and minimum elements of matrices::
* Matrix properties::
* Example programs for matrices::
* References and Further Reading: References and Further Reading<4>.


File: gsl-ref.info,  Node: Matrix allocation,  Next: Accessing matrix elements,  Up: Matrices

8.4.1 Matrix allocation
-----------------------

The functions for allocating memory to a matrix follow the style of
‘malloc’ and ‘free’.  They also perform their own error checking.  If
there is insufficient memory available to allocate a matrix then the
functions call the GSL error handler (with an error number of *note
GSL_ENOMEM: 2a.) in addition to returning a null pointer.  Thus if you
use the library error handler to abort your program then it isn’t
necessary to check every ‘alloc’.

 -- Function: *note gsl_matrix: 3a2. *gsl_matrix_alloc (size_t n1,
          size_t n2)

     This function creates a matrix of size *note n1: 3a4. rows by *note
     n2: 3a4. columns, returning a pointer to a newly initialized matrix
     struct.  A new block is allocated for the elements of the matrix,
     and stored in the ‘block’ component of the matrix struct.  The
     block is “owned” by the matrix, and will be deallocated when the
     matrix is deallocated.  Requesting zero for *note n1: 3a4. or *note
     n2: 3a4. is valid and returns a non-null result.

 -- Function: *note gsl_matrix: 3a2. *gsl_matrix_calloc (size_t n1,
          size_t n2)

     This function allocates memory for a matrix of size *note n1: 3a5.
     rows by *note n2: 3a5. columns and initializes all the elements of
     the matrix to zero.

 -- Function: void gsl_matrix_free (gsl_matrix *m)

     This function frees a previously allocated matrix *note m: 3a6.  If
     the matrix was created using *note gsl_matrix_alloc(): 3a4. then
     the block underlying the matrix will also be deallocated.  If the
     matrix has been created from another object then the memory is
     still owned by that object and will not be deallocated.


File: gsl-ref.info,  Node: Accessing matrix elements,  Next: Initializing matrix elements,  Prev: Matrix allocation,  Up: Matrices

8.4.2 Accessing matrix elements
-------------------------------

The functions for accessing the elements of a matrix use the same range
checking system as vectors.  You can turn off range checking by
recompiling your program with the preprocessor definition *note
GSL_RANGE_CHECK_OFF: 367.

The elements of the matrix are stored in “C-order”, where the second
index moves continuously through memory.  More precisely, the element
accessed by the function ‘gsl_matrix_get(m,i,j)’ and
‘gsl_matrix_set(m,i,j,x)’ is:

     m->data[i * m->tda + j]

where ‘tda’ is the physical row-length of the matrix.

 -- Function: double gsl_matrix_get (const gsl_matrix *m, const size_t
          i, const size_t j)

     This function returns the (i,j)-th element of a matrix *note m:
     3a8.  If *note i: 3a8. or *note j: 3a8. lie outside the allowed
     range of 0 to ‘n1 - 1’ and 0 to ‘n2 - 1’ then the error handler is
     invoked and 0 is returned.  An inline version of this function is
     used when ‘HAVE_INLINE’ is defined.

 -- Function: void gsl_matrix_set (gsl_matrix *m, const size_t i, const
          size_t j, double x)

     This function sets the value of the (i,j)-th element of a matrix
     *note m: 3a9. to *note x: 3a9.  If *note i: 3a9. or *note j: 3a9.
     lies outside the allowed range of 0 to ‘n1 - 1’ and 0 to ‘n2 - 1’
     then the error handler is invoked.  An inline version of this
     function is used when ‘HAVE_INLINE’ is defined.

 -- Function: double *gsl_matrix_ptr (gsl_matrix *m, size_t i, size_t j)
 -- Function: const double *gsl_matrix_const_ptr (const gsl_matrix *m,
          size_t i, size_t j)

     These functions return a pointer to the (i,j)-th element of a
     matrix *note m: 3ab.  If *note i: 3ab. or *note j: 3ab. lie outside
     the allowed range of 0 to ‘n1 - 1’ and 0 to ‘n2 - 1’ then the error
     handler is invoked and a null pointer is returned.  Inline versions
     of these functions are used when ‘HAVE_INLINE’ is defined.


File: gsl-ref.info,  Node: Initializing matrix elements,  Next: Reading and writing matrices,  Prev: Accessing matrix elements,  Up: Matrices

8.4.3 Initializing matrix elements
----------------------------------

 -- Function: void gsl_matrix_set_all (gsl_matrix *m, double x)

     This function sets all the elements of the matrix *note m: 3ad. to
     the value *note x: 3ad.

 -- Function: void gsl_matrix_set_zero (gsl_matrix *m)

     This function sets all the elements of the matrix *note m: 3ae. to
     zero.

 -- Function: void gsl_matrix_set_identity (gsl_matrix *m)

     This function sets the elements of the matrix *note m: 3af. to the
     corresponding elements of the identity matrix, m(i,j) =
     \delta(i,j), i.e.  a unit diagonal with all off-diagonal elements
     zero.  This applies to both square and rectangular matrices.


File: gsl-ref.info,  Node: Reading and writing matrices,  Next: Matrix views,  Prev: Initializing matrix elements,  Up: Matrices

8.4.4 Reading and writing matrices
----------------------------------

The library provides functions for reading and writing matrices to a
file as binary data or formatted text.

 -- Function: int gsl_matrix_fwrite (FILE *stream, const gsl_matrix *m)

     This function writes the elements of the matrix *note m: 3b1. to
     the stream *note stream: 3b1. in binary format.  The return value
     is 0 for success and ‘GSL_EFAILED’ if there was a problem writing
     to the file.  Since the data is written in the native binary format
     it may not be portable between different architectures.

 -- Function: int gsl_matrix_fread (FILE *stream, gsl_matrix *m)

     This function reads into the matrix *note m: 3b2. from the open
     stream *note stream: 3b2. in binary format.  The matrix *note m:
     3b2. must be preallocated with the correct dimensions since the
     function uses the size of *note m: 3b2. to determine how many bytes
     to read.  The return value is 0 for success and ‘GSL_EFAILED’ if
     there was a problem reading from the file.  The data is assumed to
     have been written in the native binary format on the same
     architecture.

 -- Function: int gsl_matrix_fprintf (FILE *stream, const gsl_matrix *m,
          const char *format)

     This function writes the elements of the matrix *note m: 3b3.
     line-by-line to the stream *note stream: 3b3. using the format
     specifier *note format: 3b3, which should be one of the ‘%g’, ‘%e’
     or ‘%f’ formats for floating point numbers and ‘%d’ for integers.
     The function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem writing to the file.

 -- Function: int gsl_matrix_fscanf (FILE *stream, gsl_matrix *m)

     This function reads formatted data from the stream *note stream:
     3b4. into the matrix *note m: 3b4.  The matrix *note m: 3b4. must
     be preallocated with the correct dimensions since the function uses
     the size of *note m: 3b4. to determine how many numbers to read.
     The function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem reading from the file.


File: gsl-ref.info,  Node: Matrix views,  Next: Creating row and column views,  Prev: Reading and writing matrices,  Up: Matrices

8.4.5 Matrix views
------------------

 -- Type: gsl_matrix_view
 -- Type: gsl_matrix_const_view

     A matrix view is a temporary object, stored on the stack, which can
     be used to operate on a subset of matrix elements.  Matrix views
     can be defined for both constant and non-constant matrices using
     separate types that preserve constness.  A matrix view has the type
     *note gsl_matrix_view: 3b6. and a constant matrix view has the type
     *note gsl_matrix_const_view: 3b7.  In both cases the elements of
     the view can by accessed using the ‘matrix’ component of the view
     object.  A pointer ‘gsl_matrix *’ or ‘const gsl_matrix *’ can be
     obtained by taking the address of the ‘matrix’ component with the
     ‘&’ operator.  In addition to matrix views it is also possible to
     create vector views of a matrix, such as row or column views.

 -- Function: *note gsl_matrix_view: 3b6. gsl_matrix_submatrix
          (gsl_matrix *m, size_t k1, size_t k2, size_t n1, size_t n2)
 -- Function: *note gsl_matrix_const_view: 3b7.
          gsl_matrix_const_submatrix (const gsl_matrix *m, size_t k1,
          size_t k2, size_t n1, size_t n2)

     These functions return a matrix view of a submatrix of the matrix
     *note m: 3b9.  The upper-left element of the submatrix is the
     element (*note k1: 3b9, *note k2: 3b9.) of the original matrix.
     The submatrix has *note n1: 3b9. rows and *note n2: 3b9. columns.
     The physical number of columns in memory given by ‘tda’ is
     unchanged.  Mathematically, the (i,j)-th element of the new matrix
     is given by:

          m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The ‘data’ pointer of the returned matrix struct is set to null if
     the combined parameters (‘i’, ‘j’, *note n1: 3b9, *note n2: 3b9,
     ‘tda’) overrun the ends of the original matrix.

     The new matrix view is only a view of the block underlying the
     existing matrix, *note m: 3b9.  The block containing the elements
     of *note m: 3b9. is not owned by the new matrix view.  When the
     view goes out of scope the original matrix *note m: 3b9. and its
     block will continue to exist.  The original memory can only be
     deallocated by freeing the original matrix.  Of course, the
     original matrix should not be deallocated while the view is still
     in use.

     The function *note gsl_matrix_const_submatrix(): 3b9. is equivalent
     to *note gsl_matrix_submatrix(): 3b8. but can be used for matrices
     which are declared ‘const’.

 -- Function: *note gsl_matrix_view: 3b6. gsl_matrix_view_array (double
          *base, size_t n1, size_t n2)
 -- Function: *note gsl_matrix_const_view: 3b7.
          gsl_matrix_const_view_array (const double *base, size_t n1,
          size_t n2)

     These functions return a matrix view of the array *note base: 3bb.
     The matrix has *note n1: 3bb. rows and *note n2: 3bb. columns.  The
     physical number of columns in memory is also given by *note n2:
     3bb.  Mathematically, the (i,j)-th element of the new matrix is
     given by:

          m'(i,j) = base[i*n2 + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the array *note base: 3bb.  When
     the view goes out of scope the original array *note base: 3bb. will
     continue to exist.  The original memory can only be deallocated by
     freeing the original array.  Of course, the original array should
     not be deallocated while the view is still in use.

     The function *note gsl_matrix_const_view_array(): 3bb. is
     equivalent to *note gsl_matrix_view_array(): 3ba. but can be used
     for matrices which are declared ‘const’.

 -- Function: *note gsl_matrix_view: 3b6. gsl_matrix_view_array_with_tda
          (double *base, size_t n1, size_t n2, size_t tda)
 -- Function: *note gsl_matrix_const_view: 3b7.
          gsl_matrix_const_view_array_with_tda (const double *base,
          size_t n1, size_t n2, size_t tda)

     These functions return a matrix view of the array *note base: 3bd.
     with a physical number of columns *note tda: 3bd. which may differ
     from the corresponding dimension of the matrix.  The matrix has
     *note n1: 3bd. rows and *note n2: 3bd. columns, and the physical
     number of columns in memory is given by *note tda: 3bd.
     Mathematically, the (i,j)-th element of the new matrix is given by:

          m'(i,j) = base[i*tda + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the array *note base: 3bd.  When
     the view goes out of scope the original array *note base: 3bd. will
     continue to exist.  The original memory can only be deallocated by
     freeing the original array.  Of course, the original array should
     not be deallocated while the view is still in use.

     The function *note gsl_matrix_const_view_array_with_tda(): 3bd. is
     equivalent to *note gsl_matrix_view_array_with_tda(): 3bc. but can
     be used for matrices which are declared ‘const’.

 -- Function: *note gsl_matrix_view: 3b6. gsl_matrix_view_vector
          (gsl_vector *v, size_t n1, size_t n2)
 -- Function: *note gsl_matrix_const_view: 3b7.
          gsl_matrix_const_view_vector (const gsl_vector *v, size_t n1,
          size_t n2)

     These functions return a matrix view of the vector *note v: 3bf.
     The matrix has *note n1: 3bf. rows and *note n2: 3bf. columns.  The
     vector must have unit stride.  The physical number of columns in
     memory is also given by *note n2: 3bf.  Mathematically, the
     (i,j)-th element of the new matrix is given by:

          m'(i,j) = v->data[i*n2 + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the vector *note v: 3bf.  When the
     view goes out of scope the original vector *note v: 3bf. will
     continue to exist.  The original memory can only be deallocated by
     freeing the original vector.  Of course, the original vector should
     not be deallocated while the view is still in use.

     The function *note gsl_matrix_const_view_vector(): 3bf. is
     equivalent to *note gsl_matrix_view_vector(): 3be. but can be used
     for matrices which are declared ‘const’.

 -- Function: *note gsl_matrix_view: 3b6.
          gsl_matrix_view_vector_with_tda (gsl_vector *v, size_t n1,
          size_t n2, size_t tda)
 -- Function: *note gsl_matrix_const_view: 3b7.
          gsl_matrix_const_view_vector_with_tda (const gsl_vector *v,
          size_t n1, size_t n2, size_t tda)

     These functions return a matrix view of the vector *note v: 3c1.
     with a physical number of columns *note tda: 3c1. which may differ
     from the corresponding matrix dimension.  The vector must have unit
     stride.  The matrix has *note n1: 3c1. rows and *note n2: 3c1.
     columns, and the physical number of columns in memory is given by
     *note tda: 3c1.  Mathematically, the (i,j)-th element of the new
     matrix is given by:

          m'(i,j) = v->data[i*tda + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the vector *note v: 3c1.  When the
     view goes out of scope the original vector *note v: 3c1. will
     continue to exist.  The original memory can only be deallocated by
     freeing the original vector.  Of course, the original vector should
     not be deallocated while the view is still in use.

     The function *note gsl_matrix_const_view_vector_with_tda(): 3c1. is
     equivalent to *note gsl_matrix_view_vector_with_tda(): 3c0. but can
     be used for matrices which are declared ‘const’.


File: gsl-ref.info,  Node: Creating row and column views,  Next: Copying matrices,  Prev: Matrix views,  Up: Matrices

8.4.6 Creating row and column views
-----------------------------------

In general there are two ways to access an object, by reference or by
copying.  The functions described in this section create vector views
which allow access to a row or column of a matrix by reference.
Modifying elements of the view is equivalent to modifying the matrix,
since both the vector view and the matrix point to the same memory
block.

 -- Function: *note gsl_vector_view: 376. gsl_matrix_row (gsl_matrix *m,
          size_t i)
 -- Function: *note gsl_vector_const_view: 377. gsl_matrix_const_row
          (const gsl_matrix *m, size_t i)

     These functions return a vector view of the *note i: 3c4.-th row of
     the matrix *note m: 3c4.  The ‘data’ pointer of the new vector is
     set to null if *note i: 3c4. is out of range.

     The function *note gsl_matrix_const_row(): 3c4. is equivalent to
     *note gsl_matrix_row(): 3c3. but can be used for matrices which are
     declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_matrix_column (gsl_matrix
          *m, size_t j)
 -- Function: *note gsl_vector_const_view: 377. gsl_matrix_const_column
          (const gsl_matrix *m, size_t j)

     These functions return a vector view of the *note j: 3c6.-th column
     of the matrix *note m: 3c6.  The ‘data’ pointer of the new vector
     is set to null if *note j: 3c6. is out of range.

     The function *note gsl_matrix_const_column(): 3c6. is equivalent to
     *note gsl_matrix_column(): 3c5. but can be used for matrices which
     are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_matrix_subrow (gsl_matrix
          *m, size_t i, size_t offset, size_t n)
 -- Function: *note gsl_vector_const_view: 377. gsl_matrix_const_subrow
          (const gsl_matrix *m, size_t i, size_t offset, size_t n)

     These functions return a vector view of the *note i: 3c8.-th row of
     the matrix *note m: 3c8. beginning at *note offset: 3c8. elements
     past the first column and containing *note n: 3c8. elements.  The
     ‘data’ pointer of the new vector is set to null if *note i: 3c8,
     *note offset: 3c8, or *note n: 3c8. are out of range.

     The function *note gsl_matrix_const_subrow(): 3c8. is equivalent to
     *note gsl_matrix_subrow(): 3c7. but can be used for matrices which
     are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_matrix_subcolumn
          (gsl_matrix *m, size_t j, size_t offset, size_t n)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_matrix_const_subcolumn (const gsl_matrix *m, size_t j,
          size_t offset, size_t n)

     These functions return a vector view of the *note j: 3ca.-th column
     of the matrix *note m: 3ca. beginning at *note offset: 3ca.
     elements past the first row and containing *note n: 3ca. elements.
     The ‘data’ pointer of the new vector is set to null if *note j:
     3ca, *note offset: 3ca, or *note n: 3ca. are out of range.

     The function *note gsl_matrix_const_subcolumn(): 3ca. is equivalent
     to *note gsl_matrix_subcolumn(): 3c9. but can be used for matrices
     which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_matrix_diagonal
          (gsl_matrix *m)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_matrix_const_diagonal (const gsl_matrix *m)

     These functions return a vector view of the diagonal of the matrix
     *note m: 3cc.  The matrix *note m: 3cc. is not required to be
     square.  For a rectangular matrix the length of the diagonal is the
     same as the smaller dimension of the matrix.

     The function *note gsl_matrix_const_diagonal(): 3cc. is equivalent
     to *note gsl_matrix_diagonal(): 3cb. but can be used for matrices
     which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_matrix_subdiagonal
          (gsl_matrix *m, size_t k)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_matrix_const_subdiagonal (const gsl_matrix *m, size_t k)

     These functions return a vector view of the *note k: 3ce.-th
     subdiagonal of the matrix *note m: 3ce.  The matrix *note m: 3ce.
     is not required to be square.  The diagonal of the matrix
     corresponds to k = 0.

     The function *note gsl_matrix_const_subdiagonal(): 3ce. is
     equivalent to *note gsl_matrix_subdiagonal(): 3cd. but can be used
     for matrices which are declared ‘const’.

 -- Function: *note gsl_vector_view: 376. gsl_matrix_superdiagonal
          (gsl_matrix *m, size_t k)
 -- Function: *note gsl_vector_const_view: 377.
          gsl_matrix_const_superdiagonal (const gsl_matrix *m, size_t k)

     These functions return a vector view of the *note k: 3d0.-th
     superdiagonal of the matrix *note m: 3d0.  The matrix *note m: 3d0.
     is not required to be square.  The diagonal of the matrix
     corresponds to k = 0.

     The function *note gsl_matrix_const_superdiagonal(): 3d0. is
     equivalent to *note gsl_matrix_superdiagonal(): 3cf. but can be
     used for matrices which are declared ‘const’.


File: gsl-ref.info,  Node: Copying matrices,  Next: Copying rows and columns,  Prev: Creating row and column views,  Up: Matrices

8.4.7 Copying matrices
----------------------

 -- Function: int gsl_matrix_memcpy (gsl_matrix *dest, const gsl_matrix
          *src)

     This function copies the elements of the matrix *note src: 3d2.
     into the matrix *note dest: 3d2.  The two matrices must have the
     same size.

 -- Function: int gsl_matrix_swap (gsl_matrix *m1, gsl_matrix *m2)

     This function exchanges the elements of the matrices *note m1: 3d3.
     and *note m2: 3d3. by copying.  The two matrices must have the same
     size.


File: gsl-ref.info,  Node: Copying rows and columns,  Next: Exchanging rows and columns,  Prev: Copying matrices,  Up: Matrices

8.4.8 Copying rows and columns
------------------------------

The functions described in this section copy a row or column of a matrix
into a vector.  This allows the elements of the vector and the matrix to
be modified independently.  Note that if the matrix and the vector point
to overlapping regions of memory then the result will be undefined.  The
same effect can be achieved with more generality using *note
gsl_vector_memcpy(): 385. with vector views of rows and columns.

 -- Function: int gsl_matrix_get_row (gsl_vector *v, const gsl_matrix
          *m, size_t i)

     This function copies the elements of the *note i: 3d5.-th row of
     the matrix *note m: 3d5. into the vector *note v: 3d5.  The length
     of the vector must be the same as the length of the row.

 -- Function: int gsl_matrix_get_col (gsl_vector *v, const gsl_matrix
          *m, size_t j)

     This function copies the elements of the *note j: 3d6.-th column of
     the matrix *note m: 3d6. into the vector *note v: 3d6.  The length
     of the vector must be the same as the length of the column.

 -- Function: int gsl_matrix_set_row (gsl_matrix *m, size_t i, const
          gsl_vector *v)

     This function copies the elements of the vector *note v: 3d7. into
     the *note i: 3d7.-th row of the matrix *note m: 3d7.  The length of
     the vector must be the same as the length of the row.

 -- Function: int gsl_matrix_set_col (gsl_matrix *m, size_t j, const
          gsl_vector *v)

     This function copies the elements of the vector *note v: 3d8. into
     the *note j: 3d8.-th column of the matrix *note m: 3d8.  The length
     of the vector must be the same as the length of the column.


File: gsl-ref.info,  Node: Exchanging rows and columns,  Next: Matrix operations,  Prev: Copying rows and columns,  Up: Matrices

8.4.9 Exchanging rows and columns
---------------------------------

The following functions can be used to exchange the rows and columns of
a matrix.

 -- Function: int gsl_matrix_swap_rows (gsl_matrix *m, size_t i, size_t
          j)

     This function exchanges the *note i: 3da.-th and *note j: 3da.-th
     rows of the matrix *note m: 3da. in-place.

 -- Function: int gsl_matrix_swap_columns (gsl_matrix *m, size_t i,
          size_t j)

     This function exchanges the *note i: 3db.-th and *note j: 3db.-th
     columns of the matrix *note m: 3db. in-place.

 -- Function: int gsl_matrix_swap_rowcol (gsl_matrix *m, size_t i,
          size_t j)

     This function exchanges the *note i: 3dc.-th row and *note j:
     3dc.-th column of the matrix *note m: 3dc. in-place.  The matrix
     must be square for this operation to be possible.

 -- Function: int gsl_matrix_transpose_memcpy (gsl_matrix *dest, const
          gsl_matrix *src)

     This function makes the matrix *note dest: 3dd. the transpose of
     the matrix *note src: 3dd. by copying the elements of *note src:
     3dd. into *note dest: 3dd.  This function works for all matrices
     provided that the dimensions of the matrix *note dest: 3dd. match
     the transposed dimensions of the matrix *note src: 3dd.

 -- Function: int gsl_matrix_transpose (gsl_matrix *m)

     This function replaces the matrix *note m: 3de. by its transpose by
     copying the elements of the matrix in-place.  The matrix must be
     square for this operation to be possible.

 -- Function: int gsl_matrix_complex_conjtrans_memcpy (gsl_matrix *dest,
          const gsl_matrix *src)

     This function makes the matrix *note dest: 3df. the conjugate
     transpose of the matrix *note src: 3df. by copying the complex
     conjugate elements of *note src: 3df. into *note dest: 3df.  This
     function works for all complex matrices provided that the
     dimensions of the matrix *note dest: 3df. match the transposed
     dimensions of the matrix *note src: 3df.


File: gsl-ref.info,  Node: Matrix operations,  Next: Finding maximum and minimum elements of matrices,  Prev: Exchanging rows and columns,  Up: Matrices

8.4.10 Matrix operations
------------------------

The following operations are defined for real and complex matrices.

 -- Function: int gsl_matrix_add (gsl_matrix *a, const gsl_matrix *b)

     This function adds the elements of matrix *note b: 3e1. to the
     elements of matrix *note a: 3e1.  The result a(i,j) \leftarrow
     a(i,j) + b(i,j) is stored in *note a: 3e1. and *note b: 3e1.
     remains unchanged.  The two matrices must have the same dimensions.

 -- Function: int gsl_matrix_sub (gsl_matrix *a, const gsl_matrix *b)

     This function subtracts the elements of matrix *note b: 3e2. from
     the elements of matrix *note a: 3e2.  The result a(i,j) \leftarrow
     a(i,j) - b(i,j) is stored in *note a: 3e2. and *note b: 3e2.
     remains unchanged.  The two matrices must have the same dimensions.

 -- Function: int gsl_matrix_mul_elements (gsl_matrix *a, const
          gsl_matrix *b)

     This function multiplies the elements of matrix *note a: 3e3. by
     the elements of matrix *note b: 3e3.  The result a(i,j) \leftarrow
     a(i,j) * b(i,j) is stored in *note a: 3e3. and *note b: 3e3.
     remains unchanged.  The two matrices must have the same dimensions.

 -- Function: int gsl_matrix_div_elements (gsl_matrix *a, const
          gsl_matrix *b)

     This function divides the elements of matrix *note a: 3e4. by the
     elements of matrix *note b: 3e4.  The result a(i,j) \leftarrow
     a(i,j) / b(i,j) is stored in *note a: 3e4. and *note b: 3e4.
     remains unchanged.  The two matrices must have the same dimensions.

 -- Function: int gsl_matrix_scale (gsl_matrix *a, const double x)

     This function multiplies the elements of matrix *note a: 3e5. by
     the constant factor *note x: 3e5.  The result a(i,j) \leftarrow x
     a(i,j) is stored in *note a: 3e5.

 -- Function: int gsl_matrix_scale_columns (gsl_matrix *A, const
          gsl_vector *x)

     This function scales the columns of the M-by-N matrix *note A: 3e6.
     by the elements of the vector *note x: 3e6, of length N. The j-th
     column of *note A: 3e6. is multiplied by x_j.  This is equivalent
     to forming

          A \rightarrow A X

     where X = \textrm{diag}(x).

 -- Function: int gsl_matrix_scale_rows (gsl_matrix *A, const gsl_vector
          *x)

     This function scales the rows of the M-by-N matrix *note A: 3e7. by
     the elements of the vector *note x: 3e7, of length M. The i-th row
     of *note A: 3e7. is multiplied by x_i.  This is equivalent to
     forming

          A \rightarrow X A

     where X = \textrm{diag}(x).

 -- Function: int gsl_matrix_add_constant (gsl_matrix *a, const double
          x)

     This function adds the constant value *note x: 3e8. to the elements
     of the matrix *note a: 3e8.  The result a(i,j) \leftarrow a(i,j) +
     x is stored in *note a: 3e8.


File: gsl-ref.info,  Node: Finding maximum and minimum elements of matrices,  Next: Matrix properties,  Prev: Matrix operations,  Up: Matrices

8.4.11 Finding maximum and minimum elements of matrices
-------------------------------------------------------

The following operations are only defined for real matrices.

 -- Function: double gsl_matrix_max (const gsl_matrix *m)

     This function returns the maximum value in the matrix *note m: 3ea.

 -- Function: double gsl_matrix_min (const gsl_matrix *m)

     This function returns the minimum value in the matrix *note m: 3eb.

 -- Function: void gsl_matrix_minmax (const gsl_matrix *m, double
          *min_out, double *max_out)

     This function returns the minimum and maximum values in the matrix
     *note m: 3ec, storing them in *note min_out: 3ec. and *note
     max_out: 3ec.

 -- Function: void gsl_matrix_max_index (const gsl_matrix *m, size_t
          *imax, size_t *jmax)

     This function returns the indices of the maximum value in the
     matrix *note m: 3ed, storing them in *note imax: 3ed. and *note
     jmax: 3ed.  When there are several equal maximum elements then the
     first element found is returned, searching in row-major order.

 -- Function: void gsl_matrix_min_index (const gsl_matrix *m, size_t
          *imin, size_t *jmin)

     This function returns the indices of the minimum value in the
     matrix *note m: 3ee, storing them in *note imin: 3ee. and *note
     jmin: 3ee.  When there are several equal minimum elements then the
     first element found is returned, searching in row-major order.

 -- Function: void gsl_matrix_minmax_index (const gsl_matrix *m, size_t
          *imin, size_t *jmin, size_t *imax, size_t *jmax)

     This function returns the indices of the minimum and maximum values
     in the matrix *note m: 3ef, storing them in (*note imin: 3ef, *note
     jmin: 3ef.) and (*note imax: 3ef, *note jmax: 3ef.).  When there
     are several equal minimum or maximum elements then the first
     elements found are returned, searching in row-major order.


File: gsl-ref.info,  Node: Matrix properties,  Next: Example programs for matrices,  Prev: Finding maximum and minimum elements of matrices,  Up: Matrices

8.4.12 Matrix properties
------------------------

The following functions are defined for real and complex matrices.  For
complex matrices both the real and imaginary parts must satisfy the
conditions.

 -- Function: int gsl_matrix_isnull (const gsl_matrix *m)
 -- Function: int gsl_matrix_ispos (const gsl_matrix *m)
 -- Function: int gsl_matrix_isneg (const gsl_matrix *m)
 -- Function: int gsl_matrix_isnonneg (const gsl_matrix *m)

     These functions return 1 if all the elements of the matrix *note m:
     3f4. are zero, strictly positive, strictly negative, or
     non-negative respectively, and 0 otherwise.  To test whether a
     matrix is positive-definite, use the *note Cholesky decomposition:
     3f5.

 -- Function: int gsl_matrix_equal (const gsl_matrix *a, const
          gsl_matrix *b)

     This function returns 1 if the matrices *note a: 3f6. and *note b:
     3f6. are equal (by comparison of element values) and 0 otherwise.

 -- Function: double gsl_matrix_norm1 (const gsl_matrix *A)

     This function returns the 1-norm of the m-by-n matrix *note A: 3f7,
     defined as the maximum column sum,

          ||A||_1 = \textrm{max}_{1 \le j \le n} \sum_{i=1}^m |A_{ij}|


File: gsl-ref.info,  Node: Example programs for matrices,  Next: References and Further Reading<4>,  Prev: Matrix properties,  Up: Matrices

8.4.13 Example programs for matrices
------------------------------------

The program below shows how to allocate, initialize and read from a
matrix using the functions *note gsl_matrix_alloc(): 3a4, *note
gsl_matrix_set(): 3a9. and *note gsl_matrix_get(): 3a8.

     #include <stdio.h>
     #include <gsl/gsl_matrix.h>

     int
     main (void)
     {
       int i, j;
       gsl_matrix * m = gsl_matrix_alloc (10, 3);

       for (i = 0; i < 10; i++)
         for (j = 0; j < 3; j++)
           gsl_matrix_set (m, i, j, 0.23 + 100*i + j);

       for (i = 0; i < 100; i++)  /* OUT OF RANGE ERROR */
         for (j = 0; j < 3; j++)
           printf ("m(%d,%d) = %g\n", i, j,
                   gsl_matrix_get (m, i, j));

       gsl_matrix_free (m);

       return 0;
     }

Here is the output from the program.  The final loop attempts to read
outside the range of the matrix ‘m’, and the error is trapped by the
range-checking code in *note gsl_matrix_get(): 3a8.

     $ ./a.out
     m(0,0) = 0.23
     m(0,1) = 1.23
     m(0,2) = 2.23
     m(1,0) = 100.23
     m(1,1) = 101.23
     m(1,2) = 102.23
     ...
     m(9,2) = 902.23
     gsl: matrix_source.c:13: ERROR: first index out of range
     Default GSL error handler invoked.
     Aborted (core dumped)

The next program shows how to write a matrix to a file.

     #include <stdio.h>
     #include <gsl/gsl_matrix.h>

     int
     main (void)
     {
       int i, j, k = 0;
       gsl_matrix * m = gsl_matrix_alloc (100, 100);
       gsl_matrix * a = gsl_matrix_alloc (100, 100);

       for (i = 0; i < 100; i++)
         for (j = 0; j < 100; j++)
           gsl_matrix_set (m, i, j, 0.23 + i + j);

       {
          FILE * f = fopen ("test.dat", "wb");
          gsl_matrix_fwrite (f, m);
          fclose (f);
       }

       {
          FILE * f = fopen ("test.dat", "rb");
          gsl_matrix_fread (f, a);
          fclose (f);
       }

       for (i = 0; i < 100; i++)
         for (j = 0; j < 100; j++)
           {
             double mij = gsl_matrix_get (m, i, j);
             double aij = gsl_matrix_get (a, i, j);
             if (mij != aij) k++;
           }

       gsl_matrix_free (m);
       gsl_matrix_free (a);

       printf ("differences = %d (should be zero)\n", k);
       return (k > 0);
     }

After running this program the file ‘test.dat’ should contain the
elements of ‘m’, written in binary format.  The matrix which is read
back in using the function *note gsl_matrix_fread(): 3b2. should be
exactly equal to the original matrix.

The following program demonstrates the use of vector views.  The program
computes the column norms of a matrix.

     #include <math.h>
     #include <stdio.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>

     int
     main (void)
     {
       size_t i,j;

       gsl_matrix *m = gsl_matrix_alloc (10, 10);

       for (i = 0; i < 10; i++)
         for (j = 0; j < 10; j++)
           gsl_matrix_set (m, i, j, sin (i) + cos (j));

       for (j = 0; j < 10; j++)
         {
           gsl_vector_view column = gsl_matrix_column (m, j);
           double d;

           d = gsl_blas_dnrm2 (&column.vector);

           printf ("matrix column %zu, norm = %g\n", j, d);
         }

       gsl_matrix_free (m);

       return 0;
     }

Here is the output of the program,

     matrix column 0, norm = 4.31461
     matrix column 1, norm = 3.1205
     matrix column 2, norm = 2.19316
     matrix column 3, norm = 3.26114
     matrix column 4, norm = 2.53416
     matrix column 5, norm = 2.57281
     matrix column 6, norm = 4.20469
     matrix column 7, norm = 3.65202
     matrix column 8, norm = 2.08524
     matrix column 9, norm = 3.07313

The results can be confirmed using GNU octave:

     $ octave
     GNU Octave, version 2.0.16.92
     octave> m = sin(0:9)' * ones(1,10)
                    + ones(10,1) * cos(0:9);
     octave> sqrt(sum(m.^2))
     ans =
       4.3146  3.1205  2.1932  3.2611  2.5342  2.5728
       4.2047  3.6520  2.0852  3.0731


File: gsl-ref.info,  Node: References and Further Reading<4>,  Prev: Example programs for matrices,  Up: Matrices

8.4.14 References and Further Reading
-------------------------------------

The block, vector and matrix objects in GSL follow the ‘valarray’ model
of C++.  A description of this model can be found in the following
reference,

   * B. Stroustrup, The C++ Programming Language (3rd Ed), Section 22.4
     Vector Arithmetic.  Addison-Wesley 1997, ISBN 0-201-88954-4.


File: gsl-ref.info,  Node: Permutations,  Next: Combinations,  Prev: Vectors and Matrices,  Up: Top

9 Permutations
**************

This chapter describes functions for creating and manipulating
permutations.  A permutation p is represented by an array of n integers
in the range 0 to n-1, where each value p_i occurs once and only once.
The application of a permutation p to a vector v yields a new vector v'
where v'_i = v_{p_i}.  For example, the array (0,1,3,2) represents a
permutation which exchanges the last two elements of a four element
vector.  The corresponding identity permutation is (0,1,2,3).

Note that the permutations produced by the linear algebra routines
correspond to the exchange of matrix columns, and so should be
considered as applying to row-vectors in the form v' = v P rather than
column-vectors, when permuting the elements of a vector.

The functions described in this chapter are defined in the header file
‘gsl_permutation.h’.

* Menu:

* The Permutation struct::
* Permutation allocation::
* Accessing permutation elements::
* Permutation properties::
* Permutation functions::
* Applying Permutations::
* Reading and writing permutations::
* Permutations in cyclic form::
* Examples: Examples<4>.
* References and Further Reading: References and Further Reading<5>.


File: gsl-ref.info,  Node: The Permutation struct,  Next: Permutation allocation,  Up: Permutations

9.1 The Permutation struct
==========================

 -- Type: gsl_permutation

     A permutation is defined by a structure containing two components,
     the size of the permutation and a pointer to the permutation array.
     The elements of the permutation array are all of type ‘size_t’.
     The *note gsl_permutation: 3fd. structure looks like this:

          typedef struct
          {
            size_t size;
            size_t * data;
          } gsl_permutation;


File: gsl-ref.info,  Node: Permutation allocation,  Next: Accessing permutation elements,  Prev: The Permutation struct,  Up: Permutations

9.2 Permutation allocation
==========================

 -- Function: *note gsl_permutation: 3fd. *gsl_permutation_alloc (size_t
          n)

     This function allocates memory for a new permutation of size *note
     n: 3ff.  The permutation is not initialized and its elements are
     undefined.  Use the function *note gsl_permutation_calloc(): 400.
     if you want to create a permutation which is initialized to the
     identity.  A null pointer is returned if insufficient memory is
     available to create the permutation.

 -- Function: *note gsl_permutation: 3fd. *gsl_permutation_calloc
          (size_t n)

     This function allocates memory for a new permutation of size *note
     n: 400. and initializes it to the identity.  A null pointer is
     returned if insufficient memory is available to create the
     permutation.

 -- Function: void gsl_permutation_init (gsl_permutation *p)

     This function initializes the permutation *note p: 401. to the
     identity, i.e.  (0, 1, 2, \dots, n - 1).

 -- Function: void gsl_permutation_free (gsl_permutation *p)

     This function frees all the memory used by the permutation *note p:
     402.

 -- Function: int gsl_permutation_memcpy (gsl_permutation *dest, const
          gsl_permutation *src)

     This function copies the elements of the permutation *note src:
     403. into the permutation *note dest: 403.  The two permutations
     must have the same size.


File: gsl-ref.info,  Node: Accessing permutation elements,  Next: Permutation properties,  Prev: Permutation allocation,  Up: Permutations

9.3 Accessing permutation elements
==================================

The following functions can be used to access and manipulate
permutations.

 -- Function: size_t gsl_permutation_get (const gsl_permutation *p,
          const size_t i)

     This function returns the value of the *note i: 405.-th element of
     the permutation *note p: 405.  If *note i: 405. lies outside the
     allowed range of 0 to n - 1 then the error handler is invoked and 0
     is returned.  An inline version of this function is used when
     ‘HAVE_INLINE’ is defined.

 -- Function: int gsl_permutation_swap (gsl_permutation *p, const size_t
          i, const size_t j)

     This function exchanges the *note i: 406.-th and *note j: 406.-th
     elements of the permutation *note p: 406.


File: gsl-ref.info,  Node: Permutation properties,  Next: Permutation functions,  Prev: Accessing permutation elements,  Up: Permutations

9.4 Permutation properties
==========================

 -- Function: size_t gsl_permutation_size (const gsl_permutation *p)

     This function returns the size of the permutation *note p: 408.

 -- Function: size_t *gsl_permutation_data (const gsl_permutation *p)

     This function returns a pointer to the array of elements in the
     permutation *note p: 409.

 -- Function: int gsl_permutation_valid (const gsl_permutation *p)

     This function checks that the permutation *note p: 40a. is valid.
     The ‘n’ elements should contain each of the numbers 0 to ‘n - 1’
     once and only once.


File: gsl-ref.info,  Node: Permutation functions,  Next: Applying Permutations,  Prev: Permutation properties,  Up: Permutations

9.5 Permutation functions
=========================

 -- Function: void gsl_permutation_reverse (gsl_permutation *p)

     This function reverses the elements of the permutation *note p:
     40c.

 -- Function: int gsl_permutation_inverse (gsl_permutation *inv, const
          gsl_permutation *p)

     This function computes the inverse of the permutation *note p: 40d,
     storing the result in *note inv: 40d.

 -- Function: int gsl_permutation_next (gsl_permutation *p)

     This function advances the permutation *note p: 40e. to the next
     permutation in lexicographic order and returns ‘GSL_SUCCESS’.  If
     no further permutations are available it returns ‘GSL_FAILURE’ and
     leaves *note p: 40e. unmodified.  Starting with the identity
     permutation and repeatedly applying this function will iterate
     through all possible permutations of a given order.

 -- Function: int gsl_permutation_prev (gsl_permutation *p)

     This function steps backwards from the permutation *note p: 40f. to
     the previous permutation in lexicographic order, returning
     ‘GSL_SUCCESS’.  If no previous permutation is available it returns
     ‘GSL_FAILURE’ and leaves *note p: 40f. unmodified.


File: gsl-ref.info,  Node: Applying Permutations,  Next: Reading and writing permutations,  Prev: Permutation functions,  Up: Permutations

9.6 Applying Permutations
=========================

The following functions are defined in the header files ‘gsl_permute.h’
and ‘gsl_permute_vector.h’.

 -- Function: int gsl_permute (const size_t *p, double *data, size_t
          stride, size_t n)

     This function applies the permutation *note p: 411. to the array
     *note data: 411. of size *note n: 411. with stride *note stride:
     411.

 -- Function: int gsl_permute_inverse (const size_t *p, double *data,
          size_t stride, size_t n)

     This function applies the inverse of the permutation *note p: 412.
     to the array *note data: 412. of size *note n: 412. with stride
     *note stride: 412.

 -- Function: int gsl_permute_vector (const gsl_permutation *p,
          gsl_vector *v)

     This function applies the permutation *note p: 413. to the elements
     of the vector *note v: 413, considered as a row-vector acted on by
     a permutation matrix from the right, v' = v P. The j-th column of
     the permutation matrix P is given by the p_j-th column of the
     identity matrix.  The permutation *note p: 413. and the vector
     *note v: 413. must have the same length.

 -- Function: int gsl_permute_vector_inverse (const gsl_permutation *p,
          gsl_vector *v)

     This function applies the inverse of the permutation *note p: 414.
     to the elements of the vector *note v: 414, considered as a
     row-vector acted on by an inverse permutation matrix from the
     right, v' = v P^T. Note that for permutation matrices the inverse
     is the same as the transpose.  The j-th column of the permutation
     matrix P is given by the p_j-th column of the identity matrix.  The
     permutation *note p: 414. and the vector *note v: 414. must have
     the same length.

 -- Function: int gsl_permute_matrix (const gsl_permutation *p,
          gsl_matrix *A)

     This function applies the permutation *note p: 415. to the matrix
     *note A: 415. from the right, A' = A P. The j-th column of the
     permutation matrix P is given by the p_j-th column of the identity
     matrix.  This effectively permutes the columns of *note A: 415.
     according to the permutation *note p: 415, and so the number of
     columns of *note A: 415. must equal the size of the permutation
     *note p: 415.

 -- Function: int gsl_permutation_mul (gsl_permutation *p, const
          gsl_permutation *pa, const gsl_permutation *pb)

     This function combines the two permutations *note pa: 416. and
     *note pb: 416. into a single permutation *note p: 416, where p = pa
     * pb The permutation *note p: 416. is equivalent to applying *note
     pb: 416. first and then *note pa: 416.


File: gsl-ref.info,  Node: Reading and writing permutations,  Next: Permutations in cyclic form,  Prev: Applying Permutations,  Up: Permutations

9.7 Reading and writing permutations
====================================

The library provides functions for reading and writing permutations to a
file as binary data or formatted text.

 -- Function: int gsl_permutation_fwrite (FILE *stream, const
          gsl_permutation *p)

     This function writes the elements of the permutation *note p: 418.
     to the stream *note stream: 418. in binary format.  The function
     returns ‘GSL_EFAILED’ if there was a problem writing to the file.
     Since the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_permutation_fread (FILE *stream, gsl_permutation
          *p)

     This function reads into the permutation *note p: 419. from the
     open stream *note stream: 419. in binary format.  The permutation
     *note p: 419. must be preallocated with the correct length since
     the function uses the size of *note p: 419. to determine how many
     bytes to read.  The function returns ‘GSL_EFAILED’ if there was a
     problem reading from the file.  The data is assumed to have been
     written in the native binary format on the same architecture.

 -- Function: int gsl_permutation_fprintf (FILE *stream, const
          gsl_permutation *p, const char *format)

     This function writes the elements of the permutation *note p: 41a.
     line-by-line to the stream *note stream: 41a. using the format
     specifier *note format: 41a, which should be suitable for a type of
     ‘size_t’.  In ISO C99 the type modifier ‘z’ represents ‘size_t’, so
     ‘"%zu\n"’ is a suitable format (1).  The function returns
     ‘GSL_EFAILED’ if there was a problem writing to the file.

 -- Function: int gsl_permutation_fscanf (FILE *stream, gsl_permutation
          *p)

     This function reads formatted data from the stream *note stream:
     41b. into the permutation *note p: 41b.  The permutation *note p:
     41b. must be preallocated with the correct length since the
     function uses the size of *note p: 41b. to determine how many
     numbers to read.  The function returns ‘GSL_EFAILED’ if there was a
     problem reading from the file.

   ---------- Footnotes ----------

   (1) (1) In versions of the GNU C library prior to the ISO C99
standard, the type modifier ‘Z’ was used instead.


File: gsl-ref.info,  Node: Permutations in cyclic form,  Next: Examples<4>,  Prev: Reading and writing permutations,  Up: Permutations

9.8 Permutations in cyclic form
===============================

A permutation can be represented in both `linear' and `cyclic'
notations.  The functions described in this section convert between the
two forms.  The linear notation is an index mapping, and has already
been described above.  The cyclic notation expresses a permutation as a
series of circular rearrangements of groups of elements, or `cycles'.

For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced
by 3 and 3 is replaced by 1 in a circular fashion.  Cycles of different
sets of elements can be combined independently, for example (1 2 3) (4
5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange
of elements 4 and 5.  A cycle of length one represents an element which
is unchanged by the permutation and is referred to as a `singleton'.

It can be shown that every permutation can be decomposed into
combinations of cycles.  The decomposition is not unique, but can always
be rearranged into a standard `canonical form' by a reordering of
elements.  The library uses the canonical form defined in Knuth’s `Art
of Computer Programming' (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.

The procedure for obtaining the canonical form given by Knuth is,

  1. Write all singleton cycles explicitly

  2. Within each cycle, put the smallest number first

  3. Order the cycles in decreasing order of the first number in the
     cycle.

For example, the linear representation (2 4 3 0 1) is represented as (1
4) (0 2 3) in canonical form.  The permutation corresponds to an
exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.

The important property of the canonical form is that it can be
reconstructed from the contents of each cycle without the brackets.  In
addition, by removing the brackets it can be considered as a linear
representation of a different permutation.  In the example given above
the permutation (2 4 3 0 1) would become (1 4 0 2 3).  This mapping has
many applications in the theory of permutations.

 -- Function: int gsl_permutation_linear_to_canonical (gsl_permutation
          *q, const gsl_permutation *p)

     This function computes the canonical form of the permutation *note
     p: 41d. and stores it in the output argument *note q: 41d.

 -- Function: int gsl_permutation_canonical_to_linear (gsl_permutation
          *p, const gsl_permutation *q)

     This function converts a permutation *note q: 41e. in canonical
     form back into linear form storing it in the output argument *note
     p: 41e.

 -- Function: size_t gsl_permutation_inversions (const gsl_permutation
          *p)

     This function counts the number of inversions in the permutation
     *note p: 41f.  An inversion is any pair of elements that are not in
     order.  For example, the permutation 2031 has three inversions,
     corresponding to the pairs (2,0) (2,1) and (3,1).  The identity
     permutation has no inversions.

 -- Function: size_t gsl_permutation_linear_cycles (const
          gsl_permutation *p)

     This function counts the number of cycles in the permutation *note
     p: 420, given in linear form.

 -- Function: size_t gsl_permutation_canonical_cycles (const
          gsl_permutation *q)

     This function counts the number of cycles in the permutation *note
     q: 421, given in canonical form.


File: gsl-ref.info,  Node: Examples<4>,  Next: References and Further Reading<5>,  Prev: Permutations in cyclic form,  Up: Permutations

9.9 Examples
============

The example program below creates a random permutation (by shuffling the
elements of the identity) and finds its inverse.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_permutation.h>

     int
     main (void)
     {
       const size_t N = 10;
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_permutation * p = gsl_permutation_alloc (N);
       gsl_permutation * q = gsl_permutation_alloc (N);

       gsl_rng_env_setup();
       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       printf ("initial permutation:");
       gsl_permutation_init (p);
       gsl_permutation_fprintf (stdout, p, " %u");
       printf ("\n");

       printf (" random permutation:");
       gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
       gsl_permutation_fprintf (stdout, p, " %u");
       printf ("\n");

       printf ("inverse permutation:");
       gsl_permutation_inverse (q, p);
       gsl_permutation_fprintf (stdout, q, " %u");
       printf ("\n");

       gsl_permutation_free (p);
       gsl_permutation_free (q);
       gsl_rng_free (r);

       return 0;
     }

Here is the output from the program:

     $ ./a.out
     initial permutation: 0 1 2 3 4 5 6 7 8 9
      random permutation: 1 3 5 2 7 6 0 4 9 8
     inverse permutation: 6 0 3 1 7 2 5 4 9 8

The random permutation ‘p[i]’ and its inverse ‘q[i]’ are related through
the identity ‘p[q[i]] = i’, which can be verified from the output.

The next example program steps forwards through all possible third order
permutations, starting from the identity,

     #include <stdio.h>
     #include <gsl/gsl_permutation.h>

     int
     main (void)
     {
       gsl_permutation * p = gsl_permutation_alloc (3);

       gsl_permutation_init (p);

       do
        {
           gsl_permutation_fprintf (stdout, p, " %u");
           printf ("\n");
        }
       while (gsl_permutation_next(p) == GSL_SUCCESS);

       gsl_permutation_free (p);

       return 0;
     }

Here is the output from the program:

     $ ./a.out
      0 1 2
      0 2 1
      1 0 2
      1 2 0
      2 0 1
      2 1 0

The permutations are generated in lexicographic order.  To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace *note gsl_permutation_next(): 40e. with *note
gsl_permutation_prev(): 40f.


File: gsl-ref.info,  Node: References and Further Reading<5>,  Prev: Examples<4>,  Up: Permutations

9.10 References and Further Reading
===================================

The subject of permutations is covered extensively in the following,

   * Donald E. Knuth, The Art of Computer Programming: Sorting and
     Searching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

For the definition of the `canonical form' see,

   * Donald E. Knuth, The Art of Computer Programming: Fundamental
     Algorithms (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
     Section 1.3.3, An Unusual Correspondence, p.178–179.


File: gsl-ref.info,  Node: Combinations,  Next: Multisets,  Prev: Permutations,  Up: Top

10 Combinations
***************

This chapter describes functions for creating and manipulating
combinations.  A combination c is represented by an array of k integers
in the range 0 to n - 1, where each value c_i occurs at most once.  The
combination c corresponds to indices of k elements chosen from an n
element vector.  Combinations are useful for iterating over all
k-element subsets of a set.

The functions described in this chapter are defined in the header file
‘gsl_combination.h’.

* Menu:

* The Combination struct::
* Combination allocation::
* Accessing combination elements::
* Combination properties::
* Combination functions::
* Reading and writing combinations::
* Examples: Examples<5>.
* References and Further Reading: References and Further Reading<6>.


File: gsl-ref.info,  Node: The Combination struct,  Next: Combination allocation,  Up: Combinations

10.1 The Combination struct
===========================

 -- Type: gsl_combination

     A combination is defined by a structure containing three
     components, the values of n and k, and a pointer to the combination
     array.  The elements of the combination array are all of type
     ‘size_t’, and are stored in increasing order.  The *note
     gsl_combination: 427. structure looks like this:

          typedef struct
          {
            size_t n;
            size_t k;
            size_t *data;
          } gsl_combination;


File: gsl-ref.info,  Node: Combination allocation,  Next: Accessing combination elements,  Prev: The Combination struct,  Up: Combinations

10.2 Combination allocation
===========================

 -- Function: *note gsl_combination: 427. *gsl_combination_alloc (size_t
          n, size_t k)

     This function allocates memory for a new combination with
     parameters *note n: 429, *note k: 429.  The combination is not
     initialized and its elements are undefined.  Use the function *note
     gsl_combination_calloc(): 42a. if you want to create a combination
     which is initialized to the lexicographically first combination.  A
     null pointer is returned if insufficient memory is available to
     create the combination.

 -- Function: *note gsl_combination: 427. *gsl_combination_calloc
          (size_t n, size_t k)

     This function allocates memory for a new combination with
     parameters *note n: 42a, *note k: 42a. and initializes it to the
     lexicographically first combination.  A null pointer is returned if
     insufficient memory is available to create the combination.

 -- Function: void gsl_combination_init_first (gsl_combination *c)

     This function initializes the combination *note c: 42b. to the
     lexicographically first combination, i.e.  (0, 1, 2, \dots, k - 1).

 -- Function: void gsl_combination_init_last (gsl_combination *c)

     This function initializes the combination *note c: 42c. to the
     lexicographically last combination, i.e.  (n - k, n - k + 1, \dots,
     n - 1).

 -- Function: void gsl_combination_free (gsl_combination *c)

     This function frees all the memory used by the combination *note c:
     42d.

 -- Function: int gsl_combination_memcpy (gsl_combination *dest, const
          gsl_combination *src)

     This function copies the elements of the combination *note src:
     42e. into the combination *note dest: 42e.  The two combinations
     must have the same size.


File: gsl-ref.info,  Node: Accessing combination elements,  Next: Combination properties,  Prev: Combination allocation,  Up: Combinations

10.3 Accessing combination elements
===================================

The following function can be used to access the elements of a
combination.

 -- Function: size_t gsl_combination_get (const gsl_combination *c,
          const size_t i)

     This function returns the value of the *note i: 430.-th element of
     the combination *note c: 430.  If *note i: 430. lies outside the
     allowed range of 0 to k - 1 then the error handler is invoked and 0
     is returned.  An inline version of this function is used when
     ‘HAVE_INLINE’ is defined.


File: gsl-ref.info,  Node: Combination properties,  Next: Combination functions,  Prev: Accessing combination elements,  Up: Combinations

10.4 Combination properties
===========================

 -- Function: size_t gsl_combination_n (const gsl_combination *c)

     This function returns the range (n) of the combination c.

 -- Function: size_t gsl_combination_k (const gsl_combination *c)

     This function returns the number of elements (k) in the combination
     *note c: 433.

 -- Function: size_t *gsl_combination_data (const gsl_combination *c)

     This function returns a pointer to the array of elements in the
     combination *note c: 434.

 -- Function: int gsl_combination_valid (gsl_combination *c)

     This function checks that the combination *note c: 435. is valid.
     The ‘k’ elements should lie in the range 0 to n - 1, with each
     value occurring once at most and in increasing order.


File: gsl-ref.info,  Node: Combination functions,  Next: Reading and writing combinations,  Prev: Combination properties,  Up: Combinations

10.5 Combination functions
==========================

 -- Function: int gsl_combination_next (gsl_combination *c)

     This function advances the combination *note c: 437. to the next
     combination in lexicographic order and returns ‘GSL_SUCCESS’.  If
     no further combinations are available it returns ‘GSL_FAILURE’ and
     leaves *note c: 437. unmodified.  Starting with the first
     combination and repeatedly applying this function will iterate
     through all possible combinations of a given order.

 -- Function: int gsl_combination_prev (gsl_combination *c)

     This function steps backwards from the combination *note c: 438. to
     the previous combination in lexicographic order, returning
     ‘GSL_SUCCESS’.  If no previous combination is available it returns
     ‘GSL_FAILURE’ and leaves *note c: 438. unmodified.


File: gsl-ref.info,  Node: Reading and writing combinations,  Next: Examples<5>,  Prev: Combination functions,  Up: Combinations

10.6 Reading and writing combinations
=====================================

The library provides functions for reading and writing combinations to a
file as binary data or formatted text.

 -- Function: int gsl_combination_fwrite (FILE *stream, const
          gsl_combination *c)

     This function writes the elements of the combination *note c: 43a.
     to the stream *note stream: 43a. in binary format.  The function
     returns ‘GSL_EFAILED’ if there was a problem writing to the file.
     Since the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_combination_fread (FILE *stream, gsl_combination
          *c)

     This function reads elements from the open stream *note stream:
     43b. into the combination *note c: 43b. in binary format.  The
     combination *note c: 43b. must be preallocated with correct values
     of n and k since the function uses the size of *note c: 43b. to
     determine how many bytes to read.  The function returns
     ‘GSL_EFAILED’ if there was a problem reading from the file.  The
     data is assumed to have been written in the native binary format on
     the same architecture.

 -- Function: int gsl_combination_fprintf (FILE *stream, const
          gsl_combination *c, const char *format)

     This function writes the elements of the combination *note c: 43c.
     line-by-line to the stream *note stream: 43c. using the format
     specifier *note format: 43c, which should be suitable for a type of
     ‘size_t’.  In ISO C99 the type modifier ‘z’ represents ‘size_t’, so
     ‘"%zu\n"’ is a suitable format (1).  The function returns
     ‘GSL_EFAILED’ if there was a problem writing to the file.

 -- Function: int gsl_combination_fscanf (FILE *stream, gsl_combination
          *c)

     This function reads formatted data from the stream *note stream:
     43d. into the combination *note c: 43d.  The combination *note c:
     43d. must be preallocated with correct values of n and k since the
     function uses the size of *note c: 43d. to determine how many
     numbers to read.  The function returns ‘GSL_EFAILED’ if there was a
     problem reading from the file.

   ---------- Footnotes ----------

   (1) (1) In versions of the GNU C library prior to the ISO C99
standard, the type modifier ‘Z’ was used instead.


File: gsl-ref.info,  Node: Examples<5>,  Next: References and Further Reading<6>,  Prev: Reading and writing combinations,  Up: Combinations

10.7 Examples
=============

The example program below prints all subsets of the set {0,1,2,3}
ordered by size.  Subsets of the same size are ordered
lexicographically.

     #include <stdio.h>
     #include <gsl/gsl_combination.h>

     int
     main (void)
     {
       gsl_combination * c;
       size_t i;

       printf ("All subsets of {0,1,2,3} by size:\n") ;
       for (i = 0; i <= 4; i++)
         {
           c = gsl_combination_calloc (4, i);
           do
             {
               printf ("{");
               gsl_combination_fprintf (stdout, c, " %u");
               printf (" }\n");
             }
           while (gsl_combination_next (c) == GSL_SUCCESS);
           gsl_combination_free (c);
         }

       return 0;
     }

Here is the output from the program,

     All subsets of {0,1,2,3} by size:
     { }
     { 0 }
     { 1 }
     { 2 }
     { 3 }
     { 0 1 }
     { 0 2 }
     { 0 3 }
     { 1 2 }
     { 1 3 }
     { 2 3 }
     { 0 1 2 }
     { 0 1 3 }
     { 0 2 3 }
     { 1 2 3 }
     { 0 1 2 3 }

All 16 subsets are generated, and the subsets of each size are sorted
lexicographically.


File: gsl-ref.info,  Node: References and Further Reading<6>,  Prev: Examples<5>,  Up: Combinations

10.8 References and Further Reading
===================================

Further information on combinations can be found in,

   * Donald L. Kreher, Douglas R. Stinson, Combinatorial Algorithms:
     Generation, Enumeration and Search, 1998, CRC Press LLC, ISBN
     084933988X


File: gsl-ref.info,  Node: Multisets,  Next: Sorting,  Prev: Combinations,  Up: Top

11 Multisets
************

This chapter describes functions for creating and manipulating
multisets.  A multiset c is represented by an array of k integers in the
range 0 to n - 1, where each value c_i may occur more than once.  The
multiset c corresponds to indices of k elements chosen from an n element
vector with replacement.  In mathematical terms, n is the cardinality of
the multiset while k is the maximum multiplicity of any value.
Multisets are useful, for example, when iterating over the indices of a
k-th order symmetric tensor in n-space.

The functions described in this chapter are defined in the header file
‘gsl_multiset.h’.

* Menu:

* The Multiset struct::
* Multiset allocation::
* Accessing multiset elements::
* Multiset properties::
* Multiset functions::
* Reading and writing multisets::
* Examples: Examples<6>.


File: gsl-ref.info,  Node: The Multiset struct,  Next: Multiset allocation,  Up: Multisets

11.1 The Multiset struct
========================

 -- Type: gsl_multiset

     A multiset is defined by a structure containing three components,
     the values of n and k, and a pointer to the multiset array.  The
     elements of the multiset array are all of type ‘size_t’, and are
     stored in increasing order.  The *note gsl_multiset: 443. structure
     looks like this:

          typedef struct
          {
            size_t n;
            size_t k;
            size_t *data;
          } gsl_multiset;


File: gsl-ref.info,  Node: Multiset allocation,  Next: Accessing multiset elements,  Prev: The Multiset struct,  Up: Multisets

11.2 Multiset allocation
========================

 -- Function: *note gsl_multiset: 443. *gsl_multiset_alloc (size_t n,
          size_t k)

     This function allocates memory for a new multiset with parameters
     *note n: 445, *note k: 445.  The multiset is not initialized and
     its elements are undefined.  Use the function *note
     gsl_multiset_calloc(): 446. if you want to create a multiset which
     is initialized to the lexicographically first multiset element.  A
     null pointer is returned if insufficient memory is available to
     create the multiset.

 -- Function: *note gsl_multiset: 443. *gsl_multiset_calloc (size_t n,
          size_t k)

     This function allocates memory for a new multiset with parameters
     *note n: 446, *note k: 446. and initializes it to the
     lexicographically first multiset element.  A null pointer is
     returned if insufficient memory is available to create the
     multiset.

 -- Function: void gsl_multiset_init_first (gsl_multiset *c)

     This function initializes the multiset *note c: 447. to the
     lexicographically first multiset element, i.e.  0 repeated k times.

 -- Function: void gsl_multiset_init_last (gsl_multiset *c)

     This function initializes the multiset *note c: 448. to the
     lexicographically last multiset element, i.e.  n-1 repeated k
     times.

 -- Function: void gsl_multiset_free (gsl_multiset *c)

     This function frees all the memory used by the multiset *note c:
     449.

 -- Function: int gsl_multiset_memcpy (gsl_multiset *dest, const
          gsl_multiset *src)

     This function copies the elements of the multiset *note src: 44a.
     into the multiset *note dest: 44a.  The two multisets must have the
     same size.


File: gsl-ref.info,  Node: Accessing multiset elements,  Next: Multiset properties,  Prev: Multiset allocation,  Up: Multisets

11.3 Accessing multiset elements
================================

The following function can be used to access the elements of a multiset.

 -- Function: size_t gsl_multiset_get (const gsl_multiset *c, const
          size_t i)

     This function returns the value of the *note i: 44c.-th element of
     the multiset *note c: 44c.  If *note i: 44c. lies outside the
     allowed range of 0 to k - 1 then the error handler is invoked and 0
     is returned.  An inline version of this function is used when
     ‘HAVE_INLINE’ is defined.


File: gsl-ref.info,  Node: Multiset properties,  Next: Multiset functions,  Prev: Accessing multiset elements,  Up: Multisets

11.4 Multiset properties
========================

 -- Function: size_t gsl_multiset_n (const gsl_multiset *c)

     This function returns the range (n) of the multiset *note c: 44e.

 -- Function: size_t gsl_multiset_k (const gsl_multiset *c)

     This function returns the number of elements (k) in the multiset
     *note c: 44f.

 -- Function: size_t *gsl_multiset_data (const gsl_multiset *c)

     This function returns a pointer to the array of elements in the
     multiset *note c: 450.

 -- Function: int gsl_multiset_valid (gsl_multiset *c)

     This function checks that the multiset *note c: 451. is valid.  The
     ‘k’ elements should lie in the range 0 to n - 1, with each value
     occurring in nondecreasing order.


File: gsl-ref.info,  Node: Multiset functions,  Next: Reading and writing multisets,  Prev: Multiset properties,  Up: Multisets

11.5 Multiset functions
=======================

 -- Function: int gsl_multiset_next (gsl_multiset *c)

     This function advances the multiset *note c: 453. to the next
     multiset element in lexicographic order and returns ‘GSL_SUCCESS’.
     If no further multisets elements are available it returns
     ‘GSL_FAILURE’ and leaves *note c: 453. unmodified.  Starting with
     the first multiset and repeatedly applying this function will
     iterate through all possible multisets of a given order.

 -- Function: int gsl_multiset_prev (gsl_multiset *c)

     This function steps backwards from the multiset *note c: 454. to
     the previous multiset element in lexicographic order, returning
     ‘GSL_SUCCESS’.  If no previous multiset is available it returns
     ‘GSL_FAILURE’ and leaves *note c: 454. unmodified.


File: gsl-ref.info,  Node: Reading and writing multisets,  Next: Examples<6>,  Prev: Multiset functions,  Up: Multisets

11.6 Reading and writing multisets
==================================

The library provides functions for reading and writing multisets to a
file as binary data or formatted text.

 -- Function: int gsl_multiset_fwrite (FILE *stream, const gsl_multiset
          *c)

     This function writes the elements of the multiset *note c: 456. to
     the stream *note stream: 456. in binary format.  The function
     returns ‘GSL_EFAILED’ if there was a problem writing to the file.
     Since the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_multiset_fread (FILE *stream, gsl_multiset *c)

     This function reads elements from the open stream *note stream:
     457. into the multiset *note c: 457. in binary format.  The
     multiset *note c: 457. must be preallocated with correct values of
     n and k since the function uses the size of *note c: 457. to
     determine how many bytes to read.  The function returns
     ‘GSL_EFAILED’ if there was a problem reading from the file.  The
     data is assumed to have been written in the native binary format on
     the same architecture.

 -- Function: int gsl_multiset_fprintf (FILE *stream, const gsl_multiset
          *c, const char *format)

     This function writes the elements of the multiset *note c: 458.
     line-by-line to the stream *note stream: 458. using the format
     specifier *note format: 458, which should be suitable for a type of
     ‘size_t’.  In ISO C99 the type modifier ‘z’ represents ‘size_t’, so
     ‘"%zu\n"’ is a suitable format (1).  The function returns
     ‘GSL_EFAILED’ if there was a problem writing to the file.

 -- Function: int gsl_multiset_fscanf (FILE *stream, gsl_multiset *c)

     This function reads formatted data from the stream *note stream:
     459. into the multiset *note c: 459.  The multiset *note c: 459.
     must be preallocated with correct values of n and k since the
     function uses the size of *note c: 459. to determine how many
     numbers to read.  The function returns ‘GSL_EFAILED’ if there was a
     problem reading from the file.

   ---------- Footnotes ----------

   (1) (1) In versions of the GNU C library prior to the ISO C99
standard, the type modifier ‘Z’ was used instead.


File: gsl-ref.info,  Node: Examples<6>,  Prev: Reading and writing multisets,  Up: Multisets

11.7 Examples
=============

The example program below prints all multisets elements containing the
values {0,1,2,3} ordered by size.  Multiset elements of the same size
are ordered lexicographically.

     #include <stdio.h>
     #include <gsl/gsl_multiset.h>

     int
     main (void)
     {
       gsl_multiset * c;
       size_t i;

       printf ("All multisets of {0,1,2,3} by size:\n") ;
       for (i = 0; i <= 4; i++)
         {
           c = gsl_multiset_calloc (4, i);
           do
             {
               printf ("{");
               gsl_multiset_fprintf (stdout, c, " %u");
               printf (" }\n");
             }
           while (gsl_multiset_next (c) == GSL_SUCCESS);
           gsl_multiset_free (c);
         }

       return 0;
     }

Here is the output from the program,

     All multisets of {0,1,2,3} by size:
     { }
     { 0 }
     { 1 }
     { 2 }
     { 3 }
     { 0 0 }
     { 0 1 }
     { 0 2 }
     { 0 3 }
     { 1 1 }
     { 1 2 }
     { 1 3 }
     { 2 2 }
     { 2 3 }
     { 3 3 }
     { 0 0 0 }
     { 0 0 1 }
     { 0 0 2 }
     { 0 0 3 }
     { 0 1 1 }
     { 0 1 2 }
     { 0 1 3 }
     { 0 2 2 }
     { 0 2 3 }
     { 0 3 3 }
     { 1 1 1 }
     { 1 1 2 }
     { 1 1 3 }
     { 1 2 2 }
     { 1 2 3 }
     { 1 3 3 }
     { 2 2 2 }
     { 2 2 3 }
     { 2 3 3 }
     { 3 3 3 }
     { 0 0 0 0 }
     { 0 0 0 1 }
     { 0 0 0 2 }
     { 0 0 0 3 }
     { 0 0 1 1 }
     { 0 0 1 2 }
     { 0 0 1 3 }
     { 0 0 2 2 }
     { 0 0 2 3 }
     { 0 0 3 3 }
     { 0 1 1 1 }
     { 0 1 1 2 }
     { 0 1 1 3 }
     { 0 1 2 2 }
     { 0 1 2 3 }
     { 0 1 3 3 }
     { 0 2 2 2 }
     { 0 2 2 3 }
     { 0 2 3 3 }
     { 0 3 3 3 }
     { 1 1 1 1 }
     { 1 1 1 2 }
     { 1 1 1 3 }
     { 1 1 2 2 }
     { 1 1 2 3 }
     { 1 1 3 3 }
     { 1 2 2 2 }
     { 1 2 2 3 }
     { 1 2 3 3 }
     { 1 3 3 3 }
     { 2 2 2 2 }
     { 2 2 2 3 }
     { 2 2 3 3 }
     { 2 3 3 3 }
     { 3 3 3 3 }

All 70 multisets are generated and sorted lexicographically.


File: gsl-ref.info,  Node: Sorting,  Next: BLAS Support,  Prev: Multisets,  Up: Top

12 Sorting
**********

This chapter describes functions for sorting data, both directly and
indirectly (using an index).  All the functions use the `heapsort'
algorithm.  Heapsort is an O(N \log N) algorithm which operates in-place
and does not require any additional storage.  It also provides
consistent performance, the running time for its worst-case (ordered
data) being not significantly longer than the average and best cases.
Note that the heapsort algorithm does not preserve the relative ordering
of equal elements—it is an `unstable' sort.  However the resulting order
of equal elements will be consistent across different platforms when
using these functions.

* Menu:

* Sorting objects::
* Sorting vectors::
* Selecting the k smallest or largest elements::
* Computing the rank::
* Examples: Examples<7>.
* References and Further Reading: References and Further Reading<7>.


File: gsl-ref.info,  Node: Sorting objects,  Next: Sorting vectors,  Up: Sorting

12.1 Sorting objects
====================

The following function provides a simple alternative to the standard
library function ‘qsort()’.  It is intended for systems lacking
‘qsort()’, not as a replacement for it.  The function ‘qsort()’ should
be used whenever possible, as it will be faster and can provide stable
ordering of equal elements.  Documentation for ‘qsort()’ is available in
the GNU C Library Reference Manual.

The functions described in this section are defined in the header file
‘gsl_heapsort.h’.

 -- Function: void gsl_heapsort (void *array, size_t count, size_t size,
          gsl_comparison_fn_t compare)

     This function sorts the *note count: 45e. elements of the array
     *note array: 45e, each of size *note size: 45e, into ascending
     order using the comparison function *note compare: 45e.  The type
     of the comparison function is defined by

      -- Type: gsl_comparison_fn_t

               int (*gsl_comparison_fn_t) (const void * a, const void * b)

     A comparison function should return a negative integer if the first
     argument is less than the second argument, ‘0’ if the two arguments
     are equal and a positive integer if the first argument is greater
     than the second argument.

     For example, the following function can be used to sort doubles
     into ascending numerical order.

          int
          compare_doubles (const double * a, const double * b)
          {
            if (*a > *b)
              return 1;
            else if (*a < *b)
              return -1;
            else
              return 0;
          }

     The appropriate function call to perform the sort is:

          gsl_heapsort (array, count, sizeof(double), compare_doubles);

     Note that unlike ‘qsort()’ the heapsort algorithm cannot be made
     into a stable sort by pointer arithmetic.  The trick of comparing
     pointers for equal elements in the comparison function does not
     work for the heapsort algorithm.  The heapsort algorithm performs
     an internal rearrangement of the data which destroys its initial
     ordering.

 -- Function: int gsl_heapsort_index (size_t *p, const void *array,
          size_t count, size_t size, gsl_comparison_fn_t compare)

     This function indirectly sorts the *note count: 460. elements of
     the array *note array: 460, each of size *note size: 460, into
     ascending order using the comparison function *note compare: 460.
     The resulting permutation is stored in *note p: 460, an array of
     length ‘n’.  The elements of *note p: 460. give the index of the
     array element which would have been stored in that position if the
     array had been sorted in place.  The first element of *note p: 460.
     gives the index of the least element in *note array: 460, and the
     last element of *note p: 460. gives the index of the greatest
     element in *note array: 460.  The array itself is not changed.


File: gsl-ref.info,  Node: Sorting vectors,  Next: Selecting the k smallest or largest elements,  Prev: Sorting objects,  Up: Sorting

12.2 Sorting vectors
====================

The following functions will sort the elements of an array or vector,
either directly or indirectly.  They are defined for all real and
integer types using the normal suffix rules.  For example, the ‘float’
versions of the array functions are ‘gsl_sort_float()’ and
‘gsl_sort_float_index()’.  The corresponding vector functions are
‘gsl_sort_vector_float()’ and ‘gsl_sort_vector_float_index()’.  The
prototypes are available in the header files ‘gsl_sort_float.h’
‘gsl_sort_vector_float.h’.  The complete set of prototypes can be
included using the header files ‘gsl_sort.h’ and ‘gsl_sort_vector.h’.

There are no functions for sorting complex arrays or vectors, since the
ordering of complex numbers is not uniquely defined.  To sort a complex
vector by magnitude compute a real vector containing the magnitudes of
the complex elements, and sort this vector indirectly.  The resulting
index gives the appropriate ordering of the original complex vector.

 -- Function: void gsl_sort (double *data, const size_t stride, size_t
          n)

     This function sorts the *note n: 462. elements of the array *note
     data: 462. with stride *note stride: 462. into ascending numerical
     order.

 -- Function: void gsl_sort2 (double *data1, const size_t stride1,
          double *data2, const size_t stride2, size_t n)

     This function sorts the *note n: 463. elements of the array *note
     data1: 463. with stride *note stride1: 463. into ascending
     numerical order, while making the same rearrangement of the array
     *note data2: 463. with stride *note stride2: 463, also of size
     *note n: 463.

 -- Function: void gsl_sort_vector (gsl_vector *v)

     This function sorts the elements of the vector *note v: 464. into
     ascending numerical order.

 -- Function: void gsl_sort_vector2 (gsl_vector *v1, gsl_vector *v2)

     This function sorts the elements of the vector *note v1: 465. into
     ascending numerical order, while making the same rearrangement of
     the vector *note v2: 465.

 -- Function: void gsl_sort_index (size_t *p, const double *data, size_t
          stride, size_t n)

     This function indirectly sorts the *note n: 466. elements of the
     array *note data: 466. with stride *note stride: 466. into
     ascending order, storing the resulting permutation in *note p: 466.
     The array *note p: 466. must be allocated with a sufficient length
     to store the *note n: 466. elements of the permutation.  The
     elements of *note p: 466. give the index of the array element which
     would have been stored in that position if the array had been
     sorted in place.  The array *note data: 466. is not changed.

 -- Function: int gsl_sort_vector_index (gsl_permutation *p, const
          gsl_vector *v)

     This function indirectly sorts the elements of the vector *note v:
     467. into ascending order, storing the resulting permutation in
     *note p: 467.  The elements of *note p: 467. give the index of the
     vector element which would have been stored in that position if the
     vector had been sorted in place.  The first element of *note p:
     467. gives the index of the least element in *note v: 467, and the
     last element of *note p: 467. gives the index of the greatest
     element in *note v: 467.  The vector *note v: 467. is not changed.


File: gsl-ref.info,  Node: Selecting the k smallest or largest elements,  Next: Computing the rank,  Prev: Sorting vectors,  Up: Sorting

12.3 Selecting the k smallest or largest elements
=================================================

The functions described in this section select the k smallest or largest
elements of a data set of size N. The routines use an O(kN) direct
insertion algorithm which is suited to subsets that are small compared
with the total size of the dataset.  For example, the routines are
useful for selecting the 10 largest values from one million data points,
but not for selecting the largest 100,000 values.  If the subset is a
significant part of the total dataset it may be faster to sort all the
elements of the dataset directly with an O(N \log N) algorithm and
obtain the smallest or largest values that way.

 -- Function: int gsl_sort_smallest (double *dest, size_t k, const
          double *src, size_t stride, size_t n)

     This function copies the *note k: 469. smallest elements of the
     array *note src: 469, of size *note n: 469. and stride *note
     stride: 469, in ascending numerical order into the array *note
     dest: 469.  The size *note k: 469. of the subset must be less than
     or equal to *note n: 469.  The data *note src: 469. is not modified
     by this operation.

 -- Function: int gsl_sort_largest (double *dest, size_t k, const double
          *src, size_t stride, size_t n)

     This function copies the *note k: 46a. largest elements of the
     array *note src: 46a, of size *note n: 46a. and stride *note
     stride: 46a, in descending numerical order into the array *note
     dest: 46a.  *note k: 46a. must be less than or equal to *note n:
     46a.  The data *note src: 46a. is not modified by this operation.

 -- Function: int gsl_sort_vector_smallest (double *dest, size_t k,
          const gsl_vector *v)
 -- Function: int gsl_sort_vector_largest (double *dest, size_t k, const
          gsl_vector *v)

     These functions copy the *note k: 46c. smallest or largest elements
     of the vector *note v: 46c. into the array *note dest: 46c.  *note
     k: 46c. must be less than or equal to the length of the vector
     *note v: 46c.

The following functions find the indices of the k smallest or largest
elements of a dataset.

 -- Function: int gsl_sort_smallest_index (size_t *p, size_t k, const
          double *src, size_t stride, size_t n)

     This function stores the indices of the *note k: 46d. smallest
     elements of the array *note src: 46d, of size *note n: 46d. and
     stride *note stride: 46d, in the array *note p: 46d.  The indices
     are chosen so that the corresponding data is in ascending numerical
     order.  *note k: 46d. must be less than or equal to *note n: 46d.
     The data *note src: 46d. is not modified by this operation.

 -- Function: int gsl_sort_largest_index (size_t *p, size_t k, const
          double *src, size_t stride, size_t n)

     This function stores the indices of the *note k: 46e. largest
     elements of the array *note src: 46e, of size *note n: 46e. and
     stride *note stride: 46e, in the array *note p: 46e.  The indices
     are chosen so that the corresponding data is in descending
     numerical order.  *note k: 46e. must be less than or equal to *note
     n: 46e.  The data *note src: 46e. is not modified by this
     operation.

 -- Function: int gsl_sort_vector_smallest_index (size_t *p, size_t k,
          const gsl_vector *v)
 -- Function: int gsl_sort_vector_largest_index (size_t *p, size_t k,
          const gsl_vector *v)

     These functions store the indices of the *note k: 470. smallest or
     largest elements of the vector *note v: 470. in the array *note p:
     470.  *note k: 470. must be less than or equal to the length of the
     vector *note v: 470.


File: gsl-ref.info,  Node: Computing the rank,  Next: Examples<7>,  Prev: Selecting the k smallest or largest elements,  Up: Sorting

12.4 Computing the rank
=======================

The `rank' of an element is its order in the sorted data.  The rank is
the inverse of the index permutation, p.  It can be computed using the
following algorithm:

     for (i = 0; i < p->size; i++)
       {
         size_t pi = p->data[i];
         rank->data[pi] = i;
       }

This can be computed directly from the function
‘gsl_permutation_inverse(rank,p)’.

The following function will print the rank of each element of the vector
v:

     void
     print_rank (gsl_vector * v)
     {
       size_t i;
       size_t n = v->size;
       gsl_permutation * perm = gsl_permutation_alloc(n);
       gsl_permutation * rank = gsl_permutation_alloc(n);

       gsl_sort_vector_index (perm, v);
       gsl_permutation_inverse (rank, perm);

       for (i = 0; i < n; i++)
         {
           double vi = gsl_vector_get(v, i);
           printf ("element = %d, value = %g, rank = %d\n",
                    i, vi, rank->data[i]);
         }

       gsl_permutation_free (perm);
       gsl_permutation_free (rank);
     }


File: gsl-ref.info,  Node: Examples<7>,  Next: References and Further Reading<7>,  Prev: Computing the rank,  Up: Sorting

12.5 Examples
=============

The following example shows how to use the permutation p to print the
elements of the vector v in ascending order:

     gsl_sort_vector_index (p, v);

     for (i = 0; i < v->size; i++)
       {
         double vpi = gsl_vector_get (v, p->data[i]);
         printf ("order = %d, value = %g\n", i, vpi);
       }

The next example uses the function *note gsl_sort_smallest(): 469. to
select the 5 smallest numbers from 100000 uniform random variates stored
in an array,

     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_sort_double.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       size_t i, k = 5, N = 100000;

       double * x = malloc (N * sizeof(double));
       double * small = malloc (k * sizeof(double));

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < N; i++)
         {
           x[i] = gsl_rng_uniform(r);
         }

       gsl_sort_smallest (small, k, x, 1, N);

       printf ("%zu smallest values from %zu\n", k, N);

       for (i = 0; i < k; i++)
         {
           printf ("%zu: %.18f\n", i, small[i]);
         }

       free (x);
       free (small);
       gsl_rng_free (r);
       return 0;
     }

The output lists the 5 smallest values, in ascending order,

     5 smallest values from 100000
     0: 0.000003489200025797
     1: 0.000008199829608202
     2: 0.000008953968062997
     3: 0.000010712770745158
     4: 0.000033531803637743


File: gsl-ref.info,  Node: References and Further Reading<7>,  Prev: Examples<7>,  Up: Sorting

12.6 References and Further Reading
===================================

The subject of sorting is covered extensively in the following,

   * Donald E. Knuth, The Art of Computer Programming: Sorting and
     Searching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

The Heapsort algorithm is described in the following book,

   * Robert Sedgewick, Algorithms in C, Addison-Wesley, ISBN 0201514257.


File: gsl-ref.info,  Node: BLAS Support,  Next: Linear Algebra,  Prev: Sorting,  Up: Top

13 BLAS Support
***************

The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental
operations on vectors and matrices which can be used to create optimized
higher-level linear algebra functionality.

The library provides a low-level layer which corresponds directly to the
C-language BLAS standard, referred to here as “CBLAS”, and a
higher-level interface for operations on GSL vectors and matrices.
Users who are interested in simple operations on GSL vector and matrix
objects should use the high-level layer described in this chapter.  The
functions are declared in the file ‘gsl_blas.h’ and should satisfy the
needs of most users.

Note that GSL matrices are implemented using dense-storage so the
interface only includes the corresponding dense-storage BLAS functions.
The full BLAS functionality for band-format and packed-format matrices
is available through the low-level CBLAS interface.  Similarly, GSL
vectors are restricted to positive strides, whereas the low-level CBLAS
interface supports negative strides as specified in the BLAS standard
(1).

The interface for the ‘gsl_cblas’ layer is specified in the file
‘gsl_cblas.h’.  This interface corresponds to the BLAS Technical Forum’s
standard for the C interface to legacy BLAS implementations.  Users who
have access to other conforming CBLAS implementations can use these in
place of the version provided by the library.  Note that users who have
only a Fortran BLAS library can use a CBLAS conformant wrapper to
convert it into a CBLAS library.  A reference CBLAS wrapper for legacy
Fortran implementations exists as part of the CBLAS standard and can be
obtained from Netlib.  The complete set of CBLAS functions is listed in
an *note appendix: 476.

There are three levels of BLAS operations,

`Level 1'       Vector operations, e.g.  y = \alpha x + y
                
                
`Level 2'       Matrix-vector operations, e.g.  y = \alpha A x + \beta y
                
                
`Level 3'       Matrix-matrix operations, e.g.  C = \alpha A B + C
                

Each routine has a name which specifies the operation, the type of
matrices involved and their precisions.  Some of the most common
operations and their names are given below,

`DOT'        scalar product, x^T y
             
             
`AXPY'       vector sum, \alpha x + y
             
             
`MV'         matrix-vector product, A x
             
             
`SV'         matrix-vector solve, inv(A) x
             
             
`MM'         matrix-matrix product, A B
             
             
`SM'         matrix-matrix solve, inv(A) B
             

The types of matrices are,

`GE'       general
           
           
`GB'       general band
           
           
`SY'       symmetric
           
           
`SB'       symmetric band
           
           
`SP'       symmetric packed
           
           
`HE'       hermitian
           
           
`HB'       hermitian band
           
           
`HP'       hermitian packed
           
           
`TR'       triangular
           
           
`TB'       triangular band
           
           
`TP'       triangular packed
           

Each operation is defined for four precisions,

`S'       single real
          
          
`D'       double real
          
          
`C'       single complex
          
          
`Z'       double complex
          

Thus, for example, the name SGEMM stands for “single-precision general
matrix-matrix multiply” and ZGEMM stands for “double-precision complex
matrix-matrix multiply”.

Note that the vector and matrix arguments to BLAS functions must not be
aliased, as the results are undefined when the underlying arrays overlap
(*note Aliasing of arrays: 1f.).

* Menu:

* GSL BLAS Interface::
* Examples: Examples<8>.
* References and Further Reading: References and Further Reading<8>.

   ---------- Footnotes ----------

   (1) (1) In the low-level CBLAS interface, a negative stride accesses
the vector elements in reverse order, i.e.  the i-th element is given by
(N-i)*|incx| for incx < 0.


File: gsl-ref.info,  Node: GSL BLAS Interface,  Next: Examples<8>,  Up: BLAS Support

13.1 GSL BLAS Interface
=======================

GSL provides dense vector and matrix objects, based on the relevant
built-in types.  The library provides an interface to the BLAS
operations which apply to these objects.  The interface to this
functionality is given in the file ‘gsl_blas.h’.

* Menu:

* Level 1::
* Level 2::
* Level 3::


File: gsl-ref.info,  Node: Level 1,  Next: Level 2,  Up: GSL BLAS Interface

13.1.1 Level 1
--------------

 -- Function: int gsl_blas_sdsdot (float alpha, const gsl_vector_float
          *x, const gsl_vector_float *y, float *result)

     This function computes the sum \alpha + x^T y for the vectors *note
     x: 479. and *note y: 479, returning the result in *note result:
     479.

 -- Function: int gsl_blas_sdot (const gsl_vector_float *x, const
          gsl_vector_float *y, float *result)
 -- Function: int gsl_blas_dsdot (const gsl_vector_float *x, const
          gsl_vector_float *y, double *result)
 -- Function: int gsl_blas_ddot (const gsl_vector *x, const gsl_vector
          *y, double *result)

     These functions compute the scalar product x^T y for the vectors
     *note x: 47c. and *note y: 47c, returning the result in *note
     result: 47c.

 -- Function: int gsl_blas_cdotu (const gsl_vector_complex_float *x,
          const gsl_vector_complex_float *y, gsl_complex_float *dotu)
 -- Function: int gsl_blas_zdotu (const gsl_vector_complex *x, const
          gsl_vector_complex *y, gsl_complex *dotu)

     These functions compute the complex scalar product x^T y for the
     vectors *note x: 47e. and *note y: 47e, returning the result in
     *note dotu: 47e.

 -- Function: int gsl_blas_cdotc (const gsl_vector_complex_float *x,
          const gsl_vector_complex_float *y, gsl_complex_float *dotc)
 -- Function: int gsl_blas_zdotc (const gsl_vector_complex *x, const
          gsl_vector_complex *y, gsl_complex *dotc)

     These functions compute the complex conjugate scalar product x^H y
     for the vectors *note x: 480. and *note y: 480, returning the
     result in *note dotc: 480.

 -- Function: float gsl_blas_snrm2 (const gsl_vector_float *x)
 -- Function: double gsl_blas_dnrm2 (const gsl_vector *x)

     These functions compute the Euclidean norm ||x||_2 = \sqrt{\sum
     x_i^2} of the vector *note x: 482.

 -- Function: float gsl_blas_scnrm2 (const gsl_vector_complex_float *x)
 -- Function: double gsl_blas_dznrm2 (const gsl_vector_complex *x)

     These functions compute the Euclidean norm of the complex vector
     *note x: 484,

          ||x||_2 = \sqrt{\sum (\Re(x_i)^2 + \Im(x_i)^2)}.

 -- Function: float gsl_blas_sasum (const gsl_vector_float *x)
 -- Function: double gsl_blas_dasum (const gsl_vector *x)

     These functions compute the absolute sum \sum |x_i| of the elements
     of the vector *note x: 486.

 -- Function: float gsl_blas_scasum (const gsl_vector_complex_float *x)
 -- Function: double gsl_blas_dzasum (const gsl_vector_complex *x)

     These functions compute the sum of the magnitudes of the real and
     imaginary parts of the complex vector *note x: 488, \sum \left(
     |\Re(x_i)| + |\Im(x_i)| \right).

 -- Function: CBLAS_INDEX_t gsl_blas_isamax (const gsl_vector_float *x)
 -- Function: CBLAS_INDEX_t gsl_blas_idamax (const gsl_vector *x)
 -- Function: CBLAS_INDEX_t gsl_blas_icamax (const
          gsl_vector_complex_float *x)
 -- Function: CBLAS_INDEX_t gsl_blas_izamax (const gsl_vector_complex
          *x)

     These functions return the index of the largest element of the
     vector *note x: 48c.  The largest element is determined by its
     absolute magnitude for real vectors and by the sum of the
     magnitudes of the real and imaginary parts |\Re(x_i)| + |\Im(x_i)|
     for complex vectors.  If the largest value occurs several times
     then the index of the first occurrence is returned.

 -- Function: int gsl_blas_sswap (gsl_vector_float *x, gsl_vector_float
          *y)
 -- Function: int gsl_blas_dswap (gsl_vector *x, gsl_vector *y)
 -- Function: int gsl_blas_cswap (gsl_vector_complex_float *x,
          gsl_vector_complex_float *y)
 -- Function: int gsl_blas_zswap (gsl_vector_complex *x,
          gsl_vector_complex *y)

     These functions exchange the elements of the vectors *note x: 490.
     and *note y: 490.

 -- Function: int gsl_blas_scopy (const gsl_vector_float *x,
          gsl_vector_float *y)
 -- Function: int gsl_blas_dcopy (const gsl_vector *x, gsl_vector *y)
 -- Function: int gsl_blas_ccopy (const gsl_vector_complex_float *x,
          gsl_vector_complex_float *y)
 -- Function: int gsl_blas_zcopy (const gsl_vector_complex *x,
          gsl_vector_complex *y)

     These functions copy the elements of the vector *note x: 494. into
     the vector *note y: 494.

 -- Function: int gsl_blas_saxpy (float alpha, const gsl_vector_float
          *x, gsl_vector_float *y)
 -- Function: int gsl_blas_daxpy (double alpha, const gsl_vector *x,
          gsl_vector *y)
 -- Function: int gsl_blas_caxpy (const gsl_complex_float alpha, const
          gsl_vector_complex_float *x, gsl_vector_complex_float *y)
 -- Function: int gsl_blas_zaxpy (const gsl_complex alpha, const
          gsl_vector_complex *x, gsl_vector_complex *y)

     These functions compute the sum y = \alpha x + y for the vectors
     *note x: 498. and *note y: 498.

 -- Function: void gsl_blas_sscal (float alpha, gsl_vector_float *x)
 -- Function: void gsl_blas_dscal (double alpha, gsl_vector *x)
 -- Function: void gsl_blas_cscal (const gsl_complex_float alpha,
          gsl_vector_complex_float *x)
 -- Function: void gsl_blas_zscal (const gsl_complex alpha,
          gsl_vector_complex *x)
 -- Function: void gsl_blas_csscal (float alpha,
          gsl_vector_complex_float *x)
 -- Function: void gsl_blas_zdscal (double alpha, gsl_vector_complex *x)

     These functions rescale the vector *note x: 49e. by the
     multiplicative factor *note alpha: 49e.

 -- Function: int gsl_blas_srotg (float a[], float b[], float c[], float
          s[])
 -- Function: int gsl_blas_drotg (double a[], double b[], double c[],
          double s[])

     These functions compute a Givens rotation (c,s) which zeroes the
     vector (a,b),

          [  c  s ] [ a ] = [ r ]
          [ -s  c ] [ b ]   [ 0 ]

     The variables *note a: 4a0. and *note b: 4a0. are overwritten by
     the routine.

 -- Function: int gsl_blas_srot (gsl_vector_float *x, gsl_vector_float
          *y, float c, float s)
 -- Function: int gsl_blas_drot (gsl_vector *x, gsl_vector *y, const
          double c, const double s)

     These functions apply a Givens rotation (x', y') = (c x + s y, -s x
     + c y) to the vectors *note x: 4a2, *note y: 4a2.

 -- Function: int gsl_blas_srotmg (float d1[], float d2[], float b1[],
          float b2, float P[])
 -- Function: int gsl_blas_drotmg (double d1[], double d2[], double
          b1[], double b2, double P[])

     These functions compute a modified Givens transformation.  The
     modified Givens transformation is defined in the original Level-1
     BLAS specification, given in the references.

 -- Function: int gsl_blas_srotm (gsl_vector_float *x, gsl_vector_float
          *y, const float P[])
 -- Function: int gsl_blas_drotm (gsl_vector *x, gsl_vector *y, const
          double P[])

     These functions apply a modified Givens transformation.


File: gsl-ref.info,  Node: Level 2,  Next: Level 3,  Prev: Level 1,  Up: GSL BLAS Interface

13.1.2 Level 2
--------------

 -- Function: int gsl_blas_sgemv (CBLAS_TRANSPOSE_t TransA, float alpha,
          const gsl_matrix_float *A, const gsl_vector_float *x, float
          beta, gsl_vector_float *y)
 -- Function: int gsl_blas_dgemv (CBLAS_TRANSPOSE_t TransA, double
          alpha, const gsl_matrix *A, const gsl_vector *x, double beta,
          gsl_vector *y)
 -- Function: int gsl_blas_cgemv (CBLAS_TRANSPOSE_t TransA, const
          gsl_complex_float alpha, const gsl_matrix_complex_float *A,
          const gsl_vector_complex_float *x, const gsl_complex_float
          beta, gsl_vector_complex_float *y)
 -- Function: int gsl_blas_zgemv (CBLAS_TRANSPOSE_t TransA, const
          gsl_complex alpha, const gsl_matrix_complex *A, const
          gsl_vector_complex *x, const gsl_complex beta,
          gsl_vector_complex *y)

     These functions compute the matrix-vector product and sum y =
     \alpha op(A) x + \beta y, where op(A) = A, A^T, A^H for *note
     TransA: 4ab. = ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’.

 -- Function: int gsl_blas_strmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix_float *A,
          gsl_vector_float *x)
 -- Function: int gsl_blas_dtrmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix *A, gsl_vector *x)
 -- Function: int gsl_blas_ctrmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex_float *A,
          gsl_vector_complex_float *x)
 -- Function: int gsl_blas_ztrmv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex *A,
          gsl_vector_complex *x)

     These functions compute the matrix-vector product x = op(A) x for
     the triangular matrix *note A: 4af, where op(A) = A, A^T, A^H for
     *note TransA: 4af. = ‘CblasNoTrans’, ‘CblasTrans’,
     ‘CblasConjTrans’.  When *note Uplo: 4af. is ‘CblasUpper’ then the
     upper triangle of *note A: 4af. is used, and when *note Uplo: 4af.
     is ‘CblasLower’ then the lower triangle of *note A: 4af. is used.
     If *note Diag: 4af. is ‘CblasNonUnit’ then the diagonal of the
     matrix is used, but if *note Diag: 4af. is ‘CblasUnit’ then the
     diagonal elements of the matrix *note A: 4af. are taken as unity
     and are not referenced.

 -- Function: int gsl_blas_strsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix_float *A,
          gsl_vector_float *x)
 -- Function: int gsl_blas_dtrsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix *A, gsl_vector *x)
 -- Function: int gsl_blas_ctrsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex_float *A,
          gsl_vector_complex_float *x)
 -- Function: int gsl_blas_ztrsv (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          TransA, CBLAS_DIAG_t Diag, const gsl_matrix_complex *A,
          gsl_vector_complex *x)

     These functions compute inv(op(A)) x for *note x: 4b3, where op(A)
     = A, A^T, A^H for *note TransA: 4b3. = ‘CblasNoTrans’,
     ‘CblasTrans’, ‘CblasConjTrans’.  When *note Uplo: 4b3. is
     ‘CblasUpper’ then the upper triangle of *note A: 4b3. is used, and
     when *note Uplo: 4b3. is ‘CblasLower’ then the lower triangle of
     *note A: 4b3. is used.  If *note Diag: 4b3. is ‘CblasNonUnit’ then
     the diagonal of the matrix is used, but if *note Diag: 4b3. is
     ‘CblasUnit’ then the diagonal elements of the matrix *note A: 4b3.
     are taken as unity and are not referenced.

 -- Function: int gsl_blas_ssymv (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_matrix_float *A, const gsl_vector_float *x, float beta,
          gsl_vector_float *y)
 -- Function: int gsl_blas_dsymv (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_matrix *A, const gsl_vector *x, double beta, gsl_vector
          *y)

     These functions compute the matrix-vector product and sum y =
     \alpha A x + \beta y for the symmetric matrix *note A: 4b5.  Since
     the matrix *note A: 4b5. is symmetric only its upper half or lower
     half need to be stored.  When *note Uplo: 4b5. is ‘CblasUpper’ then
     the upper triangle and diagonal of *note A: 4b5. are used, and when
     *note Uplo: 4b5. is ‘CblasLower’ then the lower triangle and
     diagonal of *note A: 4b5. are used.

 -- Function: int gsl_blas_chemv (CBLAS_UPLO_t Uplo, const
          gsl_complex_float alpha, const gsl_matrix_complex_float *A,
          const gsl_vector_complex_float *x, const gsl_complex_float
          beta, gsl_vector_complex_float *y)
 -- Function: int gsl_blas_zhemv (CBLAS_UPLO_t Uplo, const gsl_complex
          alpha, const gsl_matrix_complex *A, const gsl_vector_complex
          *x, const gsl_complex beta, gsl_vector_complex *y)

     These functions compute the matrix-vector product and sum y =
     \alpha A x + \beta y for the hermitian matrix *note A: 4b7.  Since
     the matrix *note A: 4b7. is hermitian only its upper half or lower
     half need to be stored.  When *note Uplo: 4b7. is ‘CblasUpper’ then
     the upper triangle and diagonal of *note A: 4b7. are used, and when
     *note Uplo: 4b7. is ‘CblasLower’ then the lower triangle and
     diagonal of *note A: 4b7. are used.  The imaginary elements of the
     diagonal are automatically assumed to be zero and are not
     referenced.

 -- Function: int gsl_blas_sger (float alpha, const gsl_vector_float *x,
          const gsl_vector_float *y, gsl_matrix_float *A)
 -- Function: int gsl_blas_dger (double alpha, const gsl_vector *x,
          const gsl_vector *y, gsl_matrix *A)
 -- Function: int gsl_blas_cgeru (const gsl_complex_float alpha, const
          gsl_vector_complex_float *x, const gsl_vector_complex_float
          *y, gsl_matrix_complex_float *A)
 -- Function: int gsl_blas_zgeru (const gsl_complex alpha, const
          gsl_vector_complex *x, const gsl_vector_complex *y,
          gsl_matrix_complex *A)

     These functions compute the rank-1 update A = \alpha x y^T + A of
     the matrix *note A: 4bb.

 -- Function: int gsl_blas_cgerc (const gsl_complex_float alpha, const
          gsl_vector_complex_float *x, const gsl_vector_complex_float
          *y, gsl_matrix_complex_float *A)
 -- Function: int gsl_blas_zgerc (const gsl_complex alpha, const
          gsl_vector_complex *x, const gsl_vector_complex *y,
          gsl_matrix_complex *A)

     These functions compute the conjugate rank-1 update A = \alpha x
     y^H + A of the matrix *note A: 4bd.

 -- Function: int gsl_blas_ssyr (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_vector_float *x, gsl_matrix_float *A)
 -- Function: int gsl_blas_dsyr (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_vector *x, gsl_matrix *A)

     These functions compute the symmetric rank-1 update A = \alpha x
     x^T + A of the symmetric matrix *note A: 4bf.  Since the matrix
     *note A: 4bf. is symmetric only its upper half or lower half need
     to be stored.  When *note Uplo: 4bf. is ‘CblasUpper’ then the upper
     triangle and diagonal of *note A: 4bf. are used, and when *note
     Uplo: 4bf. is ‘CblasLower’ then the lower triangle and diagonal of
     *note A: 4bf. are used.

 -- Function: int gsl_blas_cher (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_vector_complex_float *x, gsl_matrix_complex_float *A)
 -- Function: int gsl_blas_zher (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_vector_complex *x, gsl_matrix_complex *A)

     These functions compute the hermitian rank-1 update A = \alpha x
     x^H + A of the hermitian matrix *note A: 4c1.  Since the matrix
     *note A: 4c1. is hermitian only its upper half or lower half need
     to be stored.  When *note Uplo: 4c1. is ‘CblasUpper’ then the upper
     triangle and diagonal of *note A: 4c1. are used, and when *note
     Uplo: 4c1. is ‘CblasLower’ then the lower triangle and diagonal of
     *note A: 4c1. are used.  The imaginary elements of the diagonal are
     automatically set to zero.

 -- Function: int gsl_blas_ssyr2 (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_vector_float *x, const gsl_vector_float *y,
          gsl_matrix_float *A)
 -- Function: int gsl_blas_dsyr2 (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_vector *x, const gsl_vector *y, gsl_matrix *A)

     These functions compute the symmetric rank-2 update A = \alpha x
     y^T + \alpha y x^T + A of the symmetric matrix *note A: 4c3.  Since
     the matrix *note A: 4c3. is symmetric only its upper half or lower
     half need to be stored.  When *note Uplo: 4c3. is ‘CblasUpper’ then
     the upper triangle and diagonal of *note A: 4c3. are used, and when
     *note Uplo: 4c3. is ‘CblasLower’ then the lower triangle and
     diagonal of *note A: 4c3. are used.

 -- Function: int gsl_blas_cher2 (CBLAS_UPLO_t Uplo, const
          gsl_complex_float alpha, const gsl_vector_complex_float *x,
          const gsl_vector_complex_float *y, gsl_matrix_complex_float
          *A)
 -- Function: int gsl_blas_zher2 (CBLAS_UPLO_t Uplo, const gsl_complex
          alpha, const gsl_vector_complex *x, const gsl_vector_complex
          *y, gsl_matrix_complex *A)

     These functions compute the hermitian rank-2 update A = \alpha x
     y^H + \alpha^* y x^H + A of the hermitian matrix *note A: 4c5.
     Since the matrix *note A: 4c5. is hermitian only its upper half or
     lower half need to be stored.  When *note Uplo: 4c5. is
     ‘CblasUpper’ then the upper triangle and diagonal of *note A: 4c5.
     are used, and when *note Uplo: 4c5. is ‘CblasLower’ then the lower
     triangle and diagonal of *note A: 4c5. are used.  The imaginary
     elements of the diagonal are automatically set to zero.


File: gsl-ref.info,  Node: Level 3,  Prev: Level 2,  Up: GSL BLAS Interface

13.1.3 Level 3
--------------

 -- Function: int gsl_blas_sgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, float alpha, const gsl_matrix_float
          *A, const gsl_matrix_float *B, float beta, gsl_matrix_float
          *C)
 -- Function: int gsl_blas_dgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, double alpha, const gsl_matrix *A,
          const gsl_matrix *B, double beta, gsl_matrix *C)
 -- Function: int gsl_blas_cgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, const gsl_complex_float alpha, const
          gsl_matrix_complex_float *A, const gsl_matrix_complex_float
          *B, const gsl_complex_float beta, gsl_matrix_complex_float *C)
 -- Function: int gsl_blas_zgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, const gsl_complex alpha, const
          gsl_matrix_complex *A, const gsl_matrix_complex *B, const
          gsl_complex beta, gsl_matrix_complex *C)

     These functions compute the matrix-matrix product and sum C =
     \alpha op(A) op(B) + \beta C where op(A) = A, A^T, A^H for *note
     TransA: 4ca. = ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’ and
     similarly for the parameter *note TransB: 4ca.

 -- Function: int gsl_blas_ssymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          float alpha, const gsl_matrix_float *A, const gsl_matrix_float
          *B, float beta, gsl_matrix_float *C)
 -- Function: int gsl_blas_dsymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          double alpha, const gsl_matrix *A, const gsl_matrix *B, double
          beta, gsl_matrix *C)
 -- Function: int gsl_blas_csymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex_float alpha, const gsl_matrix_complex_float
          *A, const gsl_matrix_complex_float *B, const gsl_complex_float
          beta, gsl_matrix_complex_float *C)
 -- Function: int gsl_blas_zsymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex alpha, const gsl_matrix_complex *A, const
          gsl_matrix_complex *B, const gsl_complex beta,
          gsl_matrix_complex *C)

     These functions compute the matrix-matrix product and sum C =
     \alpha A B + \beta C for *note Side: 4ce. is ‘CblasLeft’ and C =
     \alpha B A + \beta C for *note Side: 4ce. is ‘CblasRight’, where
     the matrix *note A: 4ce. is symmetric.  When *note Uplo: 4ce. is
     ‘CblasUpper’ then the upper triangle and diagonal of *note A: 4ce.
     are used, and when *note Uplo: 4ce. is ‘CblasLower’ then the lower
     triangle and diagonal of *note A: 4ce. are used.

 -- Function: int gsl_blas_chemm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex_float alpha, const gsl_matrix_complex_float
          *A, const gsl_matrix_complex_float *B, const gsl_complex_float
          beta, gsl_matrix_complex_float *C)
 -- Function: int gsl_blas_zhemm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex alpha, const gsl_matrix_complex *A, const
          gsl_matrix_complex *B, const gsl_complex beta,
          gsl_matrix_complex *C)

     These functions compute the matrix-matrix product and sum C =
     \alpha A B + \beta C for *note Side: 4d0. is ‘CblasLeft’ and C =
     \alpha B A + \beta C for *note Side: 4d0. is ‘CblasRight’, where
     the matrix *note A: 4d0. is hermitian.  When *note Uplo: 4d0. is
     ‘CblasUpper’ then the upper triangle and diagonal of *note A: 4d0.
     are used, and when *note Uplo: 4d0. is ‘CblasLower’ then the lower
     triangle and diagonal of *note A: 4d0. are used.  The imaginary
     elements of the diagonal are automatically set to zero.

 -- Function: int gsl_blas_strmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, float alpha,
          const gsl_matrix_float *A, gsl_matrix_float *B)
 -- Function: int gsl_blas_dtrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, double alpha,
          const gsl_matrix *A, gsl_matrix *B)
 -- Function: int gsl_blas_ctrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_complex_float alpha, const gsl_matrix_complex_float *A,
          gsl_matrix_complex_float *B)
 -- Function: int gsl_blas_ztrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_complex
          alpha, const gsl_matrix_complex *A, gsl_matrix_complex *B)

     These functions compute the matrix-matrix product B = \alpha op(A)
     B for *note Side: 4d4. is ‘CblasLeft’ and B = \alpha B op(A) for
     *note Side: 4d4. is ‘CblasRight’.  The matrix *note A: 4d4. is
     triangular and op(A) = A, A^T, A^H for *note TransA: 4d4. =
     ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’.  When *note Uplo:
     4d4. is ‘CblasUpper’ then the upper triangle of *note A: 4d4. is
     used, and when *note Uplo: 4d4. is ‘CblasLower’ then the lower
     triangle of *note A: 4d4. is used.  If *note Diag: 4d4. is
     ‘CblasNonUnit’ then the diagonal of *note A: 4d4. is used, but if
     *note Diag: 4d4. is ‘CblasUnit’ then the diagonal elements of the
     matrix *note A: 4d4. are taken as unity and are not referenced.

 -- Function: int gsl_blas_strsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, float alpha,
          const gsl_matrix_float *A, gsl_matrix_float *B)
 -- Function: int gsl_blas_dtrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, double alpha,
          const gsl_matrix *A, gsl_matrix *B)
 -- Function: int gsl_blas_ctrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_complex_float alpha, const gsl_matrix_complex_float *A,
          gsl_matrix_complex_float *B)
 -- Function: int gsl_blas_ztrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_complex
          alpha, const gsl_matrix_complex *A, gsl_matrix_complex *B)

     These functions compute the inverse-matrix matrix product B =
     \alpha op(inv(A))B for *note Side: 4d8. is ‘CblasLeft’ and B =
     \alpha B op(inv(A)) for *note Side: 4d8. is ‘CblasRight’.  The
     matrix *note A: 4d8. is triangular and op(A) = A, A^T, A^H for
     *note TransA: 4d8. = ‘CblasNoTrans’, ‘CblasTrans’,
     ‘CblasConjTrans’.  When *note Uplo: 4d8. is ‘CblasUpper’ then the
     upper triangle of *note A: 4d8. is used, and when *note Uplo: 4d8.
     is ‘CblasLower’ then the lower triangle of *note A: 4d8. is used.
     If *note Diag: 4d8. is ‘CblasNonUnit’ then the diagonal of *note A:
     4d8. is used, but if *note Diag: 4d8. is ‘CblasUnit’ then the
     diagonal elements of the matrix *note A: 4d8. are taken as unity
     and are not referenced.

 -- Function: int gsl_blas_ssyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, float alpha, const gsl_matrix_float *A, float beta,
          gsl_matrix_float *C)
 -- Function: int gsl_blas_dsyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, double alpha, const gsl_matrix *A, double beta,
          gsl_matrix *C)
 -- Function: int gsl_blas_csyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, const gsl_complex_float alpha, const
          gsl_matrix_complex_float *A, const gsl_complex_float beta,
          gsl_matrix_complex_float *C)
 -- Function: int gsl_blas_zsyrk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, const gsl_complex alpha, const gsl_matrix_complex *A,
          const gsl_complex beta, gsl_matrix_complex *C)

     These functions compute a rank-k update of the symmetric matrix
     *note C: 4dc, C = \alpha A A^T + \beta C when *note Trans: 4dc. is
     ‘CblasNoTrans’ and C = \alpha A^T A + \beta C when *note Trans:
     4dc. is ‘CblasTrans’.  Since the matrix *note C: 4dc. is symmetric
     only its upper half or lower half need to be stored.  When *note
     Uplo: 4dc. is ‘CblasUpper’ then the upper triangle and diagonal of
     *note C: 4dc. are used, and when *note Uplo: 4dc. is ‘CblasLower’
     then the lower triangle and diagonal of *note C: 4dc. are used.

 -- Function: int gsl_blas_cherk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, float alpha, const gsl_matrix_complex_float *A, float
          beta, gsl_matrix_complex_float *C)
 -- Function: int gsl_blas_zherk (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, double alpha, const gsl_matrix_complex *A, double beta,
          gsl_matrix_complex *C)

     These functions compute a rank-k update of the hermitian matrix
     *note C: 4de, C = \alpha A A^H + \beta C when *note Trans: 4de. is
     ‘CblasNoTrans’ and C = \alpha A^H A + \beta C when *note Trans:
     4de. is ‘CblasConjTrans’.  Since the matrix *note C: 4de. is
     hermitian only its upper half or lower half need to be stored.
     When *note Uplo: 4de. is ‘CblasUpper’ then the upper triangle and
     diagonal of *note C: 4de. are used, and when *note Uplo: 4de. is
     ‘CblasLower’ then the lower triangle and diagonal of *note C: 4de.
     are used.  The imaginary elements of the diagonal are automatically
     set to zero.

 -- Function: int gsl_blas_ssyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, float alpha, const gsl_matrix_float *A, const
          gsl_matrix_float *B, float beta, gsl_matrix_float *C)
 -- Function: int gsl_blas_dsyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, double alpha, const gsl_matrix *A, const gsl_matrix *B,
          double beta, gsl_matrix *C)
 -- Function: int gsl_blas_csyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, const gsl_complex_float alpha, const
          gsl_matrix_complex_float *A, const gsl_matrix_complex_float
          *B, const gsl_complex_float beta, gsl_matrix_complex_float *C)
 -- Function: int gsl_blas_zsyr2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, const gsl_complex alpha, const gsl_matrix_complex *A,
          const gsl_matrix_complex *B, const gsl_complex beta,
          gsl_matrix_complex *C)

     These functions compute a rank-2k update of the symmetric matrix
     *note C: 4e2, C = \alpha A B^T + \alpha B A^T + \beta C when *note
     Trans: 4e2. is ‘CblasNoTrans’ and C = \alpha A^T B + \alpha B^T A +
     \beta C when *note Trans: 4e2. is ‘CblasTrans’.  Since the matrix
     *note C: 4e2. is symmetric only its upper half or lower half need
     to be stored.  When *note Uplo: 4e2. is ‘CblasUpper’ then the upper
     triangle and diagonal of *note C: 4e2. are used, and when *note
     Uplo: 4e2. is ‘CblasLower’ then the lower triangle and diagonal of
     *note C: 4e2. are used.

 -- Function: int gsl_blas_cher2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, const gsl_complex_float alpha, const
          gsl_matrix_complex_float *A, const gsl_matrix_complex_float
          *B, float beta, gsl_matrix_complex_float *C)
 -- Function: int gsl_blas_zher2k (CBLAS_UPLO_t Uplo, CBLAS_TRANSPOSE_t
          Trans, const gsl_complex alpha, const gsl_matrix_complex *A,
          const gsl_matrix_complex *B, double beta, gsl_matrix_complex
          *C)

     These functions compute a rank-2k update of the hermitian matrix
     *note C: 4e4, C = \alpha A B^H + \alpha^* B A^H + \beta C when
     *note Trans: 4e4. is ‘CblasNoTrans’ and C = \alpha A^H B + \alpha^*
     B^H A + \beta C when *note Trans: 4e4. is ‘CblasConjTrans’.  Since
     the matrix *note C: 4e4. is hermitian only its upper half or lower
     half need to be stored.  When *note Uplo: 4e4. is ‘CblasUpper’ then
     the upper triangle and diagonal of *note C: 4e4. are used, and when
     *note Uplo: 4e4. is ‘CblasLower’ then the lower triangle and
     diagonal of *note C: 4e4. are used.  The imaginary elements of the
     diagonal are automatically set to zero.


File: gsl-ref.info,  Node: Examples<8>,  Next: References and Further Reading<8>,  Prev: GSL BLAS Interface,  Up: BLAS Support

13.2 Examples
=============

The following program computes the product of two matrices using the
Level-3 BLAS function DGEMM,

     [ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
     [ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
                         [ 1031 1032 ]

The matrices are stored in row major order, according to the C
convention for arrays.

     #include <stdio.h>
     #include <gsl/gsl_blas.h>

     int
     main (void)
     {
       double a[] = { 0.11, 0.12, 0.13,
                      0.21, 0.22, 0.23 };

       double b[] = { 1011, 1012,
                      1021, 1022,
                      1031, 1032 };

       double c[] = { 0.00, 0.00,
                      0.00, 0.00 };

       gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
       gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
       gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);

       /* Compute C = A B */

       gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
                       1.0, &A.matrix, &B.matrix,
                       0.0, &C.matrix);

       printf ("[ %g, %g\n", c[0], c[1]);
       printf ("  %g, %g ]\n", c[2], c[3]);

       return 0;
     }

Here is the output from the program,

     [ 367.76, 368.12
       674.06, 674.72 ]


File: gsl-ref.info,  Node: References and Further Reading<8>,  Prev: Examples<8>,  Up: BLAS Support

13.3 References and Further Reading
===================================

Information on the BLAS standards, including both the legacy and updated
interface standards, is available online from the BLAS Homepage and BLAS
Technical Forum web-site.

   * BLAS Homepage, ‘http://www.netlib.org/blas/’

   * BLAS Technical Forum, ‘http://www.netlib.org/blas/blast-forum/’

The following papers contain the specifications for Level 1, Level 2 and
Level 3 BLAS.

   * C. Lawson, R. Hanson, D. Kincaid, F. Krogh, “Basic Linear Algebra
     Subprograms for Fortran Usage”, ACM Transactions on Mathematical
     Software, Vol.: 5 (1979), Pages 308–325.

   * J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, “An Extended
     Set of Fortran Basic Linear Algebra Subprograms”, ACM Transactions
     on Mathematical Software, Vol.: 14, No.: 1 (1988), Pages 1–32.

   * J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, “A Set of Level 3
     Basic Linear Algebra Subprograms”, ACM Transactions on Mathematical
     Software, Vol.: 16 (1990), Pages 1–28.

Postscript versions of the latter two papers are available from
‘http://www.netlib.org/blas/’.  A CBLAS wrapper for Fortran BLAS
libraries is available from the same location.


File: gsl-ref.info,  Node: Linear Algebra,  Next: Eigensystems,  Prev: BLAS Support,  Up: Top

14 Linear Algebra
*****************

This chapter describes functions for solving linear systems.  The
library provides linear algebra operations which operate directly on the
*note gsl_vector: 35f. and *note gsl_matrix: 3a2. objects.  These
routines use the standard algorithms from Golub & Van Loan’s `Matrix
Computations' with Level-1 and Level-2 BLAS calls for efficiency.

The functions described in this chapter are declared in the header file
‘gsl_linalg.h’.

* Menu:

* LU Decomposition::
* QR Decomposition::
* QR Decomposition with Column Pivoting::
* LQ Decomposition::
* QL Decomposition::
* Complete Orthogonal Decomposition::
* Singular Value Decomposition::
* Cholesky Decomposition::
* Pivoted Cholesky Decomposition::
* Modified Cholesky Decomposition::
* LDLT Decomposition::
* Tridiagonal Decomposition of Real Symmetric Matrices::
* Tridiagonal Decomposition of Hermitian Matrices::
* Hessenberg Decomposition of Real Matrices::
* Hessenberg-Triangular Decomposition of Real Matrices::
* Bidiagonalization::
* Givens Rotations::
* Householder Transformations::
* Householder solver for linear systems::
* Tridiagonal Systems::
* Triangular Systems::
* Banded Systems::
* Balancing::
* Examples: Examples<9>.
* References and Further Reading: References and Further Reading<9>.


File: gsl-ref.info,  Node: LU Decomposition,  Next: QR Decomposition,  Up: Linear Algebra

14.1 LU Decomposition
=====================

A general M-by-N matrix A has an LU decomposition

     P A = L U

where P is an M-by-M permutation matrix, L is M-by-\min(M,N) and U is
\min(M,N)-by-N. For square matrices, L is a lower unit triangular matrix
and U is upper triangular.  For M > N, L is a unit lower trapezoidal
matrix of size M-by-N. For M < N, U is upper trapezoidal of size M-by-N.
For square matrices this decomposition can be used to convert the linear
system A x = b into a pair of triangular systems (L y = P b, U x = y),
which can be solved by forward and back-substitution.  Note that the LU
decomposition is valid for singular matrices.

 -- Function: int gsl_linalg_LU_decomp (gsl_matrix *A, gsl_permutation
          *p, int *signum)
 -- Function: int gsl_linalg_complex_LU_decomp (gsl_matrix_complex *A,
          gsl_permutation *p, int *signum)

     These functions factorize the matrix *note A: 4ed. into the LU
     decomposition PA = LU. On output the diagonal and upper triangular
     (or trapezoidal) part of the input matrix *note A: 4ed. contain the
     matrix U. The lower triangular (or trapezoidal) part of the input
     matrix (excluding the diagonal) contains L. The diagonal elements
     of L are unity, and are not stored.

     The permutation matrix P is encoded in the permutation *note p:
     4ed. on output.  The j-th column of the matrix P is given by the
     k-th column of the identity matrix, where k = p_j the j-th element
     of the permutation vector.  The sign of the permutation is given by
     *note signum: 4ed.  It has the value (-1)^n, where n is the number
     of interchanges in the permutation.

     The algorithm used in the decomposition is Gaussian Elimination
     with partial pivoting (Golub & Van Loan, `Matrix Computations',
     Algorithm 3.4.1), combined with a recursive algorithm based on
     Level 3 BLAS (Peise and Bientinesi, 2016).

 -- Function: int gsl_linalg_LU_solve (const gsl_matrix *LU, const
          gsl_permutation *p, const gsl_vector *b, gsl_vector *x)
 -- Function: int gsl_linalg_complex_LU_solve (const gsl_matrix_complex
          *LU, const gsl_permutation *p, const gsl_vector_complex *b,
          gsl_vector_complex *x)

     These functions solve the square system A x = b using the LU
     decomposition of A into (*note LU: 4ef, *note p: 4ef.) given by
     *note gsl_linalg_LU_decomp(): 4ec. or *note
     gsl_linalg_complex_LU_decomp(): 4ed. as input.

 -- Function: int gsl_linalg_LU_svx (const gsl_matrix *LU, const
          gsl_permutation *p, gsl_vector *x)
 -- Function: int gsl_linalg_complex_LU_svx (const gsl_matrix_complex
          *LU, const gsl_permutation *p, gsl_vector_complex *x)

     These functions solve the square system A x = b in-place using the
     precomputed LU decomposition of A into (*note LU: 4f1, *note p:
     4f1.).  On input *note x: 4f1. should contain the right-hand side
     b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_LU_refine (const gsl_matrix *A, const
          gsl_matrix *LU, const gsl_permutation *p, const gsl_vector *b,
          gsl_vector *x, gsl_vector *work)
 -- Function: int gsl_linalg_complex_LU_refine (const gsl_matrix_complex
          *A, const gsl_matrix_complex *LU, const gsl_permutation *p,
          const gsl_vector_complex *b, gsl_vector_complex *x,
          gsl_vector_complex *work)

     These functions apply an iterative improvement to *note x: 4f3, the
     solution of A x = b, from the precomputed LU decomposition of A
     into (*note LU: 4f3, *note p: 4f3.).  Additional workspace of
     length ‘N’ is required in *note work: 4f3.

 -- Function: int gsl_linalg_LU_invert (const gsl_matrix *LU, const
          gsl_permutation *p, gsl_matrix *inverse)
 -- Function: int gsl_linalg_complex_LU_invert (const gsl_matrix_complex
          *LU, const gsl_permutation *p, gsl_matrix_complex *inverse)

     These functions compute the inverse of a matrix A from its LU
     decomposition (*note LU: 4f5, *note p: 4f5.), storing the result in
     the matrix *note inverse: 4f5.  The inverse is computed by
     computing the inverses U^{-1}, L^{-1} and finally forming the
     product A^{-1} = U^{-1} L^{-1} P. Each step is based on Level 3
     BLAS calls.

     It is preferable to avoid direct use of the inverse whenever
     possible, as the linear solver functions can obtain the same result
     more efficiently and reliably (consult any introductory textbook on
     numerical linear algebra for details).

 -- Function: int gsl_linalg_LU_invx (gsl_matrix *LU, const
          gsl_permutation *p)
 -- Function: int gsl_linalg_complex_LU_invx (gsl_matrix_complex *LU,
          const gsl_permutation *p)

     These functions compute the inverse of a matrix A from its LU
     decomposition (*note LU: 4f7, *note p: 4f7.), storing the result
     in-place in the matrix *note LU: 4f7.  The inverse is computed by
     computing the inverses U^{-1}, L^{-1} and finally forming the
     product A^{-1} = U^{-1} L^{-1} P. Each step is based on Level 3
     BLAS calls.

     It is preferable to avoid direct use of the inverse whenever
     possible, as the linear solver functions can obtain the same result
     more efficiently and reliably (consult any introductory textbook on
     numerical linear algebra for details).

 -- Function: double gsl_linalg_LU_det (gsl_matrix *LU, int signum)
 -- Function: gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex
          *LU, int signum)

     These functions compute the determinant of a matrix A from its LU
     decomposition, *note LU: 4f9.  The determinant is computed as the
     product of the diagonal elements of U and the sign of the row
     permutation *note signum: 4f9.

 -- Function: double gsl_linalg_LU_lndet (gsl_matrix *LU)
 -- Function: double gsl_linalg_complex_LU_lndet (gsl_matrix_complex
          *LU)

     These functions compute the logarithm of the absolute value of the
     determinant of a matrix A, \ln|\det(A)|, from its LU decomposition,
     *note LU: 4fb.  This function may be useful if the direct
     computation of the determinant would overflow or underflow.

 -- Function: int gsl_linalg_LU_sgndet (gsl_matrix *LU, int signum)
 -- Function: gsl_complex gsl_linalg_complex_LU_sgndet
          (gsl_matrix_complex *LU, int signum)

     These functions compute the sign or phase factor of the determinant
     of a matrix A, \det(A)/|\det(A)|, from its LU decomposition, *note
     LU: 4fd.


File: gsl-ref.info,  Node: QR Decomposition,  Next: QR Decomposition with Column Pivoting,  Prev: LU Decomposition,  Up: Linear Algebra

14.2 QR Decomposition
=====================

A general rectangular M-by-N matrix A has a QR decomposition into the
product of a unitary M-by-M square matrix Q (where Q^{\dagger} Q = I)
and an M-by-N right-triangular matrix R,

     A = Q R

This decomposition can be used to convert the square linear system A x =
b into the triangular system R x = Q^{\dagger} b, which can be solved by
back-substitution.  Another use of the QR decomposition is to compute an
orthonormal basis for a set of vectors.  The first N columns of Q form
an orthonormal basis for the range of A, ran(A), when A has full column
rank.

When M > N, the bottom M - N rows of R are zero, and so A can be
naturally partioned as

     A = [ Q_1 Q_2 ] [ R_1  ] = Q_1 R_1
                     [  0   ]

where R_1 is N-by-N upper triangular, Q_1 is M-by-N, and Q_2 is
M-by-(M-N). Q_1 R_1 is sometimes called the `thin' or `reduced' QR
decomposition.  The solution of the least squares problem \min_x || b -
A x ||^2 when A has full rank is:

     x = R_1^{-1} c_1

where c_1 is the first N elements of Q^{\dagger} b.  If A is rank
deficient, see *note QR Decomposition with Column Pivoting: 500. and
*note Complete Orthogonal Decomposition: 501.

GSL offers two interfaces for the QR decomposition.  The first proceeds
by zeroing out columns below the diagonal of A, one column at a time
using Householder transforms.  In this method, the factor Q is
represented as a product of Householder reflectors:

     Q = H_n \cdots H_2 H_1

where each H_i = I - \tau_i v_i v_i^{\dagger} for a scalar \tau_i and
column vector v_i.  In this method, functions which compute the full
matrix Q or apply Q^{\dagger} to a right hand side vector operate by
applying the Householder matrices one at a time using Level 2 BLAS.

The second interface is based on a Level 3 BLAS block recursive
algorithm developed by Elmroth and Gustavson.  In this case, Q is
written in block form as

     Q = I - V T V^{\dagger}

where V is an M-by-N matrix of the column vectors v_i and T is an N-by-N
upper triangular matrix, whose diagonal elements are the \tau_i.
Computing the full T, while requiring more flops than the Level 2
approach, offers the advantage that all standard operations can take
advantage of cache efficient Level 3 BLAS operations, and so this method
often performs faster than the Level 2 approach.  The functions for the
recursive block algorithm have a ‘_r’ suffix, and it is recommended to
use this interface for performance critical applications.

 -- Function: int gsl_linalg_QR_decomp_r (gsl_matrix *A, gsl_matrix *T)
 -- Function: int gsl_linalg_complex_QR_decomp_r (gsl_matrix_complex *A,
          gsl_matrix_complex *T)

     These functions factor the M-by-N matrix *note A: 503. into the QR
     decomposition A = Q R using the recursive Level 3 BLAS algorithm of
     Elmroth and Gustavson.  On output the diagonal and upper triangular
     part of *note A: 503. contain the matrix R. The N-by-N matrix *note
     T: 503. stores the upper triangular factor appearing in Q. The
     matrix Q is given by Q = I - V T V^{\dagger}, where the elements
     below the diagonal of *note A: 503. contain the columns of V on
     output.

     This algorithm requires M \ge N and performs best for “tall-skinny”
     matrices, i.e.  M \gg N.

 -- Function: int gsl_linalg_QR_solve_r (const gsl_matrix *QR, const
          gsl_matrix *T, const gsl_vector *b, gsl_vector *x)
 -- Function: int gsl_linalg_complex_QR_solve_r (const
          gsl_matrix_complex *QR, const gsl_matrix_complex *T, const
          gsl_vector_complex *b, gsl_vector_complex *x)

     These functions solve the square system A x = b using the QR
     decomposition of A held in (*note QR: 505, *note T: 505.).  The
     least-squares solution for rectangular systems can be found using
     *note gsl_linalg_QR_lssolve_r(): 506. or *note
     gsl_linalg_complex_QR_lssolve_r(): 507.

 -- Function: int gsl_linalg_QR_lssolve_r (const gsl_matrix *QR, const
          gsl_matrix *T, const gsl_vector *b, gsl_vector *x, gsl_vector
          *work)
 -- Function: int gsl_linalg_complex_QR_lssolve_r (const
          gsl_matrix_complex *QR, const gsl_matrix_complex *T, const
          gsl_vector_complex *b, gsl_vector_complex *x,
          gsl_vector_complex *work)

     These functions find the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns.  The least squares solution minimizes the Euclidean
     norm of the residual, ||b - Ax||.  The routine requires as input
     the QR decomposition of A into (*note QR: 507, *note T: 507.) given
     by *note gsl_linalg_QR_decomp_r(): 502. or *note
     gsl_linalg_complex_QR_decomp_r(): 503.  The parameter *note x: 507.
     is of length M. The solution x is returned in the first N rows of
     *note x: 507, i.e.  x = ‘x[0], x[1], ..., x[N-1]’.  The last M - N
     rows of *note x: 507. contain a vector whose norm is equal to the
     residual norm || b - A x ||.  This similar to the behavior of
     LAPACK DGELS. Additional workspace of length N is required in *note
     work: 507.

 -- Function: int gsl_linalg_QR_QTvec_r (const gsl_matrix *QR, const
          gsl_matrix *T, gsl_vector *v, gsl_vector *work)
 -- Function: int gsl_linalg_complex_QR_QHvec_r (const
          gsl_matrix_complex *QR, const gsl_matrix_complex *T,
          gsl_vector_complex *v, gsl_vector_complex *work)

     These functions apply the matrix Q^T (or Q^{\dagger}) encoded in
     the decomposition (*note QR: 509, *note T: 509.) to the vector
     *note v: 509, storing the result Q^T v (or Q^{\dagger} v) in *note
     v: 509.  The matrix multiplication is carried out directly using
     the encoding of the Householder vectors without needing to form the
     full matrix Q^T (or Q^{\dagger}).  Additional workspace of size N
     is required in *note work: 509.

 -- Function: int gsl_linalg_QR_QTmat_r (const gsl_matrix *QR, const
          gsl_matrix *T, gsl_matrix *B, gsl_matrix *work)

     This function applies the matrix Q^T encoded in the decomposition
     (*note QR: 50a, *note T: 50a.) to the M-by-K matrix *note B: 50a,
     storing the result Q^T B in *note B: 50a.  The matrix
     multiplication is carried out directly using the encoding of the
     Householder vectors without needing to form the full matrix Q^T.
     Additional workspace of size N-by-K is required in *note work: 50a.

 -- Function: int gsl_linalg_QR_unpack_r (const gsl_matrix *QR, const
          gsl_matrix *T, gsl_matrix *Q, gsl_matrix *R)
 -- Function: int gsl_linalg_complex_QR_unpack_r (const
          gsl_matrix_complex *QR, const gsl_matrix_complex *T,
          gsl_matrix_complex *Q, gsl_matrix_complex *R)

     These functions unpack the encoded QR decomposition (*note QR: 50c,
     *note T: 50c.) as output from *note gsl_linalg_QR_decomp_r(): 502.
     or *note gsl_linalg_complex_QR_decomp_r(): 503. into the matrices
     *note Q: 50c. and *note R: 50c, where *note Q: 50c. is M-by-M and
     *note R: 50c. is N-by-N. Note that the full R matrix is M-by-N,
     however the lower trapezoidal portion is zero, so only the upper
     triangular factor is stored.

 -- Function: int gsl_linalg_QR_rcond (const gsl_matrix *QR, double
          *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the R factor, stored in the upper triangle of *note QR:
     50d.  The reciprocal condition number estimate, defined as 1 /
     (||R||_1 \cdot ||R^{-1}||_1), is stored in *note rcond: 50d.
     Additional workspace of size 3 N is required in *note work: 50d.

* Menu:

* Level 2 Interface::
* Triangle on Top of Rectangle::
* Triangle on Top of Triangle::
* Triangle on Top of Trapezoidal::
* Triangle on Top of Diagonal::


File: gsl-ref.info,  Node: Level 2 Interface,  Next: Triangle on Top of Rectangle,  Up: QR Decomposition

14.2.1 Level 2 Interface
------------------------

The functions below are for the slower Level 2 interface to the QR
decomposition.  It is recommended to use these functions only for M < N,
since the Level 3 interface above performs much faster for M \ge N.

 -- Function: int gsl_linalg_QR_decomp (gsl_matrix *A, gsl_vector *tau)
 -- Function: int gsl_linalg_complex_QR_decomp (gsl_matrix_complex *A,
          gsl_vector_complex *tau)

     These functions factor the M-by-N matrix *note A: 510. into the QR
     decomposition A = Q R. On output the diagonal and upper triangular
     part of the input matrix contain the matrix R. The vector *note
     tau: 510. and the columns of the lower triangular part of the
     matrix *note A: 510. contain the Householder coefficients and
     Householder vectors which encode the orthogonal matrix ‘Q’.  The
     vector *note tau: 510. must be of length N. The matrix Q is related
     to these components by the product of k=min(M,N) reflector
     matrices, Q = H_k ... H_2 H_1 where H_i = I - \tau_i v_i
     v_i^{\dagger} and v_i is the Householder vector v_i =
     (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)).  This is the same storage
     scheme as used by LAPACK.

     The algorithm used to perform the decomposition is Householder QR
     (Golub & Van Loan, “Matrix Computations”, Algorithm 5.2.1).

 -- Function: int gsl_linalg_QR_solve (const gsl_matrix *QR, const
          gsl_vector *tau, const gsl_vector *b, gsl_vector *x)
 -- Function: int gsl_linalg_complex_QR_solve (const gsl_matrix_complex
          *QR, const gsl_vector_complex *tau, const gsl_vector_complex
          *b, gsl_vector_complex *x)

     These functions solve the square system A x = b using the QR
     decomposition of A held in (*note QR: 512, *note tau: 512.).  The
     least-squares solution for rectangular systems can be found using
     *note gsl_linalg_QR_lssolve(): 513.

 -- Function: int gsl_linalg_QR_svx (const gsl_matrix *QR, const
          gsl_vector *tau, gsl_vector *x)
 -- Function: int gsl_linalg_complex_QR_svx (const gsl_matrix_complex
          *QR, const gsl_vector_complex *tau, gsl_vector_complex *x)

     These functions solve the square system A x = b in-place using the
     QR decomposition of A held in (*note QR: 515, *note tau: 515.).  On
     input *note x: 515. should contain the right-hand side b, which is
     replaced by the solution on output.

 -- Function: int gsl_linalg_QR_lssolve (const gsl_matrix *QR, const
          gsl_vector *tau, const gsl_vector *b, gsl_vector *x,
          gsl_vector *residual)
 -- Function: int gsl_linalg_complex_QR_lssolve (const
          gsl_matrix_complex *QR, const gsl_vector_complex *tau, const
          gsl_vector_complex *b, gsl_vector_complex *x,
          gsl_vector_complex *residual)

     These functions find the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns.  The least squares solution minimizes the Euclidean
     norm of the residual, ||Ax - b||.The routine requires as input the
     QR decomposition of A into (*note QR: 516, *note tau: 516.) given
     by *note gsl_linalg_QR_decomp(): 50f. or *note
     gsl_linalg_complex_QR_decomp(): 510.  The solution is returned in
     *note x: 516.  The residual is computed as a by-product and stored
     in *note residual: 516.

 -- Function: int gsl_linalg_QR_QTvec (const gsl_matrix *QR, const
          gsl_vector *tau, gsl_vector *v)
 -- Function: int gsl_linalg_complex_QR_QHvec (const gsl_matrix_complex
          *QR, const gsl_vector_complex *tau, gsl_vector_complex *v)

     These functions apply the matrix Q^T (or Q^{\dagger}) encoded in
     the decomposition (*note QR: 518, *note tau: 518.) to the vector
     *note v: 518, storing the result Q^T v (or Q^{\dagger} v) in *note
     v: 518.  The matrix multiplication is carried out directly using
     the encoding of the Householder vectors without needing to form the
     full matrix Q^T (or Q^{\dagger}).

 -- Function: int gsl_linalg_QR_Qvec (const gsl_matrix *QR, const
          gsl_vector *tau, gsl_vector *v)
 -- Function: int gsl_linalg_complex_QR_Qvec (const gsl_matrix_complex
          *QR, const gsl_vector_complex *tau, gsl_vector_complex *v)

     These functions apply the matrix Q encoded in the decomposition
     (*note QR: 51a, *note tau: 51a.) to the vector *note v: 51a,
     storing the result Q v in *note v: 51a.  The matrix multiplication
     is carried out directly using the encoding of the Householder
     vectors without needing to form the full matrix Q.

 -- Function: int gsl_linalg_QR_QTmat (const gsl_matrix *QR, const
          gsl_vector *tau, gsl_matrix *B)

     This function applies the matrix Q^T encoded in the decomposition
     (*note QR: 51b, *note tau: 51b.) to the M-by-K matrix *note B: 51b,
     storing the result Q^T B in *note B: 51b.  The matrix
     multiplication is carried out directly using the encoding of the
     Householder vectors without needing to form the full matrix Q^T.

 -- Function: int gsl_linalg_QR_Rsolve (const gsl_matrix *QR, const
          gsl_vector *b, gsl_vector *x)

     This function solves the triangular system R x = b for *note x:
     51c.  It may be useful if the product b' = Q^T b has already been
     computed using *note gsl_linalg_QR_QTvec(): 517.

 -- Function: int gsl_linalg_QR_Rsvx (const gsl_matrix *QR, gsl_vector
          *x)

     This function solves the triangular system R x = b for *note x:
     51d. in-place.  On input *note x: 51d. should contain the
     right-hand side b and is replaced by the solution on output.  This
     function may be useful if the product b' = Q^T b has already been
     computed using *note gsl_linalg_QR_QTvec(): 517.

 -- Function: int gsl_linalg_QR_unpack (const gsl_matrix *QR, const
          gsl_vector *tau, gsl_matrix *Q, gsl_matrix *R)

     This function unpacks the encoded QR decomposition (*note QR: 51e,
     *note tau: 51e.) into the matrices *note Q: 51e. and *note R: 51e,
     where *note Q: 51e. is M-by-M and *note R: 51e. is M-by-N.

 -- Function: int gsl_linalg_QR_QRsolve (gsl_matrix *Q, gsl_matrix *R,
          const gsl_vector *b, gsl_vector *x)

     This function solves the system R x = Q^T b for *note x: 51f.  It
     can be used when the QR decomposition of a matrix is available in
     unpacked form as (*note Q: 51f, *note R: 51f.).

 -- Function: int gsl_linalg_QR_update (gsl_matrix *Q, gsl_matrix *R,
          gsl_vector *w, const gsl_vector *v)

     This function performs a rank-1 update w v^T of the QR
     decomposition (*note Q: 520, *note R: 520.).  The update is given
     by Q'R' = Q (R + w v^T) where the output matrices Q and R are also
     orthogonal and right triangular.  Note that *note w: 520. is
     destroyed by the update.

 -- Function: int gsl_linalg_R_solve (const gsl_matrix *R, const
          gsl_vector *b, gsl_vector *x)

     This function solves the triangular system R x = b for the N-by-N
     matrix *note R: 521.

 -- Function: int gsl_linalg_R_svx (const gsl_matrix *R, gsl_vector *x)

     This function solves the triangular system R x = b in-place.  On
     input *note x: 522. should contain the right-hand side b, which is
     replaced by the solution on output.


File: gsl-ref.info,  Node: Triangle on Top of Rectangle,  Next: Triangle on Top of Triangle,  Prev: Level 2 Interface,  Up: QR Decomposition

14.2.2 Triangle on Top of Rectangle
-----------------------------------

This section provides routines for computing the QR decomposition of the
specialized matrix

     [ U ] = Q R
     [ A ]

where U is an N-by-N upper triangular matrix, and A is an M-by-N dense
matrix.  This type of matrix arises, for example, in the sequential TSQR
algorithm.  The Elmroth and Gustavson algorithm is used to efficiently
factor this matrix.  Due to the upper triangular factor, the Q matrix
takes the form

     Q = I - V T V^T

with

     V = [ I ]
         [ Y ]

and Y is dense and of the same dimensions as A.

 -- Function: int gsl_linalg_QR_UR_decomp (gsl_matrix *U, gsl_matrix *A,
          gsl_matrix *T)

     This function computes the QR decomposition of the matrix (U ; A),
     where U is N-by-N upper triangular and A is M-by-N dense.  On
     output, U is replaced by the R factor, and A is replaced by Y. The
     N-by-N upper triangular block reflector is stored in *note T: 524.
     on output.


File: gsl-ref.info,  Node: Triangle on Top of Triangle,  Next: Triangle on Top of Trapezoidal,  Prev: Triangle on Top of Rectangle,  Up: QR Decomposition

14.2.3 Triangle on Top of Triangle
----------------------------------

This section provides routines for computing the QR decomposition of the
specialized matrix

     [ U_1 ] = Q R
     [ U_2 ]

where U_1,U_2 are N-by-N upper triangular matrices.  The Elmroth and
Gustavson algorithm is used to efficiently factor this matrix.  The Q
matrix takes the form

     Q = I - V T V^T

with

     V = [ I ]
         [ Y ]

and Y is N-by-N upper triangular.

 -- Function: int gsl_linalg_QR_UU_decomp (gsl_matrix *U1, gsl_matrix
          *U2, gsl_matrix *T)

     This function computes the QR decomposition of the matrix (U_1 ;
     U_2), where U_1,U_2 are N-by-N upper triangular.  On output, *note
     U1: 526. is replaced by the R factor, and *note U2: 526. is
     replaced by Y. The N-by-N upper triangular block reflector is
     stored in *note T: 526. on output.

 -- Function: int gsl_linalg_QR_UU_lssolve (const gsl_matrix *R, const
          gsl_matrix *Y, const gsl_matrix *T, const gsl_vector *b,
          gsl_vector *x, gsl_vector *work)

     This function find the least squares solution to the overdetermined
     system,

          \min_x \left| \left| b - \begin{pmatrix} U_1 \\ U_2 \end{pmatrix} x \right| \right|^2

     where U_1,U_2 are N-by-N upper triangular matrices.  The routine
     requires as input the QR decomposition of (U_1; U_2) into (*note R:
     527, *note Y: 527, *note T: 527.) given by *note
     gsl_linalg_QR_UU_decomp(): 526.  The parameter *note x: 527. is of
     length 2N. The solution x is returned in the first N rows of *note
     x: 527, i.e.  x = ‘x[0], x[1], ..., x[N-1]’.  The last N rows of
     *note x: 527. contain a vector whose norm is equal to the residual
     norm || b - (U_1; U_2) x ||.  This similar to the behavior of
     LAPACK DGELS. Additional workspace of length N is required in *note
     work: 527.

 -- Function: int gsl_linalg_QR_UU_QTec (const gsl_matrix *Y, const
          gsl_matrix *T, gsl_vector *b, gsl_vector *work)

     This function computes Q^T b using the decomposition (*note Y: 528,
     *note T: 528.) previously computed by *note
     gsl_linalg_QR_UU_decomp(): 526.  On input, *note b: 528. contains
     the vector b, and on output it will contain Q^T b.  Additional
     workspace of length N is required in *note work: 528.


File: gsl-ref.info,  Node: Triangle on Top of Trapezoidal,  Next: Triangle on Top of Diagonal,  Prev: Triangle on Top of Triangle,  Up: QR Decomposition

14.2.4 Triangle on Top of Trapezoidal
-------------------------------------

This section provides routines for computing the QR decomposition of the
specialized matrix

     [ U ] = Q R
     [ A ]

where U is an N-by-N upper triangular matrix, and A is an M-by-N upper
trapezoidal matrix with M \ge N. A has the structure,

     A = [ A_d ]
         [ A_u ]

where A_d is (M-N)-by-N dense, and A_u is N-by-N upper triangular.  The
Elmroth and Gustavson algorithm is used to efficiently factor this
matrix.  The Q matrix takes the form

     Q = I - V T V^T

with

     V = [ I ]
         [ Y ]

and Y is upper trapezoidal and of the same dimensions as A.

 -- Function: int gsl_linalg_QR_UZ_decomp (gsl_matrix *U, gsl_matrix *A,
          gsl_matrix *T)

     This function computes the QR decomposition of the matrix (U ; A),
     where U is N-by-N upper triangular and A is M-by-N upper
     trapezoidal.  On output, U is replaced by the R factor, and A is
     replaced by Y. The N-by-N upper triangular block reflector is
     stored in *note T: 52a. on output.


File: gsl-ref.info,  Node: Triangle on Top of Diagonal,  Prev: Triangle on Top of Trapezoidal,  Up: QR Decomposition

14.2.5 Triangle on Top of Diagonal
----------------------------------

This section provides routines for computing the QR decomposition of the
specialized matrix

     [ U ] = Q R
     [ D ]

where U is an N-by-N upper triangular matrix and D is an N-by-N diagonal
matrix.  This type of matrix arises in regularized least squares
problems.  The Elmroth and Gustavson algorithm is used to efficiently
factor this matrix.  The Q matrix takes the form

     Q = I - V T V^T

with

     V = [ I ]
         [ Y ]

and Y is N-by-N upper triangular.

 -- Function: int gsl_linalg_QR_UD_decomp (gsl_matrix *U, const
          gsl_vector *D, gsl_matrix *Y, gsl_matrix *T)

     This function computes the QR decomposition of the matrix (U ; D),
     where U is N-by-N upper triangular and D is N-by-N diagonal.  On
     output, *note U: 52c. is replaced by the R factor and Y is stored
     in *note Y: 52c.  The N-by-N upper triangular block reflector is
     stored in *note T: 52c. on output.

 -- Function: int gsl_linalg_QR_UD_lssolve (const gsl_matrix *R, const
          gsl_matrix *Y, const gsl_matrix *T, const gsl_vector *b,
          gsl_vector *x, gsl_vector *work)

     This function find the least squares solution to the overdetermined
     system,

          \min_x \left| \left| b - \begin{pmatrix} U \\ D \end{pmatrix} x \right| \right|^2

     where U is N-by-N upper triangular and D is N-by-N diagonal.  The
     routine requires as input the QR decomposition of (U; D) into
     (*note R: 52d, *note Y: 52d, *note T: 52d.) given by *note
     gsl_linalg_QR_UD_decomp(): 52c.  The parameter *note x: 52d. is of
     length 2N. The solution x is returned in the first N rows of *note
     x: 52d, i.e.  x = ‘x[0], x[1], ..., x[N-1]’.  The last N rows of
     *note x: 52d. contain a vector whose norm is equal to the residual
     norm || b - (U; D) x ||.  This similar to the behavior of LAPACK
     DGELS. Additional workspace of length N is required in *note work:
     52d.


File: gsl-ref.info,  Node: QR Decomposition with Column Pivoting,  Next: LQ Decomposition,  Prev: QR Decomposition,  Up: Linear Algebra

14.3 QR Decomposition with Column Pivoting
==========================================

The QR decomposition of an M-by-N matrix A can be extended to the rank
deficient case by introducing a column permutation P,

     A P = Q R

The first r columns of Q form an orthonormal basis for the range of A
for a matrix with column rank r.  This decomposition can also be used to
convert the square linear system A x = b into the triangular system R y
= Q^T b, x = P y, which can be solved by back-substitution and
permutation.  We denote the QR decomposition with column pivoting by
QRP^T since A = Q R P^T. When A is rank deficient with r = {\rm
rank}(A), the matrix R can be partitioned as

     R = [ R11 R12 ] =~ [ R11 R12 ]
         [  0  R22 ]    [  0   0  ]

where R_{11} is r-by-r and nonsingular.  In this case, a `basic' least
squares solution for the overdetermined system A x = b can be obtained
as

     x = P [ R11^-1 c1 ]
           [     0     ]

where c_1 consists of the first r elements of Q^T b.  This basic
solution is not guaranteed to be the minimum norm solution unless R_{12}
= 0 (see *note Complete Orthogonal Decomposition: 501.).

 -- Function: int gsl_linalg_QRPT_decomp (gsl_matrix *A, gsl_vector
          *tau, gsl_permutation *p, int *signum, gsl_vector *norm)

     This function factorizes the M-by-N matrix *note A: 52f. into the
     QRP^T decomposition A = Q R P^T. On output the diagonal and upper
     triangular part of the input matrix contain the matrix R. The
     permutation matrix P is stored in the permutation *note p: 52f.
     The sign of the permutation is given by *note signum: 52f.  It has
     the value (-1)^n, where n is the number of interchanges in the
     permutation.  The vector *note tau: 52f. and the columns of the
     lower triangular part of the matrix *note A: 52f. contain the
     Householder coefficients and vectors which encode the orthogonal
     matrix ‘Q’.  The vector *note tau: 52f. must be of length
     k=\min(M,N). The matrix Q is related to these components by, Q =
     Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the
     Householder vector

          v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))

     This is the same storage scheme as used by LAPACK. The vector *note
     norm: 52f. is a workspace of length ‘N’ used for column pivoting.

     The algorithm used to perform the decomposition is Householder QR
     with column pivoting (Golub & Van Loan, “Matrix Computations”,
     Algorithm 5.4.1).

 -- Function: int gsl_linalg_QRPT_decomp2 (const gsl_matrix *A,
          gsl_matrix *q, gsl_matrix *r, gsl_vector *tau, gsl_permutation
          *p, int *signum, gsl_vector *norm)

     This function factorizes the matrix *note A: 530. into the
     decomposition A = Q R P^T without modifying *note A: 530. itself
     and storing the output in the separate matrices *note q: 530. and
     *note r: 530.

 -- Function: int gsl_linalg_QRPT_solve (const gsl_matrix *QR, const
          gsl_vector *tau, const gsl_permutation *p, const gsl_vector
          *b, gsl_vector *x)

     This function solves the square system A x = b using the QRP^T
     decomposition of A held in (*note QR: 531, *note tau: 531, *note p:
     531.) which must have been computed previously by *note
     gsl_linalg_QRPT_decomp(): 52f.

 -- Function: int gsl_linalg_QRPT_svx (const gsl_matrix *QR, const
          gsl_vector *tau, const gsl_permutation *p, gsl_vector *x)

     This function solves the square system A x = b in-place using the
     QRP^T decomposition of A held in (*note QR: 532, *note tau: 532,
     *note p: 532.).  On input *note x: 532. should contain the
     right-hand side b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_QRPT_lssolve (const gsl_matrix *QR, const
          gsl_vector *tau, const gsl_permutation *p, const gsl_vector
          *b, gsl_vector *x, gsl_vector *residual)

     This function finds the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns and is assumed to have full rank.  The least squares
     solution minimizes the Euclidean norm of the residual, ||b - A x||.
     The routine requires as input the QR decomposition of A into (*note
     QR: 533, *note tau: 533, *note p: 533.) given by *note
     gsl_linalg_QRPT_decomp(): 52f.  The solution is returned in *note
     x: 533.  The residual is computed as a by-product and stored in
     *note residual: 533.  For rank deficient matrices, *note
     gsl_linalg_QRPT_lssolve2(): 534. should be used instead.

 -- Function: int gsl_linalg_QRPT_lssolve2 (const gsl_matrix *QR, const
          gsl_vector *tau, const gsl_permutation *p, const gsl_vector
          *b, const size_t rank, gsl_vector *x, gsl_vector *residual)

     This function finds the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns and has rank given by the input *note rank: 534.  If
     the user does not know the rank of A, the routine *note
     gsl_linalg_QRPT_rank(): 535. can be called to estimate it.  The
     least squares solution is the so-called “basic” solution discussed
     above and may not be the minimum norm solution.  The routine
     requires as input the QR decomposition of A into (*note QR: 534,
     *note tau: 534, *note p: 534.) given by *note
     gsl_linalg_QRPT_decomp(): 52f.  The solution is returned in *note
     x: 534.  The residual is computed as a by-product and stored in
     *note residual: 534.

 -- Function: int gsl_linalg_QRPT_QRsolve (const gsl_matrix *Q, const
          gsl_matrix *R, const gsl_permutation *p, const gsl_vector *b,
          gsl_vector *x)

     This function solves the square system R P^T x = Q^T b for *note x:
     536.  It can be used when the QR decomposition of a matrix is
     available in unpacked form as (*note Q: 536, *note R: 536.).

 -- Function: int gsl_linalg_QRPT_update (gsl_matrix *Q, gsl_matrix *R,
          const gsl_permutation *p, gsl_vector *w, const gsl_vector *v)

     This function performs a rank-1 update w v^T of the QRP^T
     decomposition (*note Q: 537, *note R: 537, *note p: 537.).  The
     update is given by Q'R' = Q (R + w v^T P) where the output matrices
     Q' and R' are also orthogonal and right triangular.  Note that
     *note w: 537. is destroyed by the update.  The permutation *note p:
     537. is not changed.

 -- Function: int gsl_linalg_QRPT_Rsolve (const gsl_matrix *QR, const
          gsl_permutation *p, const gsl_vector *b, gsl_vector *x)

     This function solves the triangular system R P^T x = b for the
     N-by-N matrix R contained in *note QR: 538.

 -- Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix *QR, const
          gsl_permutation *p, gsl_vector *x)

     This function solves the triangular system R P^T x = b in-place for
     the N-by-N matrix R contained in *note QR: 539.  On input *note x:
     539. should contain the right-hand side b, which is replaced by the
     solution on output.

 -- Function: size_t gsl_linalg_QRPT_rank (const gsl_matrix *QR, const
          double tol)

     This function estimates the rank of the triangular matrix R
     contained in *note QR: 535.  The algorithm simply counts the number
     of diagonal elements of R whose absolute value is greater than the
     specified tolerance *note tol: 535.  If the input *note tol: 535.
     is negative, a default value of 20 (M + N) eps(max(|diag(R)|)) is
     used.

 -- Function: int gsl_linalg_QRPT_rcond (const gsl_matrix *QR, double
          *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the R factor, stored in the upper triangle of *note QR:
     53a.  The reciprocal condition number estimate, defined as 1 /
     (||R||_1 \cdot ||R^{-1}||_1), is stored in *note rcond: 53a.
     Additional workspace of size 3 N is required in *note work: 53a.


File: gsl-ref.info,  Node: LQ Decomposition,  Next: QL Decomposition,  Prev: QR Decomposition with Column Pivoting,  Up: Linear Algebra

14.4 LQ Decomposition
=====================

A general rectangular M-by-N matrix A has a LQ decomposition into the
product of a lower trapezoidal M-by-N matrix L and an orthogonal N-by-N
square matrix Q:

     A = L Q

If M \le N, then L can be written as L = (L_1 \quad 0) where L_1 is
M-by-M lower triangular, and

     A = \begin{pmatrix} L_1 & 0 \end{pmatrix} \begin{pmatrix} Q_1 \\ Q_2 \end{pmatrix} = L_1 Q_1

where Q_1 consists of the first M rows of Q, and Q_2 contains the
remaining N - M rows.  The LQ factorization of A is essentially the same
as the *note QR factorization: 4fe. of A^T.

The LQ factorization may be used to find the minimum norm solution of an
underdetermined system of equations A x = b, where A is M-by-N and M \le
N. The solution is

     x = Q^T \begin{pmatrix} L_1^{-1} b \\ 0 \end{pmatrix}

 -- Function: int gsl_linalg_LQ_decomp (gsl_matrix *A, gsl_vector *tau)

     This function factorizes the M-by-N matrix *note A: 53c. into the
     LQ decomposition A = L Q. On output the diagonal and lower
     trapezoidal part of the input matrix contain the matrix L. The
     vector *note tau: 53c. and the elements above the diagonal of the
     matrix *note A: 53c. contain the Householder coefficients and
     Householder vectors which encode the orthogonal matrix ‘Q’.  The
     vector *note tau: 53c. must be of length k=\min(M,N). The matrix Q
     is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i =
     I - \tau_i v_i v_i^T and v_i is the Householder vector v_i =
     (0,...,1,A(i,i+1),A(i,i+2),...,A(i,N)). This is the same storage
     scheme as used by LAPACK.

 -- Function: int gsl_linalg_LQ_lssolve (const gsl_matrix *LQ, const
          gsl_vector *tau, const gsl_vector *b, gsl_vector *x,
          gsl_vector *residual)

     This function finds the minimum norm least squares solution to the
     underdetermined system A x = b, where the M-by-N matrix ‘A’ has M
     \le N. The routine requires as input the LQ decomposition of A into
     (*note LQ: 53d, *note tau: 53d.) given by *note
     gsl_linalg_LQ_decomp(): 53c.  The solution is returned in *note x:
     53d.  The residual, b - Ax, is computed as a by-product and stored
     in *note residual: 53d.

 -- Function: int gsl_linalg_LQ_unpack (const gsl_matrix *LQ, const
          gsl_vector *tau, gsl_matrix *Q, gsl_matrix *L)

     This function unpacks the encoded LQ decomposition (*note LQ: 53e,
     *note tau: 53e.) into the matrices *note Q: 53e. and *note L: 53e,
     where *note Q: 53e. is N-by-N and *note L: 53e. is M-by-N.

 -- Function: int gsl_linalg_LQ_QTvec (const gsl_matrix *LQ, const
          gsl_vector *tau, gsl_vector *v)

     This function applies Q^T to the vector *note v: 53f, storing the
     result Q^T v in *note v: 53f. on output.


File: gsl-ref.info,  Node: QL Decomposition,  Next: Complete Orthogonal Decomposition,  Prev: LQ Decomposition,  Up: Linear Algebra

14.5 QL Decomposition
=====================

A general rectangular M-by-N matrix A has a QL decomposition into the
product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and an
M-by-N left-triangular matrix L.

When M \ge N, the decomposition is given by

     A = [ 0 ; L_1 ]

where L_1 is N-by-N lower triangular.  When M \le N, the decomposition
is given by

     A = [ L_1 L_2 ]

where L_1 is a dense M-by-N-M matrix and L_2 is a lower triangular
M-by-M matrix.

 -- Function: int gsl_linalg_QL_decomp (gsl_matrix *A, gsl_vector *tau)

     This function factorizes the M-by-N matrix *note A: 541. into the
     QL decomposition A = Q L. The vector *note tau: 541. must be of
     length N and contains the Householder coefficients on output.  The
     matrix Q is stored in packed form in *note A: 541. on output, using
     the same storage scheme as LAPACK.

 -- Function: int gsl_linalg_QL_unpack (const gsl_matrix *QL, const
          gsl_vector *tau, gsl_matrix *Q, gsl_matrix *L)

     This function unpacks the encoded QL decomposition (*note QL: 542,
     *note tau: 542.) into the matrices *note Q: 542. and *note L: 542,
     where *note Q: 542. is M-by-M and *note L: 542. is M-by-N.


File: gsl-ref.info,  Node: Complete Orthogonal Decomposition,  Next: Singular Value Decomposition,  Prev: QL Decomposition,  Up: Linear Algebra

14.6 Complete Orthogonal Decomposition
======================================

The complete orthogonal decomposition of a M-by-N matrix A is a
generalization of the QR decomposition with column pivoting, given by

     A P = Q [ R11 0 ] Z^T
             [  0  0 ]

where P is a N-by-N permutation matrix, Q is M-by-M orthogonal, R_{11}
is r-by-r upper triangular, with r = {\rm rank}(A), and Z is N-by-N
orthogonal.  If A has full rank, then R_{11} = R, Z = I and this reduces
to the QR decomposition with column pivoting.

For a rank deficient least squares problem, \min_x{|| b - Ax||^2}, the
solution vector x is not unique.  However if we further require that
||x||^2 is minimized, then the complete orthogonal decomposition gives
us the ability to compute the unique minimum norm solution, which is
given by

     x = P Z [ R11^-1 c1 ]
             [     0     ]

and the vector c_1 is the first r elements of Q^T b.

The COD also enables a straightforward solution of regularized least
squares problems in Tikhonov standard form, written as

     \min_x ||b - A x||^2 + \lambda^2 ||x||^2

where \lambda > 0 is a regularization parameter which represents a
tradeoff between minimizing the residual norm ||b - A x|| and the
solution norm ||x||.  For this system, the solution is given by

     x = P Z [ y1 ]
             [ 0  ]

where y_1 is a vector of length r which is found by solving

     [     R11     ] y_1 = [ c_1 ]
     [ \lambda I_r ]       [  0  ]

and c_1 is defined above.  The equation above can be solved efficiently
for different values of \lambda using QR factorizations of the left hand
side matrix.

 -- Function: int gsl_linalg_COD_decomp (gsl_matrix *A, gsl_vector
          *tau_Q, gsl_vector *tau_Z, gsl_permutation *p, size_t *rank,
          gsl_vector *work)
 -- Function: int gsl_linalg_COD_decomp_e (gsl_matrix *A, gsl_vector
          *tau_Q, gsl_vector *tau_Z, gsl_permutation *p, double tol,
          size_t *rank, gsl_vector *work)

     These functions factor the M-by-N matrix *note A: 545. into the
     decomposition A = Q R Z P^T. The rank of *note A: 545. is computed
     as the number of diagonal elements of R greater than the tolerance
     *note tol: 545. and output in *note rank: 545.  If *note tol: 545.
     is not specified, a default value is used (see *note
     gsl_linalg_QRPT_rank(): 535.).  On output, the permutation matrix P
     is stored in *note p: 545.  The matrix R_{11} is stored in the
     upper *note rank: 545.-by-*note rank: 545. block of *note A: 545.
     The matrices Q and Z are encoded in packed storage in *note A: 545.
     on output.  The vectors *note tau_Q: 545. and *note tau_Z: 545.
     contain the Householder scalars corresponding to the matrices Q and
     Z respectively and must be of length k = \min(M,N). The vector
     *note work: 545. is additional workspace of length N.

 -- Function: int gsl_linalg_COD_lssolve (const gsl_matrix *QRZT, const
          gsl_vector *tau_Q, const gsl_vector *tau_Z, const
          gsl_permutation *p, const size_t rank, const gsl_vector *b,
          gsl_vector *x, gsl_vector *residual)

     This function finds the unique minimum norm least squares solution
     to the overdetermined system A x = b where the matrix ‘A’ has more
     rows than columns.  The least squares solution minimizes the
     Euclidean norm of the residual, ||b - A x|| as well as the norm of
     the solution ||x||.  The routine requires as input the QRZT
     decomposition of A into (*note QRZT: 546, *note tau_Q: 546, *note
     tau_Z: 546, *note p: 546, *note rank: 546.) given by *note
     gsl_linalg_COD_decomp(): 544.  The solution is returned in *note x:
     546.  The residual, b - Ax, is computed as a by-product and stored
     in *note residual: 546.

 -- Function: int gsl_linalg_COD_lssolve2 (const double lambda, const
          gsl_matrix *QRZT, const gsl_vector *tau_Q, const gsl_vector
          *tau_Z, const gsl_permutation *p, const size_t rank, const
          gsl_vector *b, gsl_vector *x, gsl_vector *residual, gsl_matrix
          *S, gsl_vector *work)

     This function finds the solution to the regularized least squares
     problem in Tikhonov standard form, \min_x ||b - Ax||^2 + \lambda^2
     ||x||^2.  The routine requires as input the QRZT decomposition of A
     into (*note QRZT: 547, *note tau_Q: 547, *note tau_Z: 547, *note p:
     547, *note rank: 547.) given by *note gsl_linalg_COD_decomp(): 544.
     The parameter \lambda is supplied in *note lambda: 547.  The
     solution is returned in *note x: 547.  The residual, b - Ax, is
     stored in *note residual: 547. on output.  *note S: 547. is
     additional workspace of size *note rank: 547.-by-*note rank: 547.
     *note work: 547. is additional workspace of length *note rank: 547.

 -- Function: int gsl_linalg_COD_unpack (const gsl_matrix *QRZT, const
          gsl_vector *tau_Q, const gsl_vector *tau_Z, const size_t rank,
          gsl_matrix *Q, gsl_matrix *R, gsl_matrix *Z)

     This function unpacks the encoded QRZT decomposition (*note QRZT:
     548, *note tau_Q: 548, *note tau_Z: 548, *note rank: 548.) into the
     matrices *note Q: 548, *note R: 548, and *note Z: 548, where *note
     Q: 548. is M-by-M, *note R: 548. is M-by-N, and *note Z: 548. is
     N-by-N.

 -- Function: int gsl_linalg_COD_matZ (const gsl_matrix *QRZT, const
          gsl_vector *tau_Z, const size_t rank, gsl_matrix *A,
          gsl_vector *work)

     This function multiplies the input matrix *note A: 549. on the
     right by ‘Z’, A' = A Z using the encoded QRZT decomposition (*note
     QRZT: 549, *note tau_Z: 549, *note rank: 549.).  *note A: 549. must
     have N columns but may have any number of rows.  Additional
     workspace of length M is provided in *note work: 549.


File: gsl-ref.info,  Node: Singular Value Decomposition,  Next: Cholesky Decomposition,  Prev: Complete Orthogonal Decomposition,  Up: Linear Algebra

14.7 Singular Value Decomposition
=================================

A general rectangular M-by-N matrix A has a singular value decomposition
(SVD) into the product of an M-by-N orthogonal matrix U, an N-by-N
diagonal matrix of singular values S and the transpose of an N-by-N
orthogonal square matrix V,

     A = U S V^T

The singular values \sigma_i = S_{ii} are all non-negative and are
generally chosen to form a non-increasing sequence

     \sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0

The singular value decomposition of a matrix has many practical uses.
The condition number of the matrix is given by the ratio of the largest
singular value to the smallest singular value.  The presence of a zero
singular value indicates that the matrix is singular.  The number of
non-zero singular values indicates the rank of the matrix.  In practice
singular value decomposition of a rank-deficient matrix will not produce
exact zeroes for singular values, due to finite numerical precision.
Small singular values should be edited by choosing a suitable tolerance.

For a rank-deficient matrix, the null space of A is given by the columns
of V corresponding to the zero singular values.  Similarly, the range of
A is given by columns of U corresponding to the non-zero singular
values.

Note that the routines here compute the “thin” version of the SVD with U
as M-by-N orthogonal matrix.  This allows in-place computation and is
the most commonly-used form in practice.  Mathematically, the “full” SVD
is defined with U as an M-by-M orthogonal matrix and S as an M-by-N
diagonal matrix (with additional rows of zeros).

 -- Function: int gsl_linalg_SV_decomp (gsl_matrix *A, gsl_matrix *V,
          gsl_vector *S, gsl_vector *work)

     This function factorizes the M-by-N matrix *note A: 54b. into the
     singular value decomposition A = U S V^T for M \ge N. On output the
     matrix *note A: 54b. is replaced by U. The diagonal elements of the
     singular value matrix S are stored in the vector *note S: 54b.  The
     singular values are non-negative and form a non-increasing sequence
     from S_1 to S_N. The matrix *note V: 54b. contains the elements of
     V in untransposed form.  To form the product U S V^T it is
     necessary to take the transpose of *note V: 54b.  A workspace of
     length ‘N’ is required in *note work: 54b.

     This routine uses the Golub-Reinsch SVD algorithm.

 -- Function: int gsl_linalg_SV_decomp_mod (gsl_matrix *A, gsl_matrix
          *X, gsl_matrix *V, gsl_vector *S, gsl_vector *work)

     This function computes the SVD using the modified Golub-Reinsch
     algorithm, which is faster for M \gg N. It requires the vector
     *note work: 54c. of length ‘N’ and the N-by-N matrix *note X: 54c.
     as additional working space.

 -- Function: int gsl_linalg_SV_decomp_jacobi (gsl_matrix *A, gsl_matrix
          *V, gsl_vector *S)

     This function computes the SVD of the M-by-N matrix *note A: 54d.
     using one-sided Jacobi orthogonalization for M \ge N. The Jacobi
     method can compute singular values to higher relative accuracy than
     Golub-Reinsch algorithms (see references for details).

 -- Function: int gsl_linalg_SV_solve (const gsl_matrix *U, const
          gsl_matrix *V, const gsl_vector *S, const gsl_vector *b,
          gsl_vector *x)

     This function solves the system A x = b using the singular value
     decomposition (*note U: 54e, *note S: 54e, *note V: 54e.) of A
     which must have been computed previously with *note
     gsl_linalg_SV_decomp(): 54b.

     Only non-zero singular values are used in computing the solution.
     The parts of the solution corresponding to singular values of zero
     are ignored.  Other singular values can be edited out by setting
     them to zero before calling this function.

     In the over-determined case where ‘A’ has more rows than columns
     the system is solved in the least squares sense, returning the
     solution *note x: 54e. which minimizes ||A x - b||_2.

 -- Function: int gsl_linalg_SV_leverage (const gsl_matrix *U,
          gsl_vector *h)

     This function computes the statistical leverage values h_i of a
     matrix A using its singular value decomposition (*note U: 54f, ‘S’,
     ‘V’) previously computed with *note gsl_linalg_SV_decomp(): 54b.
     h_i are the diagonal values of the matrix A (A^T A)^{-1} A^T and
     depend only on the matrix *note U: 54f. which is the input to this
     function.


File: gsl-ref.info,  Node: Cholesky Decomposition,  Next: Pivoted Cholesky Decomposition,  Prev: Singular Value Decomposition,  Up: Linear Algebra

14.8 Cholesky Decomposition
===========================

A symmetric, positive definite square matrix A has a Cholesky
decomposition into a product of a lower triangular matrix L and its
transpose L^T,

     A = L L^T

This is sometimes referred to as taking the square-root of a matrix.
The Cholesky decomposition can only be carried out when all the
eigenvalues of the matrix are positive.  This decomposition can be used
to convert the linear system A x = b into a pair of triangular systems
(L y = b, L^T x = y), which can be solved by forward and
back-substitution.

If the matrix A is near singular, it is sometimes possible to reduce the
condition number and recover a more accurate solution vector x by
scaling as

     ( S A S ) ( S^(-1) x ) = S b

where S is a diagonal matrix whose elements are given by S_{ii} =
1/\sqrt{A_{ii}}.  This scaling is also known as Jacobi preconditioning.
There are routines below to solve both the scaled and unscaled systems.

 -- Function: int gsl_linalg_cholesky_decomp1 (gsl_matrix *A)
 -- Function: int gsl_linalg_complex_cholesky_decomp (gsl_matrix_complex
          *A)

     These functions factorize the symmetric, positive-definite square
     matrix *note A: 552. into the Cholesky decomposition A = L L^T (or
     A = L L^{\dagger} for the complex case).  On input, the values from
     the diagonal and lower-triangular part of the matrix *note A: 552.
     are used (the upper triangular part is ignored).  On output the
     diagonal and lower triangular part of the input matrix *note A:
     552. contain the matrix L, while the upper triangular part contains
     the original matrix.  If the matrix is not positive-definite then
     the decomposition will fail, returning the error code *note
     GSL_EDOM: 28.

     When testing whether a matrix is positive-definite, disable the
     error handler first to avoid triggering an error.  These functions
     use Level 3 BLAS to compute the Cholesky factorization (Peise and
     Bientinesi, 2016).

 -- Function: int gsl_linalg_cholesky_decomp (gsl_matrix *A)

     This function is now deprecated and is provided only for backward
     compatibility.

 -- Function: int gsl_linalg_cholesky_solve (const gsl_matrix *cholesky,
          const gsl_vector *b, gsl_vector *x)
 -- Function: int gsl_linalg_complex_cholesky_solve (const
          gsl_matrix_complex *cholesky, const gsl_vector_complex *b,
          gsl_vector_complex *x)

     These functions solve the system A x = b using the Cholesky
     decomposition of A held in the matrix *note cholesky: 555. which
     must have been previously computed by *note
     gsl_linalg_cholesky_decomp(): 553. or *note
     gsl_linalg_complex_cholesky_decomp(): 552.

 -- Function: int gsl_linalg_cholesky_svx (const gsl_matrix *cholesky,
          gsl_vector *x)
 -- Function: int gsl_linalg_complex_cholesky_svx (const
          gsl_matrix_complex *cholesky, gsl_vector_complex *x)

     These functions solve the system A x = b in-place using the
     Cholesky decomposition of A held in the matrix *note cholesky: 557.
     which must have been previously computed by *note
     gsl_linalg_cholesky_decomp(): 553. or *note
     gsl_linalg_complex_cholesky_decomp(): 552.  On input *note x: 557.
     should contain the right-hand side b, which is replaced by the
     solution on output.

 -- Function: int gsl_linalg_cholesky_invert (gsl_matrix *cholesky)
 -- Function: int gsl_linalg_complex_cholesky_invert (gsl_matrix_complex
          *cholesky)

     These functions compute the inverse of a matrix from its Cholesky
     decomposition *note cholesky: 559, which must have been previously
     computed by *note gsl_linalg_cholesky_decomp(): 553. or *note
     gsl_linalg_complex_cholesky_decomp(): 552.  On output, the inverse
     is stored in-place in *note cholesky: 559.

 -- Function: int gsl_linalg_cholesky_decomp2 (gsl_matrix *A, gsl_vector
          *S)

     This function calculates a diagonal scaling transformation S for
     the symmetric, positive-definite square matrix *note A: 55a, and
     then computes the Cholesky decomposition S A S = L L^T. On input,
     the values from the diagonal and lower-triangular part of the
     matrix *note A: 55a. are used (the upper triangular part is
     ignored).  On output the diagonal and lower triangular part of the
     input matrix *note A: 55a. contain the matrix L, while the upper
     triangular part of the input matrix is overwritten with L^T (the
     diagonal terms being identical for both L and L^T). If the matrix
     is not positive-definite then the decomposition will fail,
     returning the error code *note GSL_EDOM: 28.  The diagonal scale
     factors are stored in *note S: 55a. on output.

     When testing whether a matrix is positive-definite, disable the
     error handler first to avoid triggering an error.

 -- Function: int gsl_linalg_cholesky_solve2 (const gsl_matrix
          *cholesky, const gsl_vector *S, const gsl_vector *b,
          gsl_vector *x)

     This function solves the system (S A S) (S^{-1} x) = S b using the
     Cholesky decomposition of S A S held in the matrix *note cholesky:
     55b. which must have been previously computed by *note
     gsl_linalg_cholesky_decomp2(): 55a.

 -- Function: int gsl_linalg_cholesky_svx2 (const gsl_matrix *cholesky,
          const gsl_vector *S, gsl_vector *x)

     This function solves the system (S A S) (S^{-1} x) = S b in-place
     using the Cholesky decomposition of S A S held in the matrix *note
     cholesky: 55c. which must have been previously computed by *note
     gsl_linalg_cholesky_decomp2(): 55a.  On input *note x: 55c. should
     contain the right-hand side b, which is replaced by the solution on
     output.

 -- Function: int gsl_linalg_cholesky_scale (const gsl_matrix *A,
          gsl_vector *S)

     This function calculates a diagonal scaling transformation of the
     symmetric, positive definite matrix *note A: 55d, such that S A S
     has a condition number within a factor of N of the matrix of
     smallest possible condition number over all possible diagonal
     scalings.  On output, *note S: 55d. contains the scale factors,
     given by S_i = 1/\sqrt{A_{ii}}.  For any A_{ii} \le 0, the
     corresponding scale factor S_i is set to 1.

 -- Function: int gsl_linalg_cholesky_scale_apply (gsl_matrix *A, const
          gsl_vector *S)

     This function applies the scaling transformation *note S: 55e. to
     the matrix *note A: 55e.  On output, *note A: 55e. is replaced by S
     A S.

 -- Function: int gsl_linalg_cholesky_rcond (const gsl_matrix *cholesky,
          double *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric positive definite matrix A, using its
     Cholesky decomposition provided in *note cholesky: 55f.  The
     reciprocal condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in *note rcond: 55f.  Additional workspace
     of size 3 N is required in *note work: 55f.


File: gsl-ref.info,  Node: Pivoted Cholesky Decomposition,  Next: Modified Cholesky Decomposition,  Prev: Cholesky Decomposition,  Up: Linear Algebra

14.9 Pivoted Cholesky Decomposition
===================================

A symmetric positive semi-definite square matrix A has an alternate
Cholesky decomposition into a product of a lower unit triangular matrix
L, a diagonal matrix D and L^T, given by L D L^T. For postive definite
matrices, this is equivalent to the Cholesky formulation discussed
above, with the standard Cholesky lower triangular factor given by L
D^{1 \over 2}.  For ill-conditioned matrices, it can help to use a
pivoting strategy to prevent the entries of D and L from growing too
large, and also ensure D_1 \ge D_2 \ge \cdots \ge D_n > 0, where D_i are
the diagonal entries of D. The final decomposition is given by

     P A P^T = L D L^T

where P is a permutation matrix.

 -- Function: int gsl_linalg_pcholesky_decomp (gsl_matrix *A,
          gsl_permutation *p)

     This function factors the symmetric, positive-definite square
     matrix *note A: 561. into the Pivoted Cholesky decomposition P A
     P^T = L D L^T. On input, the values from the diagonal and
     lower-triangular part of the matrix *note A: 561. are used to
     construct the factorization.  On output the diagonal of the input
     matrix *note A: 561. stores the diagonal elements of D, and the
     lower triangular portion of *note A: 561. contains the matrix L.
     Since L has ones on its diagonal these do not need to be
     explicitely stored.  The upper triangular portion of *note A: 561.
     is unmodified.  The permutation matrix P is stored in *note p: 561.
     on output.

 -- Function: int gsl_linalg_pcholesky_solve (const gsl_matrix *LDLT,
          const gsl_permutation *p, const gsl_vector *b, gsl_vector *x)

     This function solves the system A x = b using the Pivoted Cholesky
     decomposition of A held in the matrix *note LDLT: 562. and
     permutation *note p: 562. which must have been previously computed
     by *note gsl_linalg_pcholesky_decomp(): 561.

 -- Function: int gsl_linalg_pcholesky_svx (const gsl_matrix *LDLT,
          const gsl_permutation *p, gsl_vector *x)

     This function solves the system A x = b in-place using the Pivoted
     Cholesky decomposition of A held in the matrix *note LDLT: 563. and
     permutation *note p: 563. which must have been previously computed
     by *note gsl_linalg_pcholesky_decomp(): 561.  On input, *note x:
     563. contains the right hand side vector b which is replaced by the
     solution vector on output.

 -- Function: int gsl_linalg_pcholesky_decomp2 (gsl_matrix *A,
          gsl_permutation *p, gsl_vector *S)

     This function computes the pivoted Cholesky factorization of the
     matrix S A S, where the input matrix *note A: 564. is symmetric and
     positive definite, and the diagonal scaling matrix *note S: 564. is
     computed to reduce the condition number of *note A: 564. as much as
     possible.  See *note Cholesky Decomposition: 3f5. for more
     information on the matrix *note S: 564.  The Pivoted Cholesky
     decomposition satisfies P S A S P^T = L D L^T. On input, the values
     from the diagonal and lower-triangular part of the matrix *note A:
     564. are used to construct the factorization.  On output the
     diagonal of the input matrix *note A: 564. stores the diagonal
     elements of D, and the lower triangular portion of *note A: 564.
     contains the matrix L. Since L has ones on its diagonal these do
     not need to be explicitely stored.  The upper triangular portion of
     *note A: 564. is unmodified.  The permutation matrix P is stored in
     *note p: 564. on output.  The diagonal scaling transformation is
     stored in *note S: 564. on output.

 -- Function: int gsl_linalg_pcholesky_solve2 (const gsl_matrix *LDLT,
          const gsl_permutation *p, const gsl_vector *S, const
          gsl_vector *b, gsl_vector *x)

     This function solves the system (S A S) (S^{-1} x) = S b using the
     Pivoted Cholesky decomposition of S A S held in the matrix *note
     LDLT: 565, permutation *note p: 565, and vector *note S: 565, which
     must have been previously computed by *note
     gsl_linalg_pcholesky_decomp2(): 564.

 -- Function: int gsl_linalg_pcholesky_svx2 (const gsl_matrix *LDLT,
          const gsl_permutation *p, const gsl_vector *S, gsl_vector *x)

     This function solves the system (S A S) (S^{-1} x) = S b in-place
     using the Pivoted Cholesky decomposition of S A S held in the
     matrix *note LDLT: 566, permutation *note p: 566. and vector *note
     S: 566, which must have been previously computed by *note
     gsl_linalg_pcholesky_decomp2(): 564.  On input, *note x: 566.
     contains the right hand side vector b which is replaced by the
     solution vector on output.

 -- Function: int gsl_linalg_pcholesky_invert (const gsl_matrix *LDLT,
          const gsl_permutation *p, gsl_matrix *Ainv)

     This function computes the inverse of the matrix A, using the
     Pivoted Cholesky decomposition stored in *note LDLT: 567. and *note
     p: 567.  On output, the matrix *note Ainv: 567. contains A^{-1}.

 -- Function: int gsl_linalg_pcholesky_rcond (const gsl_matrix *LDLT,
          const gsl_permutation *p, double *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric positive definite matrix A, using its
     pivoted Cholesky decomposition provided in *note LDLT: 568.  The
     reciprocal condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in *note rcond: 568.  Additional workspace
     of size 3 N is required in *note work: 568.


File: gsl-ref.info,  Node: Modified Cholesky Decomposition,  Next: LDLT Decomposition,  Prev: Pivoted Cholesky Decomposition,  Up: Linear Algebra

14.10 Modified Cholesky Decomposition
=====================================

The modified Cholesky decomposition is suitable for solving systems A x
= b where A is a symmetric indefinite matrix.  Such matrices arise in
nonlinear optimization algorithms.  The standard Cholesky decomposition
requires a positive definite matrix and would fail in this case.
Instead of resorting to a method like QR or SVD, which do not take into
account the symmetry of the matrix, we can instead introduce a small
perturbation to the matrix A to make it positive definite, and then use
a Cholesky decomposition on the perturbed matrix.  The resulting
decomposition satisfies

     P (A + E) P^T = L D L^T

where P is a permutation matrix, E is a diagonal perturbation matrix, L
is unit lower triangular, and D is diagonal.  If A is sufficiently
positive definite, then the perturbation matrix E will be zero and this
method is equivalent to the pivoted Cholesky algorithm.  For indefinite
matrices, the perturbation matrix E is computed to ensure that A + E is
positive definite and well conditioned.

 -- Function: int gsl_linalg_mcholesky_decomp (gsl_matrix *A,
          gsl_permutation *p, gsl_vector *E)

     This function factors the symmetric, indefinite square matrix *note
     A: 56a. into the Modified Cholesky decomposition P (A + E) P^T = L
     D L^T. On input, the values from the diagonal and lower-triangular
     part of the matrix *note A: 56a. are used to construct the
     factorization.  On output the diagonal of the input matrix *note A:
     56a. stores the diagonal elements of D, and the lower triangular
     portion of *note A: 56a. contains the matrix L. Since L has ones on
     its diagonal these do not need to be explicitely stored.  The upper
     triangular portion of *note A: 56a. is unmodified.  The permutation
     matrix P is stored in *note p: 56a. on output.  The diagonal
     perturbation matrix is stored in *note E: 56a. on output.  The
     parameter *note E: 56a. may be set to NULL if it is not required.

 -- Function: int gsl_linalg_mcholesky_solve (const gsl_matrix *LDLT,
          const gsl_permutation *p, const gsl_vector *b, gsl_vector *x)

     This function solves the perturbed system (A + E) x = b using the
     Cholesky decomposition of A + E held in the matrix *note LDLT: 56b.
     and permutation *note p: 56b. which must have been previously
     computed by *note gsl_linalg_mcholesky_decomp(): 56a.

 -- Function: int gsl_linalg_mcholesky_svx (const gsl_matrix *LDLT,
          const gsl_permutation *p, gsl_vector *x)

     This function solves the perturbed system (A + E) x = b in-place
     using the Cholesky decomposition of A + E held in the matrix *note
     LDLT: 56c. and permutation *note p: 56c. which must have been
     previously computed by *note gsl_linalg_mcholesky_decomp(): 56a.
     On input, *note x: 56c. contains the right hand side vector b which
     is replaced by the solution vector on output.

 -- Function: int gsl_linalg_mcholesky_rcond (const gsl_matrix *LDLT,
          const gsl_permutation *p, double *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the perturbed matrix A + E, using its pivoted Cholesky
     decomposition provided in *note LDLT: 56d.  The reciprocal
     condition number estimate, defined as 1 / (||A + E||_1 \cdot ||(A +
     E)^{-1}||_1), is stored in *note rcond: 56d.  Additional workspace
     of size 3 N is required in *note work: 56d.


File: gsl-ref.info,  Node: LDLT Decomposition,  Next: Tridiagonal Decomposition of Real Symmetric Matrices,  Prev: Modified Cholesky Decomposition,  Up: Linear Algebra

14.11 LDLT Decomposition
========================

If A is a symmetric, nonsingular square matrix, then it has a unique
factorization of the form

     A = L D L^T

where L is a unit lower triangular matrix and D is diagonal.  If A is
positive definite, then this factorization is equivalent to the Cholesky
factorization, where the lower triangular Cholesky factor is L
D^{\frac{1}{2}}.  Some indefinite matrices for which no Cholesky
decomposition exists have an L D L^T decomposition with negative entries
in D. The L D L^T algorithm is sometimes referred to as the `square root
free' Cholesky decomposition, as the algorithm does not require the
computation of square roots.  The algorithm is stable for positive
definite matrices, but is not guaranteed to be stable for indefinite
matrices.

 -- Function: int gsl_linalg_ldlt_decomp (gsl_matrix *A)

     This function factorizes the symmetric, non-singular square matrix
     *note A: 570. into the decomposition A = L D L^T. On input, the
     values from the diagonal and lower-triangular part of the matrix
     *note A: 570. are used.  The upper triangle of *note A: 570. is
     used as temporary workspace.  On output the diagonal of *note A:
     570. contains the matrix D and the lower triangle of *note A: 570.
     contains the unit lower triangular matrix L. The matrix 1-norm,
     ||A||_1 is stored in the upper right corner on output, for later
     use by *note gsl_linalg_ldlt_rcond(): 571.

     If the matrix is detected to be singular, the function returns the
     error code *note GSL_EDOM: 28.

 -- Function: int gsl_linalg_ldlt_solve (const gsl_matrix *LDLT, const
          gsl_vector *b, gsl_vector *x)

     This function solves the system A x = b using the L D L^T
     decomposition of A held in the matrix *note LDLT: 572. which must
     have been previously computed by *note gsl_linalg_ldlt_decomp():
     570.

 -- Function: int gsl_linalg_ldlt_svx (const gsl_matrix *LDLT,
          gsl_vector *x)

     This function solves the system A x = b in-place using the L D L^T
     decomposition of A held in the matrix *note LDLT: 573. which must
     have been previously computed by *note gsl_linalg_ldlt_decomp():
     570.  On input *note x: 573. should contain the right-hand side b,
     which is replaced by the solution on output.

 -- Function: int gsl_linalg_ldlt_rcond (const gsl_matrix *LDLT, double
          *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric nonsingular matrix A, using its L D L^T
     decomposition provided in *note LDLT: 571.  The reciprocal
     condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in *note rcond: 571.  Additional workspace
     of size 3 N is required in *note work: 571.


File: gsl-ref.info,  Node: Tridiagonal Decomposition of Real Symmetric Matrices,  Next: Tridiagonal Decomposition of Hermitian Matrices,  Prev: LDLT Decomposition,  Up: Linear Algebra

14.12 Tridiagonal Decomposition of Real Symmetric Matrices
==========================================================

A symmetric matrix A can be factorized by similarity transformations
into the form,

     A = Q T Q^T

where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.

 -- Function: int gsl_linalg_symmtd_decomp (gsl_matrix *A, gsl_vector
          *tau)

     This function factorizes the symmetric square matrix *note A: 575.
     into the symmetric tridiagonal decomposition Q T Q^T. On output the
     diagonal and subdiagonal part of the input matrix *note A: 575.
     contain the tridiagonal matrix T. The remaining lower triangular
     part of the input matrix contains the Householder vectors which,
     together with the Householder coefficients *note tau: 575, encode
     the orthogonal matrix Q. This storage scheme is the same as used by
     LAPACK. The upper triangular part of *note A: 575. is not
     referenced.

 -- Function: int gsl_linalg_symmtd_unpack (const gsl_matrix *A, const
          gsl_vector *tau, gsl_matrix *Q, gsl_vector *diag, gsl_vector
          *subdiag)

     This function unpacks the encoded symmetric tridiagonal
     decomposition (*note A: 576, *note tau: 576.) obtained from *note
     gsl_linalg_symmtd_decomp(): 575. into the orthogonal matrix *note
     Q: 576, the vector of diagonal elements *note diag: 576. and the
     vector of subdiagonal elements *note subdiag: 576.

 -- Function: int gsl_linalg_symmtd_unpack_T (const gsl_matrix *A,
          gsl_vector *diag, gsl_vector *subdiag)

     This function unpacks the diagonal and subdiagonal of the encoded
     symmetric tridiagonal decomposition (*note A: 577, ‘tau’) obtained
     from *note gsl_linalg_symmtd_decomp(): 575. into the vectors *note
     diag: 577. and *note subdiag: 577.


File: gsl-ref.info,  Node: Tridiagonal Decomposition of Hermitian Matrices,  Next: Hessenberg Decomposition of Real Matrices,  Prev: Tridiagonal Decomposition of Real Symmetric Matrices,  Up: Linear Algebra

14.13 Tridiagonal Decomposition of Hermitian Matrices
=====================================================

A hermitian matrix A can be factorized by similarity transformations
into the form,

     A = U T U^T

where U is a unitary matrix and T is a real symmetric tridiagonal
matrix.

 -- Function: int gsl_linalg_hermtd_decomp (gsl_matrix_complex *A,
          gsl_vector_complex *tau)

     This function factorizes the hermitian matrix *note A: 579. into
     the symmetric tridiagonal decomposition U T U^T. On output the real
     parts of the diagonal and subdiagonal part of the input matrix
     *note A: 579. contain the tridiagonal matrix T. The remaining lower
     triangular part of the input matrix contains the Householder
     vectors which, together with the Householder coefficients *note
     tau: 579, encode the unitary matrix U. This storage scheme is the
     same as used by LAPACK. The upper triangular part of *note A: 579.
     and imaginary parts of the diagonal are not referenced.

 -- Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex *A,
          const gsl_vector_complex *tau, gsl_matrix_complex *U,
          gsl_vector *diag, gsl_vector *subdiag)

     This function unpacks the encoded tridiagonal decomposition (*note
     A: 57a, *note tau: 57a.) obtained from *note
     gsl_linalg_hermtd_decomp(): 579. into the unitary matrix *note U:
     57a, the real vector of diagonal elements *note diag: 57a. and the
     real vector of subdiagonal elements *note subdiag: 57a.

 -- Function: int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex
          *A, gsl_vector *diag, gsl_vector *subdiag)

     This function unpacks the diagonal and subdiagonal of the encoded
     tridiagonal decomposition (*note A: 57b, ‘tau’) obtained from the
     *note gsl_linalg_hermtd_decomp(): 579. into the real vectors *note
     diag: 57b. and *note subdiag: 57b.


File: gsl-ref.info,  Node: Hessenberg Decomposition of Real Matrices,  Next: Hessenberg-Triangular Decomposition of Real Matrices,  Prev: Tridiagonal Decomposition of Hermitian Matrices,  Up: Linear Algebra

14.14 Hessenberg Decomposition of Real Matrices
===============================================

A general real matrix A can be decomposed by orthogonal similarity
transformations into the form

     A = U H U^T

where U is orthogonal and H is an upper Hessenberg matrix, meaning that
it has zeros below the first subdiagonal.  The Hessenberg reduction is
the first step in the Schur decomposition for the nonsymmetric
eigenvalue problem, but has applications in other areas as well.

 -- Function: int gsl_linalg_hessenberg_decomp (gsl_matrix *A,
          gsl_vector *tau)

     This function computes the Hessenberg decomposition of the matrix
     *note A: 57d. by applying the similarity transformation H = U^T A
     U. On output, H is stored in the upper portion of *note A: 57d.
     The information required to construct the matrix U is stored in the
     lower triangular portion of *note A: 57d.  U is a product of N - 2
     Householder matrices.  The Householder vectors are stored in the
     lower portion of *note A: 57d. (below the subdiagonal) and the
     Householder coefficients are stored in the vector *note tau: 57d.
     *note tau: 57d. must be of length ‘N’.

 -- Function: int gsl_linalg_hessenberg_unpack (gsl_matrix *H,
          gsl_vector *tau, gsl_matrix *U)

     This function constructs the orthogonal matrix U from the
     information stored in the Hessenberg matrix *note H: 57e. along
     with the vector *note tau: 57e.  *note H: 57e. and *note tau: 57e.
     are outputs from *note gsl_linalg_hessenberg_decomp(): 57d.

 -- Function: int gsl_linalg_hessenberg_unpack_accum (gsl_matrix *H,
          gsl_vector *tau, gsl_matrix *V)

     This function is similar to *note gsl_linalg_hessenberg_unpack():
     57e, except it accumulates the matrix ‘U’ into *note V: 57f, so
     that V' = VU. The matrix *note V: 57f. must be initialized prior to
     calling this function.  Setting *note V: 57f. to the identity
     matrix provides the same result as *note
     gsl_linalg_hessenberg_unpack(): 57e.  If *note H: 57f. is order
     ‘N’, then *note V: 57f. must have ‘N’ columns but may have any
     number of rows.

 -- Function: int gsl_linalg_hessenberg_set_zero (gsl_matrix *H)

     This function sets the lower triangular portion of *note H: 580,
     below the subdiagonal, to zero.  It is useful for clearing out the
     Householder vectors after calling *note
     gsl_linalg_hessenberg_decomp(): 57d.


File: gsl-ref.info,  Node: Hessenberg-Triangular Decomposition of Real Matrices,  Next: Bidiagonalization,  Prev: Hessenberg Decomposition of Real Matrices,  Up: Linear Algebra

14.15 Hessenberg-Triangular Decomposition of Real Matrices
==========================================================

A general real matrix pair (A, B) can be decomposed by orthogonal
similarity transformations into the form

     A = U H V^T
     B = U R V^T

where U and V are orthogonal, H is an upper Hessenberg matrix, and R is
upper triangular.  The Hessenberg-Triangular reduction is the first step
in the generalized Schur decomposition for the generalized eigenvalue
problem.

 -- Function: int gsl_linalg_hesstri_decomp (gsl_matrix *A, gsl_matrix
          *B, gsl_matrix *U, gsl_matrix *V, gsl_vector *work)

     This function computes the Hessenberg-Triangular decomposition of
     the matrix pair (*note A: 582, *note B: 582.).  On output, H is
     stored in *note A: 582, and R is stored in *note B: 582.  If *note
     U: 582. and *note V: 582. are provided (they may be null), the
     similarity transformations are stored in them.  Additional
     workspace of length N is needed in *note work: 582.


File: gsl-ref.info,  Node: Bidiagonalization,  Next: Givens Rotations,  Prev: Hessenberg-Triangular Decomposition of Real Matrices,  Up: Linear Algebra

14.16 Bidiagonalization
=======================

A general matrix A can be factorized by similarity transformations into
the form,

     A = U B V^T

where U and V are orthogonal matrices and B is a N-by-N bidiagonal
matrix with non-zero entries only on the diagonal and superdiagonal.
The size of ‘U’ is M-by-N and the size of ‘V’ is N-by-N.

 -- Function: int gsl_linalg_bidiag_decomp (gsl_matrix *A, gsl_vector
          *tau_U, gsl_vector *tau_V)

     This function factorizes the M-by-N matrix *note A: 584. into
     bidiagonal form U B V^T. The diagonal and superdiagonal of the
     matrix B are stored in the diagonal and superdiagonal of *note A:
     584.  The orthogonal matrices U and ‘V’ are stored as compressed
     Householder vectors in the remaining elements of *note A: 584.  The
     Householder coefficients are stored in the vectors *note tau_U:
     584. and *note tau_V: 584.  The length of *note tau_U: 584. must
     equal the number of elements in the diagonal of *note A: 584. and
     the length of *note tau_V: 584. should be one element shorter.

 -- Function: int gsl_linalg_bidiag_unpack (const gsl_matrix *A, const
          gsl_vector *tau_U, gsl_matrix *U, const gsl_vector *tau_V,
          gsl_matrix *V, gsl_vector *diag, gsl_vector *superdiag)

     This function unpacks the bidiagonal decomposition of *note A: 585.
     produced by *note gsl_linalg_bidiag_decomp(): 584, (*note A: 585,
     *note tau_U: 585, *note tau_V: 585.) into the separate orthogonal
     matrices *note U: 585, *note V: 585. and the diagonal vector *note
     diag: 585. and superdiagonal *note superdiag: 585.  Note that *note
     U: 585. is stored as a compact M-by-N orthogonal matrix satisfying
     U^T U = I for efficiency.

 -- Function: int gsl_linalg_bidiag_unpack2 (gsl_matrix *A, gsl_vector
          *tau_U, gsl_vector *tau_V, gsl_matrix *V)

     This function unpacks the bidiagonal decomposition of *note A: 586.
     produced by *note gsl_linalg_bidiag_decomp(): 584, (*note A: 586,
     *note tau_U: 586, *note tau_V: 586.) into the separate orthogonal
     matrices ‘U’, *note V: 586. and the diagonal vector ‘diag’ and
     superdiagonal ‘superdiag’.  The matrix ‘U’ is stored in-place in
     *note A: 586.

 -- Function: int gsl_linalg_bidiag_unpack_B (const gsl_matrix *A,
          gsl_vector *diag, gsl_vector *superdiag)

     This function unpacks the diagonal and superdiagonal of the
     bidiagonal decomposition of *note A: 587. from *note
     gsl_linalg_bidiag_decomp(): 584, into the diagonal vector *note
     diag: 587. and superdiagonal vector *note superdiag: 587.


File: gsl-ref.info,  Node: Givens Rotations,  Next: Householder Transformations,  Prev: Bidiagonalization,  Up: Linear Algebra

14.17 Givens Rotations
======================

A Givens rotation is a rotation in the plane acting on two elements of a
given vector.  It can be represented in matrix form as

where the \cos{\theta} and \sin{\theta} appear at the intersection of
the i-th and j-th rows and columns.  When acting on a vector x,
G(i,j,\theta) x performs a rotation of the (i,j) elements of x.  Givens
rotations are typically used to introduce zeros in vectors, such as
during the QR decomposition of a matrix.  In this case, it is typically
desired to find c and s such that

     [ c -s ] [ a ] = [ r ]
     [ s  c ] [ b ]   [ 0 ]

with r = \sqrt{a^2 + b^2}.

 -- Function: void gsl_linalg_givens (const double a, const double b,
          double *c, double *s)

     This function computes c = \cos{\theta} and s = \sin{\theta} so
     that the Givens matrix G(\theta) acting on the vector (a,b)
     produces (r, 0), with r = \sqrt{a^2 + b^2}.

 -- Function: void gsl_linalg_givens_gv (gsl_vector *v, const size_t i,
          const size_t j, const double c, const double s)

     This function applies the Givens rotation defined by c =
     \cos{\theta} and s = \sin{\theta} to the *note i: 58a. and *note j:
     58a. elements of *note v: 58a.  On output, (v(i),v(j)) \leftarrow
     G(\theta) (v(i),v(j)).


File: gsl-ref.info,  Node: Householder Transformations,  Next: Householder solver for linear systems,  Prev: Givens Rotations,  Up: Linear Algebra

14.18 Householder Transformations
=================================

A Householder transformation is a rank-1 modification of the identity
matrix which can be used to zero out selected elements of a vector.  A
Householder matrix H takes the form,

     H = I - \tau v v^T

where v is a vector (called the `Householder vector') and \tau = 2/(v^T
v).  The functions described in this section use the rank-1 structure of
the Householder matrix to create and apply Householder transformations
efficiently.

 -- Function: double gsl_linalg_householder_transform (gsl_vector *w)
 -- Function: gsl_complex gsl_linalg_complex_householder_transform
          (gsl_vector_complex *w)

     This function prepares a Householder transformation H = I - \tau v
     v^T which can be used to zero all the elements of the input vector
     *note w: 58d. except the first.  On output the Householder vector
     ‘v’ is stored in *note w: 58d. and the scalar \tau is returned.
     The householder vector ‘v’ is normalized so that ‘v[0] = 1’,
     however this 1 is not stored in the output vector.  Instead, ‘w[0]’
     is set to the first element of the transformed vector, so that if u
     = H w, ‘w[0] = u[0]’ on output and the remainder of u is zero.

 -- Function: int gsl_linalg_householder_hm (double tau, const
          gsl_vector *v, gsl_matrix *A)
 -- Function: int gsl_linalg_complex_householder_hm (gsl_complex tau,
          const gsl_vector_complex *v, gsl_matrix_complex *A)

     This function applies the Householder matrix H defined by the
     scalar *note tau: 58f. and the vector *note v: 58f. to the
     left-hand side of the matrix *note A: 58f.  On output the result H
     A is stored in *note A: 58f.

 -- Function: int gsl_linalg_householder_mh (double tau, const
          gsl_vector *v, gsl_matrix *A)
 -- Function: int gsl_linalg_complex_householder_mh (gsl_complex tau,
          const gsl_vector_complex *v, gsl_matrix_complex *A)

     This function applies the Householder matrix H defined by the
     scalar *note tau: 591. and the vector *note v: 591. to the
     right-hand side of the matrix *note A: 591.  On output the result A
     H is stored in *note A: 591.

 -- Function: int gsl_linalg_householder_hv (double tau, const
          gsl_vector *v, gsl_vector *w)
 -- Function: int gsl_linalg_complex_householder_hv (gsl_complex tau,
          const gsl_vector_complex *v, gsl_vector_complex *w)

     This function applies the Householder transformation H defined by
     the scalar *note tau: 593. and the vector *note v: 593. to the
     vector *note w: 593.  On output the result H w is stored in *note
     w: 593.


File: gsl-ref.info,  Node: Householder solver for linear systems,  Next: Tridiagonal Systems,  Prev: Householder Transformations,  Up: Linear Algebra

14.19 Householder solver for linear systems
===========================================

 -- Function: int gsl_linalg_HH_solve (gsl_matrix *A, const gsl_vector
          *b, gsl_vector *x)

     This function solves the system A x = b directly using Householder
     transformations.  On output the solution is stored in *note x: 595.
     and *note b: 595. is not modified.  The matrix *note A: 595. is
     destroyed by the Householder transformations.

 -- Function: int gsl_linalg_HH_svx (gsl_matrix *A, gsl_vector *x)

     This function solves the system A x = b in-place using Householder
     transformations.  On input *note x: 596. should contain the
     right-hand side b, which is replaced by the solution on output.
     The matrix *note A: 596. is destroyed by the Householder
     transformations.


File: gsl-ref.info,  Node: Tridiagonal Systems,  Next: Triangular Systems,  Prev: Householder solver for linear systems,  Up: Linear Algebra

14.20 Tridiagonal Systems
=========================

The functions described in this section efficiently solve symmetric,
non-symmetric and cyclic tridiagonal systems with minimal storage.  Note
that the current implementations of these functions use a variant of
Cholesky decomposition, so the tridiagonal matrix must be positive
definite.  For non-positive definite matrices, the functions return the
error code ‘GSL_ESING’.

 -- Function: int gsl_linalg_solve_tridiag (const gsl_vector *diag,
          const gsl_vector *e, const gsl_vector *f, const gsl_vector *b,
          gsl_vector *x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     tridiagonal (N \geq 2).  The super-diagonal and sub-diagonal
     vectors *note e: 598. and *note f: 598. must be one element shorter
     than the diagonal vector *note diag: 598.  The form of ‘A’ for the
     4-by-4 case is shown below,

          A = ( d_0 e_0  0   0  )
              ( f_0 d_1 e_1  0  )
              (  0  f_1 d_2 e_2 )
              (  0   0  f_2 d_3 )

 -- Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector *diag,
          const gsl_vector *e, const gsl_vector *b, gsl_vector *x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     symmetric tridiagonal (N \geq 2).  The off-diagonal vector *note e:
     599. must be one element shorter than the diagonal vector *note
     diag: 599.  The form of ‘A’ for the 4-by-4 case is shown below,

          A = ( d_0 e_0  0   0  )
              ( e_0 d_1 e_1  0  )
              (  0  e_1 d_2 e_2 )
              (  0   0  e_2 d_3 )

 -- Function: int gsl_linalg_solve_cyc_tridiag (const gsl_vector *diag,
          const gsl_vector *e, const gsl_vector *f, const gsl_vector *b,
          gsl_vector *x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     cyclic tridiagonal (N \geq 3).  The cyclic super-diagonal and
     sub-diagonal vectors *note e: 59a. and *note f: 59a. must have the
     same number of elements as the diagonal vector *note diag: 59a.
     The form of ‘A’ for the 4-by-4 case is shown below,

          A = ( d_0 e_0  0  f_3 )
              ( f_0 d_1 e_1  0  )
              (  0  f_1 d_2 e_2 )
              ( e_3  0  f_2 d_3 )

 -- Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector
          *diag, const gsl_vector *e, const gsl_vector *b, gsl_vector
          *x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     symmetric cyclic tridiagonal (N \geq 3).  The cyclic off-diagonal
     vector *note e: 59b. must have the same number of elements as the
     diagonal vector *note diag: 59b.  The form of ‘A’ for the 4-by-4
     case is shown below,

          A = ( d_0 e_0  0  e_3 )
              ( e_0 d_1 e_1  0  )
              (  0  e_1 d_2 e_2 )
              ( e_3  0  e_2 d_3 )


File: gsl-ref.info,  Node: Triangular Systems,  Next: Banded Systems,  Prev: Tridiagonal Systems,  Up: Linear Algebra

14.21 Triangular Systems
========================

 -- Function: int gsl_linalg_tri_invert (CBLAS_UPLO_t Uplo, CBLAS_DIAG_t
          Diag, gsl_matrix *T)
 -- Function: int gsl_linalg_complex_tri_invert (CBLAS_UPLO_t Uplo,
          CBLAS_DIAG_t Diag, gsl_matrix_complex *T)

     These functions compute the in-place inverse of the triangular
     matrix *note T: 59e, stored in the lower triangle when *note Uplo:
     59e. = ‘CblasLower’ and upper triangle when *note Uplo: 59e. =
     ‘CblasUpper’.  The parameter *note Diag: 59e. = ‘CblasUnit’,
     ‘CblasNonUnit’ specifies whether the matrix is unit triangular.

 -- Function: int gsl_linalg_tri_LTL (gsl_matrix *L)
 -- Function: int gsl_linalg_complex_tri_LHL (gsl_matrix_complex *L)

     These functions compute the product L^T L (or L^{\dagger} L)
     in-place and stores it in the lower triangle of *note L: 5a0. on
     output.

 -- Function: int gsl_linalg_tri_UL (gsl_matrix *LU)
 -- Function: int gsl_linalg_complex_tri_UL (gsl_matrix_complex *LU)

     These functions compute the product U L where U is upper triangular
     and L is unit lower triangular, stored in *note LU: 5a2, as
     computed by *note gsl_linalg_LU_decomp(): 4ec. or *note
     gsl_linalg_complex_LU_decomp(): 4ed.  The product is computed
     in-place using Level 3 BLAS.

 -- Function: int gsl_linalg_tri_rcond (CBLAS_UPLO_t Uplo, const
          gsl_matrix *A, double *rcond, gsl_vector *work)

     This function estimates the 1-norm reciprocal condition number of
     the triangular matrix *note A: 5a3, using the lower triangle when
     *note Uplo: 5a3. is ‘CblasLower’ and upper triangle when *note
     Uplo: 5a3. is ‘CblasUpper’.  The reciprocal condition number 1 /
     \left( \left|\left| A \right|\right|_1 \left|\left| A^{-1}
     \right|\right|_1 \right) is stored in *note rcond: 5a3. on output.
     Additional workspace of size 3N is required in *note work: 5a3.


File: gsl-ref.info,  Node: Banded Systems,  Next: Balancing,  Prev: Triangular Systems,  Up: Linear Algebra

14.22 Banded Systems
====================

Band matrices are sparse matrices whose non-zero entries are confined to
a diagonal `band'.  From a storage point of view, significant savings
can be achieved by storing only the non-zero diagonals of a banded
matrix.  Algorithms such as LU and Cholesky factorizations preserve the
band structure of these matrices.  Computationally, working with compact
banded matrices is preferable to working on the full dense matrix with
many zero entries.

* Menu:

* General Banded Format::
* Symmetric Banded Format::
* Banded LU Decomposition::
* Banded Cholesky Decomposition::
* Banded LDLT Decomposition::


File: gsl-ref.info,  Node: General Banded Format,  Next: Symmetric Banded Format,  Up: Banded Systems

14.22.1 General Banded Format
-----------------------------

An example of a general banded matrix is given below.

     A = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 & 0 & 0 & 0 \\ \delta_1 & \alpha_2 & \beta_2 & \gamma_2 & 0 & 0 \\ 0 & \delta_2 & \alpha_3 & \beta_3 & \gamma_3 & 0 \\ 0 & 0 & \delta_3 & \alpha_4 & \beta_4 & \gamma_4 \\ 0 & 0 & 0 & \delta_4 & \alpha_5 & \beta_5 \\ 0 & 0 & 0 & 0 & \delta_5 & \alpha_6 \end{pmatrix}

This matrix has a `lower bandwidth' of 1 and an `upper bandwidth' of 2.
The lower bandwidth is the number of non-zero subdiagonals, and the
upper bandwidth is the number of non-zero superdiagonals.  A (p,q)
banded matrix has a lower bandwidth p and upper bandwidth q.  For
example, diagonal matrices are (0,0), tridiagonal matrices are (1,1),
and upper triangular matrices are (0,N-1) banded matrices.

The corresponding 6-by-4 packed banded matrix looks like

     AB = \begin{pmatrix} * & * & \alpha_1 & \delta_1 \\ * & \beta_1 & \alpha_2 & \delta_2 \\ \gamma_1 & \beta_2 & \alpha_3 & \delta_3 \\ \gamma_2 & \beta_3 & \alpha_4 & \delta_4 \\ \gamma_3 & \beta_4 & \alpha_5 & \delta_5 \\ \gamma_4 & \beta_5 & \alpha_6 & * \end{pmatrix}

where the superdiagonals are stored in columns, followed by the
diagonal, followed by the subdiagonals.  The entries marked by * are not
referenced by the banded routines.  With this format, each row of AB
corresponds to the non-zero entries of the corresponding column of A.
For an N-by-N matrix A, the dimension of AB will be N-by-(p+q+1).


File: gsl-ref.info,  Node: Symmetric Banded Format,  Next: Banded LU Decomposition,  Prev: General Banded Format,  Up: Banded Systems

14.22.2 Symmetric Banded Format
-------------------------------

Symmetric banded matrices allow for additional storage savings.  As an
example, consider the following 6 \times 6 symmetric banded matrix with
lower bandwidth p = 2:

     A = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 & 0 & 0 & 0 \\ \beta_1 & \alpha_2 & \beta_2 & \gamma_2 & 0 & 0 \\ \gamma_1 & \beta_2 & \alpha_3 & \beta_3 & \gamma_3 & 0 \\ 0 & \gamma_2 & \beta_3 & \alpha_4 & \beta_4 & \gamma_4 \\ 0 & 0 & \gamma_3 & \beta_4 & \alpha_5 & \beta_5 \\ 0 & 0 & 0 & \gamma_4 & \beta_5 & \alpha_6 \end{pmatrix}

The packed symmetric banded 6 \times 3 matrix will look like:

     AB = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \\ \alpha_4 & \beta_4 & \gamma_4 \\ \alpha_5 & \beta_5 & * \\ \alpha_6 & * & * \end{pmatrix}

The entries marked by * are not referenced by the symmetric banded
routines.  The relationship between the packed format and original
matrix is,

     AB(i,j) = A(i, i + j) = A(i + j, i)

for i = 0, \dots, N - 1, j = 0, \dots, p.  Conversely,

     A(i,j) = AB(j, i - j)

for i = 0, \dots, N - 1, j = \textrm{max}(0,i-p), \dots, i.

     Warning: Note that this format is the transpose of the symmetric
     banded format used by LAPACK. In order to develop efficient
     routines for symmetric banded matrices, it helps to have the
     nonzero elements in each column in contiguous memory locations.
     Since C uses row-major order, GSL stores the columns in the rows of
     the packed banded format, while LAPACK, written in Fortran, uses
     the transposed format.


File: gsl-ref.info,  Node: Banded LU Decomposition,  Next: Banded Cholesky Decomposition,  Prev: Symmetric Banded Format,  Up: Banded Systems

14.22.3 Banded LU Decomposition
-------------------------------

The routines in this section are designed to factor banded M-by-N
matrices with an LU factorization, P A = L U. The matrix A is banded of
type (p,q), i.e.  a lower bandwidth of p and an upper bandwidth of q.
See *note LU Decomposition: 4eb. for more information on the
factorization.  For banded (p,q) matrices, the U factor will have an
upper bandwidth of p + q, while the L factor will have a lower bandwidth
of at most p.  Therefore, additional storage is needed to store the p
additional bands of U.

As an example, consider the M = N = 7 matrix with lower bandwidth p = 3
and upper bandwidth q = 2,

     A = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 & 0 & 0 & 0 & 0 \\ \delta_1 & \alpha_2 & \beta_2 & \gamma_2 & 0 & 0 & 0 \\ \epsilon_1 & \delta_2 & \alpha_3 & \beta_3 & \gamma_3 & 0 & 0 \\ \zeta_1 & \epsilon_2 & \delta_3 & \alpha_4 & \beta_4 & \gamma_4 & 0 \\ 0 & \zeta_2 & \epsilon_3 & \delta_4 & \alpha_5 & \beta_5 & \gamma_5 \\ 0 & 0 & \zeta_3 & \epsilon_4 & \delta_5 & \alpha_6 & \beta_6 \\ 0 & 0 & 0 & \zeta_4 & \epsilon_5 & \delta_6 & \alpha_7 \end{pmatrix}

The corresponding N-by-2p + q + 1 packed banded matrix looks like

     AB = \begin{pmatrix} * & * & * & * & * & \alpha_1 & \delta_1 & \epsilon_1 & \zeta_1 \\ * & * & * & * & \beta_1 & \alpha_2 & \delta_2 & \epsilon_2 & \zeta_2 \\ * & * & * & \gamma_1 & \beta_2 & \alpha_3 & \delta_3 & \epsilon_3 & \zeta_3 \\ * & * & - & \gamma_2 & \beta_3 & \alpha_4 & \delta_4 & \epsilon_4 & \zeta_4 \\ * & - & - & \gamma_3 & \beta_4 & \alpha_5 & \delta_5 & \epsilon_5 & * \\ - & - & - & \gamma_4 & \beta_5 & \alpha_6 & \delta_6 & * & * \\ \undermat{p}{- & - & -} & \undermat{q}{\gamma_5 & \beta_6} & \alpha_7 & \undermat{p}{* & * & * & } \end{pmatrix}

Entries marked with - are used to store the additional p diagonals of
the U factor.  Entries marked with * are not referenced by the banded
routines.

 -- Function: int gsl_linalg_LU_band_decomp (const size_t M, const
          size_t lb, const size_t ub, gsl_matrix *AB, gsl_vector_uint
          *piv)

     This function computes the LU factorization of the banded matrix
     *note AB: 5a9. which is stored in packed band format (see above)
     and has dimension N-by-2p + q + 1.  The number of rows M of the
     original matrix is provided in *note M: 5a9.  The lower bandwidth p
     is provided in *note lb: 5a9. and the upper bandwidth q is provided
     in *note ub: 5a9.  The vector *note piv: 5a9. has length
     \textrm{min}(M,N) and stores the pivot indices on output (for 0 \le
     i < \textrm{min}(M,N), row i of the matrix was interchanged with
     row ‘piv[i]’).  On output, *note AB: 5a9. contains both the L and U
     factors in packed format.

 -- Function: int gsl_linalg_LU_band_solve (const size_t lb, const
          size_t ub, const gsl_matrix *LUB, const gsl_vector_uint *piv,
          const gsl_vector *b, gsl_vector *x)

     This function solves the square system Ax = b using the banded LU
     factorization (*note LUB: 5aa, *note piv: 5aa.) computed by *note
     gsl_linalg_LU_band_decomp(): 5a9.  The lower and upper bandwidths
     are provided in *note lb: 5aa. and *note ub: 5aa. respectively.
     The right hand side vector is provided in *note b: 5aa.  The
     solution vector is stored in *note x: 5aa. on output.

 -- Function: int gsl_linalg_LU_band_svx (const size_t lb, const size_t
          ub, const gsl_matrix *LUB, const gsl_vector_uint *piv,
          gsl_vector *x)

     This function solves the square system Ax = b in-place, using the
     banded LU factorization (*note LUB: 5ab, *note piv: 5ab.) computed
     by *note gsl_linalg_LU_band_decomp(): 5a9.  The lower and upper
     bandwidths are provided in *note lb: 5ab. and *note ub: 5ab.
     respectively.  On input, the right hand side vector b is provided
     in *note x: 5ab, which is replaced by the solution vector x on
     output.

 -- Function: int gsl_linalg_LU_band_unpack (const size_t M, const
          size_t lb, const size_t ub, const gsl_matrix *LUB, const
          gsl_vector_uint *piv, gsl_matrix *L, gsl_matrix *U)

     This function unpacks the banded LU factorization (*note LUB: 5ac,
     *note piv: 5ac.) previously computed by *note
     gsl_linalg_LU_band_decomp(): 5a9. into the matrices *note L: 5ac.
     and *note U: 5ac.  The matrix *note U: 5ac. has dimension
     \textrm{min}(M,N)-by-N and stores the upper triangular factor on
     output.  The matrix *note L: 5ac. has dimension
     M-by-\textrm{min}(M,N) and stores the matrix P^T L on output.


File: gsl-ref.info,  Node: Banded Cholesky Decomposition,  Next: Banded LDLT Decomposition,  Prev: Banded LU Decomposition,  Up: Banded Systems

14.22.4 Banded Cholesky Decomposition
-------------------------------------

The routines in this section are designed to factor and solve N-by-N
linear systems of the form A x = b where A is a banded, symmetric, and
positive definite matrix with lower bandwidth p.  See *note Cholesky
Decomposition: 3f5. for more information on the factorization.  The
lower triangular factor of the Cholesky decomposition preserves the same
banded structure as the matrix A, enabling an efficient algorithm which
overwrites the original matrix with the L factor.

 -- Function: int gsl_linalg_cholesky_band_decomp (gsl_matrix *A)

     This function factorizes the symmetric, positive-definite square
     matrix *note A: 5ae. into the Cholesky decomposition A = L L^T. The
     input matrix *note A: 5ae. is given in *note symmetric banded
     format: 5a6, and has dimensions N-by-(p + 1), where p is the lower
     bandwidth of the matrix.  On output, the entries of *note A: 5ae.
     are replaced by the entries of the matrix L in the same format.  In
     addition, the lower right element of *note A: 5ae. is used to store
     the matrix 1-norm, used later by *note
     gsl_linalg_cholesky_band_rcond(): 5af. to calculate the reciprocal
     condition number.

     If the matrix is not positive-definite then the decomposition will
     fail, returning the error code *note GSL_EDOM: 28.  When testing
     whether a matrix is positive-definite, disable the error handler
     first to avoid triggering an error.

 -- Function: int gsl_linalg_cholesky_band_solve (const gsl_matrix *LLT,
          const gsl_vector *b, gsl_vector *x)
 -- Function: int gsl_linalg_cholesky_band_solvem (const gsl_matrix
          *LLT, const gsl_matrix *B, gsl_matrix *X)

     This function solves the symmetric banded system A x = b (or A X =
     B) using the Cholesky decomposition of A held in the matrix *note
     LLT: 5b1. which must have been previously computed by *note
     gsl_linalg_cholesky_band_decomp(): 5ae.

 -- Function: int gsl_linalg_cholesky_band_svx (const gsl_matrix *LLT,
          gsl_vector *x)
 -- Function: int gsl_linalg_cholesky_band_svxm (const gsl_matrix *LLT,
          gsl_matrix *X)

     This function solves the symmetric banded system A x = b (or A X =
     B) in-place using the Cholesky decomposition of A held in the
     matrix *note LLT: 5b3. which must have been previously computed by
     *note gsl_linalg_cholesky_band_decomp(): 5ae.  On input ‘x’ (or
     *note X: 5b3.) should contain the right-hand side b (or B), which
     is replaced by the solution on output.

 -- Function: int gsl_linalg_cholesky_band_invert (const gsl_matrix
          *LLT, gsl_matrix *Ainv)

     This function computes the inverse of a symmetric banded matrix
     from its Cholesky decomposition *note LLT: 5b4, which must have
     been previously computed by *note
     gsl_linalg_cholesky_band_decomp(): 5ae.  On output, the inverse is
     stored in *note Ainv: 5b4, using both the lower and upper portions.

 -- Function: int gsl_linalg_cholesky_band_unpack (const gsl_matrix
          *LLT, gsl_matrix *L)

     This function unpacks the lower triangular Cholesky factor from
     *note LLT: 5b5. and stores it in the lower triangular portion of
     the N-by-N matrix *note L: 5b5.  The upper triangular portion of
     *note L: 5b5. is not referenced.

 -- Function: int gsl_linalg_cholesky_band_scale (const gsl_matrix *A,
          gsl_vector *S)

     This function calculates a diagonal scaling transformation of the
     symmetric, positive definite banded matrix *note A: 5b6, such that
     S A S has a condition number within a factor of N of the matrix of
     smallest possible condition number over all possible diagonal
     scalings.  On output, *note S: 5b6. contains the scale factors,
     given by S_i = 1/\sqrt{A_{ii}}.  For any A_{ii} \le 0, the
     corresponding scale factor S_i is set to 1.

 -- Function: int gsl_linalg_cholesky_band_scale_apply (gsl_matrix *A,
          const gsl_vector *S)

     This function applies the scaling transformation *note S: 5b7. to
     the banded symmetric positive definite matrix *note A: 5b7.  On
     output, *note A: 5b7. is replaced by S A S.

 -- Function: int gsl_linalg_cholesky_band_rcond (const gsl_matrix *LLT,
          double *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric banded positive definite matrix A, using
     its Cholesky decomposition provided in *note LLT: 5af.  The
     reciprocal condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in *note rcond: 5af.  Additional workspace
     of size 3 N is required in *note work: 5af.


File: gsl-ref.info,  Node: Banded LDLT Decomposition,  Prev: Banded Cholesky Decomposition,  Up: Banded Systems

14.22.5 Banded LDLT Decomposition
---------------------------------

The routines in this section are designed to factor and solve N-by-N
linear systems of the form A x = b where A is a banded, symmetric, and
non-singular matrix with lower bandwidth p.  See *note LDLT
Decomposition: 56f. for more information on the factorization.  The
lower triangular factor of the L D L^T decomposition preserves the same
banded structure as the matrix A, enabling an efficient algorithm which
overwrites the original matrix with the L and D factors.

 -- Function: int gsl_linalg_ldlt_band_decomp (gsl_matrix *A)

     This function factorizes the symmetric, non-singular square matrix
     *note A: 5b9. into the decomposition A = L D L^T. The input matrix
     *note A: 5b9. is given in *note symmetric banded format: 5a6, and
     has dimensions N-by-(p + 1), where p is the lower bandwidth of the
     matrix.  On output, the entries of *note A: 5b9. are replaced by
     the entries of the matrices D and L in the same format.

     If the matrix is singular then the decomposition will fail,
     returning the error code *note GSL_EDOM: 28.

 -- Function: int gsl_linalg_ldlt_band_solve (const gsl_matrix *LDLT,
          const gsl_vector *b, gsl_vector *x)

     This function solves the symmetric banded system A x = b using the
     L D L^T decomposition of A held in the matrix *note LDLT: 5ba.
     which must have been previously computed by *note
     gsl_linalg_ldlt_band_decomp(): 5b9.

 -- Function: int gsl_linalg_ldlt_band_svx (const gsl_matrix *LDLT,
          gsl_vector *x)

     This function solves the symmetric banded system A x = b in-place
     using the L D L^T decomposition of A held in the matrix *note LDLT:
     5bb. which must have been previously computed by *note
     gsl_linalg_ldlt_band_decomp(): 5b9.  On input *note x: 5bb. should
     contain the right-hand side b, which is replaced by the solution on
     output.

 -- Function: int gsl_linalg_ldlt_band_unpack (const gsl_matrix *LDLT,
          gsl_matrix *L, gsl_vector *D)

     This function unpacks the unit lower triangular factor L from *note
     LDLT: 5bc. and stores it in the lower triangular portion of the
     N-by-N matrix *note L: 5bc.  The upper triangular portion of *note
     L: 5bc. is not referenced.  The diagonal matrix D is stored in the
     vector *note D: 5bc.

 -- Function: int gsl_linalg_ldlt_band_rcond (const gsl_matrix *LDLT,
          double *rcond, gsl_vector *work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric banded nonsingular matrix A, using its L D
     L^T decomposition provided in *note LDLT: 5bd.  The reciprocal
     condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in *note rcond: 5bd.  Additional workspace
     of size 3 N is required in *note work: 5bd.


File: gsl-ref.info,  Node: Balancing,  Next: Examples<9>,  Prev: Banded Systems,  Up: Linear Algebra

14.23 Balancing
===============

The process of balancing a matrix applies similarity transformations to
make the rows and columns have comparable norms.  This is useful, for
example, to reduce roundoff errors in the solution of eigenvalue
problems.  Balancing a matrix A consists of replacing A with a similar
matrix

     A' = D^{-1} A D

where D is a diagonal matrix whose entries are powers of the floating
point radix.

 -- Function: int gsl_linalg_balance_matrix (gsl_matrix *A, gsl_vector
          *D)

     This function replaces the matrix *note A: 5c0. with its balanced
     counterpart and stores the diagonal elements of the similarity
     transformation into the vector *note D: 5c0.


File: gsl-ref.info,  Node: Examples<9>,  Next: References and Further Reading<9>,  Prev: Balancing,  Up: Linear Algebra

14.24 Examples
==============

The following program solves the linear system A x = b.  The system to
be solved is,

     [ 0.18 0.60 0.57 0.96 ] [x0]   [1.0]
     [ 0.41 0.24 0.99 0.58 ] [x1] = [2.0]
     [ 0.14 0.30 0.97 0.66 ] [x2]   [3.0]
     [ 0.51 0.13 0.19 0.85 ] [x3]   [4.0]

and the solution is found using LU decomposition of the matrix A.

     #include <stdio.h>
     #include <gsl/gsl_linalg.h>

     int
     main (void)
     {
       double a_data[] = { 0.18, 0.60, 0.57, 0.96,
                           0.41, 0.24, 0.99, 0.58,
                           0.14, 0.30, 0.97, 0.66,
                           0.51, 0.13, 0.19, 0.85 };

       double b_data[] = { 1.0, 2.0, 3.0, 4.0 };

       gsl_matrix_view m
         = gsl_matrix_view_array (a_data, 4, 4);

       gsl_vector_view b
         = gsl_vector_view_array (b_data, 4);

       gsl_vector *x = gsl_vector_alloc (4);

       int s;

       gsl_permutation * p = gsl_permutation_alloc (4);

       gsl_linalg_LU_decomp (&m.matrix, p, &s);

       gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);

       printf ("x = \n");
       gsl_vector_fprintf (stdout, x, "%g");

       gsl_permutation_free (p);
       gsl_vector_free (x);
       return 0;
     }

Here is the output from the program,

     x =
     -4.05205
     -12.6056
     1.66091
     8.69377

This can be verified by multiplying the solution x by the original
matrix A using GNU octave,

     octave> A = [ 0.18, 0.60, 0.57, 0.96;
                   0.41, 0.24, 0.99, 0.58;
                   0.14, 0.30, 0.97, 0.66;
                   0.51, 0.13, 0.19, 0.85 ];

     octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];

     octave> A * x
     ans =
       1.0000
       2.0000
       3.0000
       4.0000

This reproduces the original right-hand side vector, b, in accordance
with the equation A x = b.


File: gsl-ref.info,  Node: References and Further Reading<9>,  Prev: Examples<9>,  Up: Linear Algebra

14.25 References and Further Reading
====================================

Further information on the algorithms described in this section can be
found in the following book,

   * G. H. Golub, C. F. Van Loan, “Matrix Computations” (3rd Ed, 1996),
     Johns Hopkins University Press, ISBN 0-8018-5414-8.

The LAPACK library is described in the following manual,

   * `LAPACK Users’ Guide' (Third Edition, 1999), Published by SIAM,
     ISBN 0-89871-447-8

The LAPACK source code can be found at ‘http://www.netlib.org/lapack’,
along with an online copy of the users guide.

Further information on recursive Level 3 BLAS algorithms may be found in
the following paper,

   * E. Peise and P. Bientinesi, “Recursive algorithms for dense linear
     algebra: the ReLAPACK collection”,
     ‘http://arxiv.org/abs/1602.06763’, 2016.

The recursive Level 3 BLAS QR decomposition is described in the
following paper,

   * E. Elmroth and F. G. Gustavson, 2000.  Applying recursion to serial
     and parallel QR factorization leads to better performance.  IBM
     Journal of Research and Development, 44(4), pp.605-624.

The Modified Golub-Reinsch algorithm is described in the following
paper,

   * T.F. Chan, “An Improved Algorithm for Computing the Singular Value
     Decomposition”, ACM Transactions on Mathematical Software, 8
     (1982), pp 72–83.

The Jacobi algorithm for singular value decomposition is described in
the following papers,

   * J.C. Nash, “A one-sided transformation method for the singular
     value decomposition and algebraic eigenproblem”, Computer Journal,
     Volume 18, Number 1 (1975), p 74–76

   * J.C. Nash and S. Shlien “Simple algorithms for the partial singular
     value decomposition”, Computer Journal, Volume 30 (1987), p
     268–275.

   * J. Demmel, K. Veselic, “Jacobi’s Method is more accurate than QR”,
     Lapack Working Note 15 (LAWN-15), October 1989.  Available from
     netlib, ‘http://www.netlib.org/lapack/’ in the ‘lawns’ or
     ‘lawnspdf’ directories.

The algorithm for estimating a matrix condition number is described in
the following paper,

   * N. J. Higham, “FORTRAN codes for estimating the one-norm of a real
     or complex matrix, with applications to condition estimation”, ACM
     Trans.  Math.  Soft., vol.  14, no.  4, pp.  381-396, December
     1988.


File: gsl-ref.info,  Node: Eigensystems,  Next: Fast Fourier Transforms FFTs,  Prev: Linear Algebra,  Up: Top

15 Eigensystems
***************

This chapter describes functions for computing eigenvalues and
eigenvectors of matrices.  There are routines for real symmetric, real
nonsymmetric, complex hermitian, real generalized symmetric-definite,
complex generalized hermitian-definite, and real generalized
nonsymmetric eigensystems.  Eigenvalues can be computed with or without
eigenvectors.  The hermitian and real symmetric matrix algorithms are
symmetric bidiagonalization followed by QR reduction.  The nonsymmetric
algorithm is the Francis QR double-shift.  The generalized nonsymmetric
algorithm is the QZ method due to Moler and Stewart.

The functions described in this chapter are declared in the header file
‘gsl_eigen.h’.

* Menu:

* Real Symmetric Matrices::
* Complex Hermitian Matrices::
* Real Nonsymmetric Matrices::
* Real Generalized Symmetric-Definite Eigensystems::
* Complex Generalized Hermitian-Definite Eigensystems::
* Real Generalized Nonsymmetric Eigensystems::
* Sorting Eigenvalues and Eigenvectors::
* Examples: Examples<10>.
* References and Further Reading: References and Further Reading<10>.


File: gsl-ref.info,  Node: Real Symmetric Matrices,  Next: Complex Hermitian Matrices,  Up: Eigensystems

15.1 Real Symmetric Matrices
============================

For real symmetric matrices, the library uses the symmetric
bidiagonalization and QR reduction method.  This is described in Golub &
van Loan, section 8.3.  The computed eigenvalues are accurate to an
absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine
precision.

 -- Type: gsl_eigen_symm_workspace

     This workspace contains internal parameters used for solving
     symmetric eigenvalue problems.

 -- Function: *note gsl_eigen_symm_workspace: 5c6. *gsl_eigen_symm_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     *note n: 5c7.-by-*note n: 5c7. real symmetric matrices.  The size
     of the workspace is O(2n).

 -- Function: void gsl_eigen_symm_free (gsl_eigen_symm_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 5c8.

 -- Function: int gsl_eigen_symm (gsl_matrix *A, gsl_vector *eval,
          gsl_eigen_symm_workspace *w)

     This function computes the eigenvalues of the real symmetric matrix
     *note A: 5c9.  Additional workspace of the appropriate size must be
     provided in *note w: 5c9.  The diagonal and lower triangular part
     of *note A: 5c9. are destroyed during the computation, but the
     strict upper triangular part is not referenced.  The eigenvalues
     are stored in the vector *note eval: 5c9. and are unordered.

 -- Type: gsl_eigen_symmv_workspace

     This workspace contains internal parameters used for solving
     symmetric eigenvalue and eigenvector problems.

 -- Function: *note gsl_eigen_symmv_workspace: 5ca.
          *gsl_eigen_symmv_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of *note n: 5cb.-by-*note n: 5cb. real symmetric
     matrices.  The size of the workspace is O(4n).

 -- Function: void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 5cc.

 -- Function: int gsl_eigen_symmv (gsl_matrix *A, gsl_vector *eval,
          gsl_matrix *evec, gsl_eigen_symmv_workspace *w)

     This function computes the eigenvalues and eigenvectors of the real
     symmetric matrix *note A: 5cd.  Additional workspace of the
     appropriate size must be provided in *note w: 5cd.  The diagonal
     and lower triangular part of *note A: 5cd. are destroyed during the
     computation, but the strict upper triangular part is not
     referenced.  The eigenvalues are stored in the vector *note eval:
     5cd. and are unordered.  The corresponding eigenvectors are stored
     in the columns of the matrix *note evec: 5cd.  For example, the
     eigenvector in the first column corresponds to the first
     eigenvalue.  The eigenvectors are guaranteed to be mutually
     orthogonal and normalised to unit magnitude.


File: gsl-ref.info,  Node: Complex Hermitian Matrices,  Next: Real Nonsymmetric Matrices,  Prev: Real Symmetric Matrices,  Up: Eigensystems

15.2 Complex Hermitian Matrices
===============================

For hermitian matrices, the library uses the complex form of the
symmetric bidiagonalization and QR reduction method.

 -- Type: gsl_eigen_herm_workspace

     This workspace contains internal parameters used for solving
     hermitian eigenvalue problems.

 -- Function: *note gsl_eigen_herm_workspace: 5cf. *gsl_eigen_herm_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     *note n: 5d0.-by-*note n: 5d0. complex hermitian matrices.  The
     size of the workspace is O(3n).

 -- Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 5d1.

 -- Function: int gsl_eigen_herm (gsl_matrix_complex *A, gsl_vector
          *eval, gsl_eigen_herm_workspace *w)

     This function computes the eigenvalues of the complex hermitian
     matrix *note A: 5d2.  Additional workspace of the appropriate size
     must be provided in *note w: 5d2.  The diagonal and lower
     triangular part of *note A: 5d2. are destroyed during the
     computation, but the strict upper triangular part is not
     referenced.  The imaginary parts of the diagonal are assumed to be
     zero and are not referenced.  The eigenvalues are stored in the
     vector *note eval: 5d2. and are unordered.

 -- Type: gsl_eigen_hermv_workspace

     This workspace contains internal parameters used for solving
     hermitian eigenvalue and eigenvector problems.

 -- Function: *note gsl_eigen_hermv_workspace: 5d3.
          *gsl_eigen_hermv_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of *note n: 5d4.-by-*note n: 5d4. complex hermitian
     matrices.  The size of the workspace is O(5n).

 -- Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 5d5.

 -- Function: int gsl_eigen_hermv (gsl_matrix_complex *A, gsl_vector
          *eval, gsl_matrix_complex *evec, gsl_eigen_hermv_workspace *w)

     This function computes the eigenvalues and eigenvectors of the
     complex hermitian matrix *note A: 5d6.  Additional workspace of the
     appropriate size must be provided in *note w: 5d6.  The diagonal
     and lower triangular part of *note A: 5d6. are destroyed during the
     computation, but the strict upper triangular part is not
     referenced.  The imaginary parts of the diagonal are assumed to be
     zero and are not referenced.  The eigenvalues are stored in the
     vector *note eval: 5d6. and are unordered.  The corresponding
     complex eigenvectors are stored in the columns of the matrix *note
     evec: 5d6.  For example, the eigenvector in the first column
     corresponds to the first eigenvalue.  The eigenvectors are
     guaranteed to be mutually orthogonal and normalised to unit
     magnitude.


File: gsl-ref.info,  Node: Real Nonsymmetric Matrices,  Next: Real Generalized Symmetric-Definite Eigensystems,  Prev: Complex Hermitian Matrices,  Up: Eigensystems

15.3 Real Nonsymmetric Matrices
===============================

The solution of the real nonsymmetric eigensystem problem for a matrix A
involves computing the Schur decomposition

     A = Z T Z^T

where Z is an orthogonal matrix of Schur vectors and T, the Schur form,
is quasi upper triangular with diagonal 1-by-1 blocks which are real
eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are
complex conjugate eigenvalues of A. The algorithm used is the
double-shift Francis method.

 -- Type: gsl_eigen_nonsymm_workspace

     This workspace contains internal parameters used for solving
     nonsymmetric eigenvalue problems.

 -- Function: *note gsl_eigen_nonsymm_workspace: 5d8.
          *gsl_eigen_nonsymm_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     *note n: 5d9.-by-*note n: 5d9. real nonsymmetric matrices.  The
     size of the workspace is O(2n).

 -- Function: void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace
          *w)

     This function frees the memory associated with the workspace *note
     w: 5da.

 -- Function: void gsl_eigen_nonsymm_params (const int compute_t, const
          int balance, gsl_eigen_nonsymm_workspace *w)

     This function sets some parameters which determine how the
     eigenvalue problem is solved in subsequent calls to *note
     gsl_eigen_nonsymm(): 5dc.

     If *note compute_t: 5db. is set to 1, the full Schur form T will be
     computed by *note gsl_eigen_nonsymm(): 5dc.  If it is set to 0, T
     will not be computed (this is the default setting).  Computing the
     full Schur form T requires approximately 1.5–2 times the number of
     flops.

     If *note balance: 5db. is set to 1, a balancing transformation is
     applied to the matrix prior to computing eigenvalues.  This
     transformation is designed to make the rows and columns of the
     matrix have comparable norms, and can result in more accurate
     eigenvalues for matrices whose entries vary widely in magnitude.
     See *note Balancing: 5be. for more information.  Note that the
     balancing transformation does not preserve the orthogonality of the
     Schur vectors, so if you wish to compute the Schur vectors with
     *note gsl_eigen_nonsymm_Z(): 5dd. you will obtain the Schur vectors
     of the balanced matrix instead of the original matrix.  The
     relationship will be

          T = Q^T D^{-1} A D Q

     where ‘Q’ is the matrix of Schur vectors for the balanced matrix,
     and ‘D’ is the balancing transformation.  Then *note
     gsl_eigen_nonsymm_Z(): 5dd. will compute a matrix ‘Z’ which
     satisfies

          T = Z^{-1} A Z

     with Z = D Q. Note that ‘Z’ will not be orthogonal.  For this
     reason, balancing is not performed by default.

 -- Function: int gsl_eigen_nonsymm (gsl_matrix *A, gsl_vector_complex
          *eval, gsl_eigen_nonsymm_workspace *w)

     This function computes the eigenvalues of the real nonsymmetric
     matrix *note A: 5dc. and stores them in the vector *note eval: 5dc.
     If T is desired, it is stored in the upper portion of *note A: 5dc.
     on output.  Otherwise, on output, the diagonal of *note A: 5dc.
     will contain the 1-by-1 real eigenvalues and 2-by-2 complex
     conjugate eigenvalue systems, and the rest of *note A: 5dc. is
     destroyed.  In rare cases, this function may fail to find all
     eigenvalues.  If this happens, an error code is returned and the
     number of converged eigenvalues is stored in ‘w->n_evals’.  The
     converged eigenvalues are stored in the beginning of *note eval:
     5dc.

 -- Function: int gsl_eigen_nonsymm_Z (gsl_matrix *A, gsl_vector_complex
          *eval, gsl_matrix *Z, gsl_eigen_nonsymm_workspace *w)

     This function is identical to *note gsl_eigen_nonsymm(): 5dc.
     except that it also computes the Schur vectors and stores them into
     *note Z: 5dd.

 -- Type: gsl_eigen_nonsymmv_workspace

     This workspace contains internal parameters used for solving
     nonsymmetric eigenvalue and eigenvector problems.

 -- Function: *note gsl_eigen_nonsymmv_workspace: 5de.
          *gsl_eigen_nonsymmv_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of *note n: 5df.-by-*note n: 5df. real nonsymmetric
     matrices.  The size of the workspace is O(5n).

 -- Function: void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace
          *w)

     This function frees the memory associated with the workspace *note
     w: 5e0.

 -- Function: void gsl_eigen_nonsymmv_params (const int balance,
          gsl_eigen_nonsymm_workspace *w)

     This function sets parameters which determine how the eigenvalue
     problem is solved in subsequent calls to *note
     gsl_eigen_nonsymmv(): 5e2.  If *note balance: 5e1. is set to 1, a
     balancing transformation is applied to the matrix.  See *note
     gsl_eigen_nonsymm_params(): 5db. for more information.  Balancing
     is turned off by default since it does not preserve the
     orthogonality of the Schur vectors.

 -- Function: int gsl_eigen_nonsymmv (gsl_matrix *A, gsl_vector_complex
          *eval, gsl_matrix_complex *evec, gsl_eigen_nonsymmv_workspace
          *w)

     This function computes eigenvalues and right eigenvectors of the
     ‘n’-by-‘n’ real nonsymmetric matrix *note A: 5e2.  It first calls
     *note gsl_eigen_nonsymm(): 5dc. to compute the eigenvalues, Schur
     form T, and Schur vectors.  Then it finds eigenvectors of T and
     backtransforms them using the Schur vectors.  The Schur vectors are
     destroyed in the process, but can be saved by using *note
     gsl_eigen_nonsymmv_Z(): 5e3.  The computed eigenvectors are
     normalized to have unit magnitude.  On output, the upper portion of
     *note A: 5e2. contains the Schur form T. If *note
     gsl_eigen_nonsymm(): 5dc. fails, no eigenvectors are computed, and
     an error code is returned.

 -- Function: int gsl_eigen_nonsymmv_Z (gsl_matrix *A,
          gsl_vector_complex *eval, gsl_matrix_complex *evec, gsl_matrix
          *Z, gsl_eigen_nonsymmv_workspace *w)

     This function is identical to *note gsl_eigen_nonsymmv(): 5e2.
     except that it also saves the Schur vectors into *note Z: 5e3.


File: gsl-ref.info,  Node: Real Generalized Symmetric-Definite Eigensystems,  Next: Complex Generalized Hermitian-Definite Eigensystems,  Prev: Real Nonsymmetric Matrices,  Up: Eigensystems

15.4 Real Generalized Symmetric-Definite Eigensystems
=====================================================

The real generalized symmetric-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that

     A x = \lambda B x

where A and B are symmetric matrices, and B is positive-definite.  This
problem reduces to the standard symmetric eigenvalue problem by applying
the Cholesky decomposition to B:

     A x = \lambda B x
     A x = \lambda L L^T x
     ( L^{-1} A L^{-T} ) L^T x = \lambda L^T x

Therefore, the problem becomes C y = \lambda y where C = L^{-1} A L^{-T}
is symmetric, and y = L^T x.  The standard symmetric eigensolver can be
applied to the matrix C. The resulting eigenvectors are backtransformed
to find the vectors of the original problem.  The eigenvalues and
eigenvectors of the generalized symmetric-definite eigenproblem are
always real.

 -- Type: gsl_eigen_gensymm_workspace

     This workspace contains internal parameters used for solving
     generalized symmetric eigenvalue problems.

 -- Function: *note gsl_eigen_gensymm_workspace: 5e5.
          *gsl_eigen_gensymm_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     *note n: 5e6.-by-*note n: 5e6. real generalized symmetric-definite
     eigensystems.  The size of the workspace is O(2n).

 -- Function: void gsl_eigen_gensymm_free (gsl_eigen_gensymm_workspace
          *w)

     This function frees the memory associated with the workspace *note
     w: 5e7.

 -- Function: int gsl_eigen_gensymm (gsl_matrix *A, gsl_matrix *B,
          gsl_vector *eval, gsl_eigen_gensymm_workspace *w)

     This function computes the eigenvalues of the real generalized
     symmetric-definite matrix pair (*note A: 5e8, *note B: 5e8.), and
     stores them in *note eval: 5e8, using the method outlined above.
     On output, *note B: 5e8. contains its Cholesky decomposition and
     *note A: 5e8. is destroyed.

 -- Type: gsl_eigen_gensymmv_workspace

     This workspace contains internal parameters used for solving
     generalized symmetric eigenvalue and eigenvector problems.

 -- Function: *note gsl_eigen_gensymmv_workspace: 5e9.
          *gsl_eigen_gensymmv_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of *note n: 5ea.-by-*note n: 5ea. real generalized
     symmetric-definite eigensystems.  The size of the workspace is
     O(4n).

 -- Function: void gsl_eigen_gensymmv_free (gsl_eigen_gensymmv_workspace
          *w)

     This function frees the memory associated with the workspace *note
     w: 5eb.

 -- Function: int gsl_eigen_gensymmv (gsl_matrix *A, gsl_matrix *B,
          gsl_vector *eval, gsl_matrix *evec,
          gsl_eigen_gensymmv_workspace *w)

     This function computes the eigenvalues and eigenvectors of the real
     generalized symmetric-definite matrix pair (*note A: 5ec, *note B:
     5ec.), and stores them in *note eval: 5ec. and *note evec: 5ec.
     respectively.  The computed eigenvectors are normalized to have
     unit magnitude.  On output, *note B: 5ec. contains its Cholesky
     decomposition and *note A: 5ec. is destroyed.


File: gsl-ref.info,  Node: Complex Generalized Hermitian-Definite Eigensystems,  Next: Real Generalized Nonsymmetric Eigensystems,  Prev: Real Generalized Symmetric-Definite Eigensystems,  Up: Eigensystems

15.5 Complex Generalized Hermitian-Definite Eigensystems
========================================================

The complex generalized hermitian-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that

     A x = \lambda B x

where A and B are hermitian matrices, and B is positive-definite.
Similarly to the real case, this can be reduced to C y = \lambda y where
C = L^{-1} A L^{-\dagger} is hermitian, and y = L^{\dagger} x.  The
standard hermitian eigensolver can be applied to the matrix C. The
resulting eigenvectors are backtransformed to find the vectors of the
original problem.  The eigenvalues of the generalized hermitian-definite
eigenproblem are always real.

 -- Type: gsl_eigen_genherm_workspace

     This workspace contains internal parameters used for solving
     generalized hermitian eigenvalue problems.

 -- Function: *note gsl_eigen_genherm_workspace: 5ee.
          *gsl_eigen_genherm_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     *note n: 5ef.-by-*note n: 5ef. complex generalized
     hermitian-definite eigensystems.  The size of the workspace is
     O(3n).

 -- Function: void gsl_eigen_genherm_free (gsl_eigen_genherm_workspace
          *w)

     This function frees the memory associated with the workspace *note
     w: 5f0.

 -- Function: int gsl_eigen_genherm (gsl_matrix_complex *A,
          gsl_matrix_complex *B, gsl_vector *eval,
          gsl_eigen_genherm_workspace *w)

     This function computes the eigenvalues of the complex generalized
     hermitian-definite matrix pair (*note A: 5f1, *note B: 5f1.), and
     stores them in *note eval: 5f1, using the method outlined above.
     On output, *note B: 5f1. contains its Cholesky decomposition and
     *note A: 5f1. is destroyed.

 -- Type: gsl_eigen_genhermv_workspace

     This workspace contains internal parameters used for solving
     generalized hermitian eigenvalue and eigenvector problems.

 -- Function: *note gsl_eigen_genhermv_workspace: 5f2.
          *gsl_eigen_genhermv_alloc (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of *note n: 5f3.-by-*note n: 5f3. complex generalized
     hermitian-definite eigensystems.  The size of the workspace is
     O(5n).

 -- Function: void gsl_eigen_genhermv_free (gsl_eigen_genhermv_workspace
          *w)

     This function frees the memory associated with the workspace *note
     w: 5f4.

 -- Function: int gsl_eigen_genhermv (gsl_matrix_complex *A,
          gsl_matrix_complex *B, gsl_vector *eval, gsl_matrix_complex
          *evec, gsl_eigen_genhermv_workspace *w)

     This function computes the eigenvalues and eigenvectors of the
     complex generalized hermitian-definite matrix pair (*note A: 5f5,
     *note B: 5f5.), and stores them in *note eval: 5f5. and *note evec:
     5f5. respectively.  The computed eigenvectors are normalized to
     have unit magnitude.  On output, *note B: 5f5. contains its
     Cholesky decomposition and *note A: 5f5. is destroyed.


File: gsl-ref.info,  Node: Real Generalized Nonsymmetric Eigensystems,  Next: Sorting Eigenvalues and Eigenvectors,  Prev: Complex Generalized Hermitian-Definite Eigensystems,  Up: Eigensystems

15.6 Real Generalized Nonsymmetric Eigensystems
===============================================

Given two square matrices (A, B), the generalized nonsymmetric
eigenvalue problem is to find eigenvalues \lambda and eigenvectors x
such that

     A x = \lambda B x

We may also define the problem as finding eigenvalues \mu and
eigenvectors y such that

     \mu A y = B y

Note that these two problems are equivalent (with \lambda = 1/\mu) if
neither \lambda nor \mu is zero.  If say, \lambda is zero, then it is
still a well defined eigenproblem, but its alternate problem involving
\mu is not.  Therefore, to allow for zero (and infinite) eigenvalues,
the problem which is actually solved is

     \beta A x = \alpha B x

The eigensolver routines below will return two values \alpha and \beta
and leave it to the user to perform the divisions \lambda = \alpha /
\beta and \mu = \beta / \alpha.

If the determinant of the matrix pencil A - \lambda B is zero for all
\lambda, the problem is said to be singular; otherwise it is called
regular.  Singularity normally leads to some \alpha = \beta = 0 which
means the eigenproblem is ill-conditioned and generally does not have
well defined eigenvalue solutions.  The routines below are intended for
regular matrix pencils and could yield unpredictable results when
applied to singular pencils.

The solution of the real generalized nonsymmetric eigensystem problem
for a matrix pair (A, B) involves computing the generalized Schur
decomposition

     A = Q S Z^T
     B = Q T Z^T

where Q and Z are orthogonal matrices of left and right Schur vectors
respectively, and (S, T) is the generalized Schur form whose diagonal
elements give the \alpha and \beta values.  The algorithm used is the QZ
method due to Moler and Stewart (see references).

 -- Type: gsl_eigen_gen_workspace

     This workspace contains internal parameters used for solving
     generalized eigenvalue problems.

 -- Function: *note gsl_eigen_gen_workspace: 5f7. *gsl_eigen_gen_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     *note n: 5f8.-by-*note n: 5f8. real generalized nonsymmetric
     eigensystems.  The size of the workspace is O(n).

 -- Function: void gsl_eigen_gen_free (gsl_eigen_gen_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 5f9.

 -- Function: void gsl_eigen_gen_params (const int compute_s, const int
          compute_t, const int balance, gsl_eigen_gen_workspace *w)

     This function sets some parameters which determine how the
     eigenvalue problem is solved in subsequent calls to *note
     gsl_eigen_gen(): 5fb.

     If *note compute_s: 5fa. is set to 1, the full Schur form S will be
     computed by *note gsl_eigen_gen(): 5fb.  If it is set to 0, S will
     not be computed (this is the default setting).  S is a quasi upper
     triangular matrix with 1-by-1 and 2-by-2 blocks on its diagonal.
     1-by-1 blocks correspond to real eigenvalues, and 2-by-2 blocks
     correspond to complex eigenvalues.

     If *note compute_t: 5fa. is set to 1, the full Schur form T will be
     computed by *note gsl_eigen_gen(): 5fb.  If it is set to 0, T will
     not be computed (this is the default setting).  T is an upper
     triangular matrix with non-negative elements on its diagonal.  Any
     2-by-2 blocks in S will correspond to a 2-by-2 diagonal block in T.

     The *note balance: 5fa. parameter is currently ignored, since
     generalized balancing is not yet implemented.

 -- Function: int gsl_eigen_gen (gsl_matrix *A, gsl_matrix *B,
          gsl_vector_complex *alpha, gsl_vector *beta,
          gsl_eigen_gen_workspace *w)

     This function computes the eigenvalues of the real generalized
     nonsymmetric matrix pair (*note A: 5fb, *note B: 5fb.), and stores
     them as pairs in (*note alpha: 5fb, *note beta: 5fb.), where *note
     alpha: 5fb. is complex and *note beta: 5fb. is real.  If \beta_i is
     non-zero, then \lambda = \alpha_i / \beta_i is an eigenvalue.
     Likewise, if \alpha_i is non-zero, then \mu = \beta_i / \alpha_i is
     an eigenvalue of the alternate problem \mu A y = B y.  The elements
     of *note beta: 5fb. are normalized to be non-negative.

     If S is desired, it is stored in *note A: 5fb. on output.  If T is
     desired, it is stored in *note B: 5fb. on output.  The ordering of
     eigenvalues in (*note alpha: 5fb, *note beta: 5fb.) follows the
     ordering of the diagonal blocks in the Schur forms S and T. In rare
     cases, this function may fail to find all eigenvalues.  If this
     occurs, an error code is returned.

 -- Function: int gsl_eigen_gen_QZ (gsl_matrix *A, gsl_matrix *B,
          gsl_vector_complex *alpha, gsl_vector *beta, gsl_matrix *Q,
          gsl_matrix *Z, gsl_eigen_gen_workspace *w)

     This function is identical to *note gsl_eigen_gen(): 5fb. except
     that it also computes the left and right Schur vectors and stores
     them into *note Q: 5fc. and *note Z: 5fc. respectively.

 -- Type: gsl_eigen_genv_workspace

     This workspace contains internal parameters used for solving
     generalized eigenvalue and eigenvector problems.

 -- Function: *note gsl_eigen_genv_workspace: 5fd. *gsl_eigen_genv_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of *note n: 5fe.-by-*note n: 5fe. real generalized
     nonsymmetric eigensystems.  The size of the workspace is O(7n).

 -- Function: void gsl_eigen_genv_free (gsl_eigen_genv_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 5ff.

 -- Function: int gsl_eigen_genv (gsl_matrix *A, gsl_matrix *B,
          gsl_vector_complex *alpha, gsl_vector *beta,
          gsl_matrix_complex *evec, gsl_eigen_genv_workspace *w)

     This function computes eigenvalues and right eigenvectors of the
     ‘n’-by-‘n’ real generalized nonsymmetric matrix pair (*note A: 600,
     *note B: 600.).  The eigenvalues are stored in (*note alpha: 600,
     *note beta: 600.) and the eigenvectors are stored in *note evec:
     600.  It first calls *note gsl_eigen_gen(): 5fb. to compute the
     eigenvalues, Schur forms, and Schur vectors.  Then it finds
     eigenvectors of the Schur forms and backtransforms them using the
     Schur vectors.  The Schur vectors are destroyed in the process, but
     can be saved by using *note gsl_eigen_genv_QZ(): 601.  The computed
     eigenvectors are normalized to have unit magnitude.  On output,
     (*note A: 600, *note B: 600.) contains the generalized Schur form
     (S, T). If *note gsl_eigen_gen(): 5fb. fails, no eigenvectors are
     computed, and an error code is returned.

 -- Function: int gsl_eigen_genv_QZ (gsl_matrix *A, gsl_matrix *B,
          gsl_vector_complex *alpha, gsl_vector *beta,
          gsl_matrix_complex *evec, gsl_matrix *Q, gsl_matrix *Z,
          gsl_eigen_genv_workspace *w)

     This function is identical to *note gsl_eigen_genv(): 600. except
     that it also computes the left and right Schur vectors and stores
     them into *note Q: 601. and *note Z: 601. respectively.


File: gsl-ref.info,  Node: Sorting Eigenvalues and Eigenvectors,  Next: Examples<10>,  Prev: Real Generalized Nonsymmetric Eigensystems,  Up: Eigensystems

15.7 Sorting Eigenvalues and Eigenvectors
=========================================

 -- Function: int gsl_eigen_symmv_sort (gsl_vector *eval, gsl_matrix
          *evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector *note eval: 603. and the corresponding real eigenvectors
     stored in the columns of the matrix *note evec: 603. into ascending
     or descending order according to the value of the parameter *note
     sort_type: 603,

      -- Type: gsl_eigen_sort_t

          ‘GSL_EIGEN_SORT_VAL_ASC’              ascending order in numerical value
                                                
                                                
          ‘GSL_EIGEN_SORT_VAL_DESC’             descending order in numerical value
                                                
                                                
          ‘GSL_EIGEN_SORT_ABS_ASC’              ascending order in magnitude
                                                
                                                
          ‘GSL_EIGEN_SORT_ABS_DESC’             descending order in magnitude
                                                

 -- Function: int gsl_eigen_hermv_sort (gsl_vector *eval,
          gsl_matrix_complex *evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector *note eval: 605. and the corresponding complex eigenvectors
     stored in the columns of the matrix *note evec: 605. into ascending
     or descending order according to the value of the parameter *note
     sort_type: 605. as shown above.

 -- Function: int gsl_eigen_nonsymmv_sort (gsl_vector_complex *eval,
          gsl_matrix_complex *evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector *note eval: 606. and the corresponding complex eigenvectors
     stored in the columns of the matrix *note evec: 606. into ascending
     or descending order according to the value of the parameter *note
     sort_type: 606. as shown above.  Only ‘GSL_EIGEN_SORT_ABS_ASC’ and
     ‘GSL_EIGEN_SORT_ABS_DESC’ are supported due to the eigenvalues
     being complex.

 -- Function: int gsl_eigen_gensymmv_sort (gsl_vector *eval, gsl_matrix
          *evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector *note eval: 607. and the corresponding real eigenvectors
     stored in the columns of the matrix *note evec: 607. into ascending
     or descending order according to the value of the parameter *note
     sort_type: 607. as shown above.

 -- Function: int gsl_eigen_genhermv_sort (gsl_vector *eval,
          gsl_matrix_complex *evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector *note eval: 608. and the corresponding complex eigenvectors
     stored in the columns of the matrix *note evec: 608. into ascending
     or descending order according to the value of the parameter *note
     sort_type: 608. as shown above.

 -- Function: int gsl_eigen_genv_sort (gsl_vector_complex *alpha,
          gsl_vector *beta, gsl_matrix_complex *evec, gsl_eigen_sort_t
          sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vectors (*note alpha: 609, *note beta: 609.) and the corresponding
     complex eigenvectors stored in the columns of the matrix *note
     evec: 609. into ascending or descending order according to the
     value of the parameter *note sort_type: 609. as shown above.  Only
     ‘GSL_EIGEN_SORT_ABS_ASC’ and ‘GSL_EIGEN_SORT_ABS_DESC’ are
     supported due to the eigenvalues being complex.


File: gsl-ref.info,  Node: Examples<10>,  Next: References and Further Reading<10>,  Prev: Sorting Eigenvalues and Eigenvectors,  Up: Eigensystems

15.8 Examples
=============

The following program computes the eigenvalues and eigenvectors of the
4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_eigen.h>

     int
     main (void)
     {
       double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
                         1/2.0, 1/3.0, 1/4.0, 1/5.0,
                         1/3.0, 1/4.0, 1/5.0, 1/6.0,
                         1/4.0, 1/5.0, 1/6.0, 1/7.0 };

       gsl_matrix_view m
         = gsl_matrix_view_array (data, 4, 4);

       gsl_vector *eval = gsl_vector_alloc (4);
       gsl_matrix *evec = gsl_matrix_alloc (4, 4);

       gsl_eigen_symmv_workspace * w =
         gsl_eigen_symmv_alloc (4);

       gsl_eigen_symmv (&m.matrix, eval, evec, w);

       gsl_eigen_symmv_free (w);

       gsl_eigen_symmv_sort (eval, evec,
                             GSL_EIGEN_SORT_ABS_ASC);

       {
         int i;

         for (i = 0; i < 4; i++)
           {
             double eval_i
                = gsl_vector_get (eval, i);
             gsl_vector_view evec_i
                = gsl_matrix_column (evec, i);

             printf ("eigenvalue = %g\n", eval_i);
             printf ("eigenvector = \n");
             gsl_vector_fprintf (stdout,
                                 &evec_i.vector, "%g");
           }
       }

       gsl_vector_free (eval);
       gsl_matrix_free (evec);

       return 0;
     }

Here is the beginning of the output from the program:

     $ ./a.out
     eigenvalue = 9.67023e-05
     eigenvector =
     -0.0291933
     0.328712
     -0.791411
     0.514553
     ...

This can be compared with the corresponding output from GNU octave:

     octave> [v,d] = eig(hilb(4));
     octave> diag(d)
     ans =

        9.6702e-05
        6.7383e-03
        1.6914e-01
        1.5002e+00

     octave> v
     v =

        0.029193   0.179186  -0.582076   0.792608
       -0.328712  -0.741918   0.370502   0.451923
        0.791411   0.100228   0.509579   0.322416
       -0.514553   0.638283   0.514048   0.252161

Note that the eigenvectors can differ by a change of sign, since the
sign of an eigenvector is arbitrary.

The following program illustrates the use of the nonsymmetric
eigensolver, by computing the eigenvalues and eigenvectors of the
Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4).

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_eigen.h>

     int
     main (void)
     {
       double data[] = { -1.0, 1.0, -1.0, 1.0,
                         -8.0, 4.0, -2.0, 1.0,
                         27.0, 9.0, 3.0, 1.0,
                         64.0, 16.0, 4.0, 1.0 };

       gsl_matrix_view m
         = gsl_matrix_view_array (data, 4, 4);

       gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
       gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);

       gsl_eigen_nonsymmv_workspace * w =
         gsl_eigen_nonsymmv_alloc (4);

       gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);

       gsl_eigen_nonsymmv_free (w);

       gsl_eigen_nonsymmv_sort (eval, evec,
                                GSL_EIGEN_SORT_ABS_DESC);

       {
         int i, j;

         for (i = 0; i < 4; i++)
           {
             gsl_complex eval_i
                = gsl_vector_complex_get (eval, i);
             gsl_vector_complex_view evec_i
                = gsl_matrix_complex_column (evec, i);

             printf ("eigenvalue = %g + %gi\n",
                     GSL_REAL(eval_i), GSL_IMAG(eval_i));
             printf ("eigenvector = \n");
             for (j = 0; j < 4; ++j)
               {
                 gsl_complex z =
                   gsl_vector_complex_get(&evec_i.vector, j);
                 printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
               }
           }
       }

       gsl_vector_complex_free(eval);
       gsl_matrix_complex_free(evec);

       return 0;
     }

Here is the beginning of the output from the program:

     $ ./a.out
     eigenvalue = -6.41391 + 0i
     eigenvector =
     -0.0998822 + 0i
     -0.111251 + 0i
     0.292501 + 0i
     0.944505 + 0i
     eigenvalue = 5.54555 + 3.08545i
     eigenvector =
     -0.043487 + -0.0076308i
     0.0642377 + -0.142127i
     -0.515253 + 0.0405118i
     -0.840592 + -0.00148565i
     ...

This can be compared with the corresponding output from GNU octave:

     octave> [v,d] = eig(vander([-1 -2 3 4]));
     octave> diag(d)
     ans =

       -6.4139 + 0.0000i
        5.5456 + 3.0854i
        5.5456 - 3.0854i
        2.3228 + 0.0000i

     octave> v
     v =

      Columns 1 through 3:

       -0.09988 + 0.00000i  -0.04350 - 0.00755i  -0.04350 + 0.00755i
       -0.11125 + 0.00000i   0.06399 - 0.14224i   0.06399 + 0.14224i
        0.29250 + 0.00000i  -0.51518 + 0.04142i  -0.51518 - 0.04142i
        0.94451 + 0.00000i  -0.84059 + 0.00000i  -0.84059 - 0.00000i

      Column 4:

       -0.14493 + 0.00000i
        0.35660 + 0.00000i
        0.91937 + 0.00000i
        0.08118 + 0.00000i

Note that the eigenvectors corresponding to the eigenvalue 5.54555 +
3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i
which is an arbitrary phase factor of magnitude 1.


File: gsl-ref.info,  Node: References and Further Reading<10>,  Prev: Examples<10>,  Up: Eigensystems

15.9 References and Further Reading
===================================

Further information on the algorithms described in this section can be
found in the following book,

   * G. H. Golub, C. F. Van Loan, “Matrix Computations” (3rd Ed, 1996),
     Johns Hopkins University Press, ISBN 0-8018-5414-8.

Further information on the generalized eigensystems QZ algorithm can be
found in this paper,

   * C. Moler, G. Stewart, “An Algorithm for Generalized Matrix
     Eigenvalue Problems”, SIAM J. Numer.  Anal., Vol 10, No 2, 1973.

Eigensystem routines for very large matrices can be found in the Fortran
library LAPACK. The LAPACK library is described in,

   * LAPACK Users’ Guide (Third Edition, 1999), Published by SIAM, ISBN
     0-89871-447-8.

The LAPACK source code can be found at the website
‘http://www.netlib.org/lapack’ along with an online copy of the users
guide.


File: gsl-ref.info,  Node: Fast Fourier Transforms FFTs,  Next: Numerical Integration,  Prev: Eigensystems,  Up: Top

16 Fast Fourier Transforms (FFTs)
*********************************

This chapter describes functions for performing Fast Fourier Transforms
(FFTs).  The library includes radix-2 routines (for lengths which are a
power of two) and mixed-radix routines (which work for any length).  For
efficiency there are separate versions of the routines for real data and
for complex data.  The mixed-radix routines are a reimplementation of
the FFTPACK library of Paul Swarztrauber.  Fortran code for FFTPACK is
available on Netlib (FFTPACK also includes some routines for sine and
cosine transforms but these are currently not available in GSL). For
details and derivations of the underlying algorithms consult the
document “GSL FFT Algorithms” (see *note References and Further Reading:
60e.)

* Menu:

* Mathematical Definitions::
* Overview of complex data FFTs::
* Radix-2 FFT routines for complex data::
* Mixed-radix FFT routines for complex data::
* Overview of real data FFTs::
* Radix-2 FFT routines for real data::
* Mixed-radix FFT routines for real data::
* References and Further Reading: References and Further Reading<11>.


File: gsl-ref.info,  Node: Mathematical Definitions,  Next: Overview of complex data FFTs,  Up: Fast Fourier Transforms FFTs

16.1 Mathematical Definitions
=============================

Fast Fourier Transforms are efficient algorithms for calculating the
discrete Fourier transform (DFT),

     x_j = \sum_{k=0}^{n-1} z_k \exp(-2 \pi i j k / n)

The DFT usually arises as an approximation to the continuous Fourier
transform when functions are sampled at discrete intervals in space or
time.  The naive evaluation of the discrete Fourier transform is a
matrix-vector multiplication W\vec{z}.  A general matrix-vector
multiplication takes O(n^2) operations for n data-points.  Fast Fourier
transform algorithms use a divide-and-conquer strategy to factorize the
matrix W into smaller sub-matrices, corresponding to the integer factors
of the length n.  If n can be factorized into a product of integers f_1
f_2 \ldots f_m then the DFT can be computed in O(n \sum f_i) operations.
For a radix-2 FFT this gives an operation count of O(n \log_2 n).

All the FFT functions offer three types of transform: forwards, inverse
and backwards, based on the same mathematical definitions.  The
definition of the `forward Fourier transform', x = \hbox{FFT}(z), is,

     x_j = \sum_{k=0}^{n-1} z_k \exp(-2 \pi i j k / n)

and the definition of the `inverse Fourier transform', x =
\hbox{IFFT}(z), is,

     z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2 \pi i j k / n).

The factor of 1/n makes this a true inverse.  For example, a call to
*note gsl_fft_complex_forward(): 610. followed by a call to *note
gsl_fft_complex_inverse(): 611. should return the original data (within
numerical errors).

In general there are two possible choices for the sign of the
exponential in the transform/ inverse-transform pair.  GSL follows the
same convention as FFTPACK, using a negative exponential for the forward
transform.  The advantage of this convention is that the inverse
transform recreates the original function with simple Fourier synthesis.
Numerical Recipes uses the opposite convention, a positive exponential
in the forward transform.

The `backwards FFT' is simply our terminology for an unscaled version of
the inverse FFT,

     z^{backwards}_j = \sum_{k=0}^{n-1} x_k \exp(2 \pi i j k / n)

When the overall scale of the result is unimportant it is often
convenient to use the backwards FFT instead of the inverse to save
unnecessary divisions.


File: gsl-ref.info,  Node: Overview of complex data FFTs,  Next: Radix-2 FFT routines for complex data,  Prev: Mathematical Definitions,  Up: Fast Fourier Transforms FFTs

16.2 Overview of complex data FFTs
==================================

The inputs and outputs for the complex FFT routines are `packed arrays'
of floating point numbers.  In a packed array the real and imaginary
parts of each complex number are placed in alternate neighboring
elements.  For example, the following definition of a packed array of
length 6:

     double x[3*2];
     gsl_complex_packed_array data = x;

can be used to hold an array of three complex numbers, ‘z[3]’, in the
following way:

     data[0] = Re(z[0])
     data[1] = Im(z[0])
     data[2] = Re(z[1])
     data[3] = Im(z[1])
     data[4] = Re(z[2])
     data[5] = Im(z[2])

The array indices for the data have the same ordering as those in the
definition of the DFT—i.e.  there are no index transformations or
permutations of the data.

A `stride' parameter allows the user to perform transforms on the
elements ‘z[stride*i]’ instead of ‘z[i]’.  A stride greater than 1 can
be used to take an in-place FFT of the column of a matrix.  A stride of
1 accesses the array without any additional spacing between elements.

To perform an FFT on a vector argument, such as ‘gsl_vector_complex *
v’, use the following definitions (or their equivalents) when calling
the functions described in this chapter:

     gsl_complex_packed_array data = v->data;
     size_t stride = v->stride;
     size_t n = v->size;

For physical applications it is important to remember that the index
appearing in the DFT does not correspond directly to a physical
frequency.  If the time-step of the DFT is \Delta then the
frequency-domain includes both positive and negative frequencies,
ranging from -1/(2\Delta) through 0 to +1/(2\Delta).  The positive
frequencies are stored from the beginning of the array up to the middle,
and the negative frequencies are stored backwards from the end of the
array.

Here is a table which shows the layout of the array ‘data’, and the
correspondence between the time-domain data z, and the frequency-domain
data x:

     index    z               x = FFT(z)

     0        z(t = 0)        x(f = 0)
     1        z(t = 1)        x(f = 1/(n Delta))
     2        z(t = 2)        x(f = 2/(n Delta))
     .        ........        ..................
     n/2      z(t = n/2)      x(f = +1/(2 Delta),
                                    -1/(2 Delta))
     .        ........        ..................
     n-3      z(t = n-3)      x(f = -3/(n Delta))
     n-2      z(t = n-2)      x(f = -2/(n Delta))
     n-1      z(t = n-1)      x(f = -1/(n Delta))

When n is even the location n/2 contains the most positive and negative
frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent.  If n
is odd then general structure of the table above still applies, but n/2
does not appear.


File: gsl-ref.info,  Node: Radix-2 FFT routines for complex data,  Next: Mixed-radix FFT routines for complex data,  Prev: Overview of complex data FFTs,  Up: Fast Fourier Transforms FFTs

16.3 Radix-2 FFT routines for complex data
==========================================

The radix-2 algorithms described in this section are simple and compact,
although not necessarily the most efficient.  They use the Cooley-Tukey
algorithm to compute in-place complex FFTs for lengths which are a power
of 2—no additional storage is required.  The corresponding self-sorting
mixed-radix routines offer better performance at the expense of
requiring additional working space.

All the functions described in this section are declared in the header
file ‘gsl_fft_complex.h’.

 -- Function: int gsl_fft_complex_radix2_forward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_transform
          (gsl_complex_packed_array data, size_t stride, size_t n,
          gsl_fft_direction sign)
 -- Function: int gsl_fft_complex_radix2_backward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_inverse
          (gsl_complex_packed_array data, size_t stride, size_t n)

     These functions compute forward, backward and inverse FFTs of
     length *note n: 616. with stride *note stride: 616, on the packed
     complex array *note data: 616. using an in-place radix-2
     decimation-in-time algorithm.  The length of the transform *note n:
     616. is restricted to powers of two.  For the ‘transform’ version
     of the function the ‘sign’ argument can be either ‘forward’ (-1) or
     ‘backward’ (+1).

     The functions return a value of ‘GSL_SUCCESS’ if no errors were
     detected, or *note GSL_EDOM: 28. if the length of the data *note n:
     616. is not a power of two.

 -- Function: int gsl_fft_complex_radix2_dif_forward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_dif_transform
          (gsl_complex_packed_array data, size_t stride, size_t n,
          gsl_fft_direction sign)
 -- Function: int gsl_fft_complex_radix2_dif_backward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_dif_inverse
          (gsl_complex_packed_array data, size_t stride, size_t n)

     These are decimation-in-frequency versions of the radix-2 FFT
     functions.

Here is an example program which computes the FFT of a short pulse in a
sample of length 128.  To make the resulting Fourier transform real the
pulse is defined for equal positive and negative times (-10 \dots 10),
where the negative times wrap around the end of the array.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>

     #define REAL(z,i) ((z)[2*(i)])
     #define IMAG(z,i) ((z)[2*(i)+1])

     int
     main (void)
     {
       int i; double data[2*128];

       for (i = 0; i < 128; i++)
         {
            REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
         }

       REAL(data,0) = 1.0;

       for (i = 1; i <= 10; i++)
         {
            REAL(data,i) = REAL(data,128-i) = 1.0;
         }

       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i), IMAG(data,i));
         }
       printf ("\n\n");

       gsl_fft_complex_radix2_forward (data, 1, 128);

       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i)/sqrt(128),
                   IMAG(data,i)/sqrt(128));
         }

       return 0;
     }

Note that we have assumed that the program is using the default error
handler (which calls ‘abort()’ for any errors).  If you are not using a
safe error handler you would need to check the return status of *note
gsl_fft_complex_radix2_forward(): 35.

The transformed data is rescaled by 1/\sqrt n so that it fits on the
same plot as the input.  Only the real part is shown, by the choice of
the input data the imaginary part is zero.  Allowing for the wrap-around
of negative times at t=128, and working in units of k/n, the DFT
approximates the continuum Fourier transform, giving a modulated sine
function.

     \int_{-a}^{+a} e^{-2 \pi i k x} dx = {\sin(2\pi k a) \over\pi k}

The output of the example program is plotted in Fig.  %s.

[gsl-ref-figures/fft-complex-radix2]
Figure: A pulse and its discrete Fourier transform, output from the
example program.


File: gsl-ref.info,  Node: Mixed-radix FFT routines for complex data,  Next: Overview of real data FFTs,  Prev: Radix-2 FFT routines for complex data,  Up: Fast Fourier Transforms FFTs

16.4 Mixed-radix FFT routines for complex data
==============================================

This section describes mixed-radix FFT algorithms for complex data.  The
mixed-radix functions work for FFTs of any length.  They are a
reimplementation of Paul Swarztrauber’s Fortran FFTPACK library.  The
theory is explained in the review article “Self-sorting Mixed-radix
FFTs” by Clive Temperton.  The routines here use the same indexing
scheme and basic algorithms as FFTPACK.

The mixed-radix algorithm is based on sub-transform modules—highly
optimized small length FFTs which are combined to create larger FFTs.
There are efficient modules for factors of 2, 3, 4, 5, 6 and 7.  The
modules for the composite factors of 4 and 6 are faster than combining
the modules for 2*2 and 2*3.

For factors which are not implemented as modules there is a fall-back to
a general length-n module which uses Singleton’s method for efficiently
computing a DFT. This module is O(n^2), and slower than a dedicated
module would be but works for any length n.  Of course, lengths which
use the general length-n module will still be factorized as much as
possible.  For example, a length of 143 will be factorized into 11*13.
Large prime factors are the worst case scenario, e.g.  as found in
n=2*3*99991, and should be avoided because their O(n^2) scaling will
dominate the run-time (consult the document “GSL FFT Algorithms”
included in the GSL distribution if you encounter this problem).

The mixed-radix initialization function *note
gsl_fft_complex_wavetable_alloc(): 61d. returns the list of factors
chosen by the library for a given length n.  It can be used to check how
well the length has been factorized, and estimate the run-time.  To a
first approximation the run-time scales as n \sum f_i, where the f_i are
the factors of n.  For programs under user control you may wish to issue
a warning that the transform will be slow when the length is poorly
factorized.  If you frequently encounter data lengths which cannot be
factorized using the existing small-prime modules consult “GSL FFT
Algorithms” for details on adding support for other factors.

All the functions described in this section are declared in the header
file ‘gsl_fft_complex.h’.

 -- Function: *note gsl_fft_complex_wavetable: 61e.
          *gsl_fft_complex_wavetable_alloc (size_t n)

     This function prepares a trigonometric lookup table for a complex
     FFT of length *note n: 61d.  The function returns a pointer to the
     newly allocated *note gsl_fft_complex_wavetable: 61e. if no errors
     were detected, and a null pointer in the case of error.  The length
     *note n: 61d. is factorized into a product of subtransforms, and
     the factors and their trigonometric coefficients are stored in the
     wavetable.  The trigonometric coefficients are computed using
     direct calls to ‘sin’ and ‘cos’, for accuracy.  Recursion relations
     could be used to compute the lookup table faster, but if an
     application performs many FFTs of the same length then this
     computation is a one-off overhead which does not affect the final
     throughput.

     The wavetable structure can be used repeatedly for any transform of
     the same length.  The table is not modified by calls to any of the
     other FFT functions.  The same wavetable can be used for both
     forward and backward (or inverse) transforms of a given length.

 -- Function: void gsl_fft_complex_wavetable_free
          (gsl_fft_complex_wavetable *wavetable)

     This function frees the memory associated with the wavetable *note
     wavetable: 61f.  The wavetable can be freed if no further FFTs of
     the same length will be needed.

These functions operate on a *note gsl_fft_complex_wavetable: 61e.
structure which contains internal parameters for the FFT. It is not
necessary to set any of the components directly but it can sometimes be
useful to examine them.  For example, the chosen factorization of the
FFT length is given and can be used to provide an estimate of the
run-time or numerical error.  The wavetable structure is declared in the
header file ‘gsl_fft_complex.h’.

 -- Type: gsl_fft_complex_wavetable

     This is a structure that holds the factorization and trigonometric
     lookup tables for the mixed radix fft algorithm.  It has the
     following components:

     ‘size_t n’                            This is the number of complex data points
                                           
                                           
     ‘size_t nf’                           This is the number of factors that the length ‘n’ was decomposed into.
                                           
                                           
     ‘size_t factor[64]’                   This is the array of factors.  Only the first ‘nf’ elements are used.
                                           
                                           
     ‘gsl_complex * trig’                  This is a pointer to a preallocated trigonometric lookup table of ‘n’ complex elements.
                                           
                                           
     ‘gsl_complex * twiddle[64]’           This is an array of pointers into ‘trig’, giving the twiddle factors for each pass.
                                           

 -- Type: gsl_fft_complex_workspace

     The mixed radix algorithms require additional working space to hold
     the intermediate steps of the transform.

 -- Function: *note gsl_fft_complex_workspace: 620.
          *gsl_fft_complex_workspace_alloc (size_t n)

     This function allocates a workspace for a complex transform of
     length *note n: 621.

 -- Function: void gsl_fft_complex_workspace_free
          (gsl_fft_complex_workspace *workspace)

     This function frees the memory associated with the workspace *note
     workspace: 622.  The workspace can be freed if no further FFTs of
     the same length will be needed.

The following functions compute the transform,

 -- Function: int gsl_fft_complex_forward (gsl_complex_packed_array
          data, size_t stride, size_t n, const gsl_fft_complex_wavetable
          *wavetable, gsl_fft_complex_workspace *work)
 -- Function: int gsl_fft_complex_transform (gsl_complex_packed_array
          data, size_t stride, size_t n, const gsl_fft_complex_wavetable
          *wavetable, gsl_fft_complex_workspace *work, gsl_fft_direction
          sign)
 -- Function: int gsl_fft_complex_backward (gsl_complex_packed_array
          data, size_t stride, size_t n, const gsl_fft_complex_wavetable
          *wavetable, gsl_fft_complex_workspace *work)
 -- Function: int gsl_fft_complex_inverse (gsl_complex_packed_array
          data, size_t stride, size_t n, const gsl_fft_complex_wavetable
          *wavetable, gsl_fft_complex_workspace *work)

     These functions compute forward, backward and inverse FFTs of
     length *note n: 611. with stride *note stride: 611, on the packed
     complex array *note data: 611, using a mixed radix
     decimation-in-frequency algorithm.  There is no restriction on the
     length *note n: 611.  Efficient modules are provided for
     subtransforms of length 2, 3, 4, 5, 6 and 7.  Any remaining factors
     are computed with a slow, O(n^2), general-n module.  The caller
     must supply a *note wavetable: 611. containing the trigonometric
     lookup tables and a workspace *note work: 611.  For the ‘transform’
     version of the function the ‘sign’ argument can be either ‘forward’
     (-1) or ‘backward’ (+1).

     The functions return a value of ‘0’ if no errors were detected.
     The following ‘gsl_errno’ conditions are defined for these
     functions:

     *note GSL_EDOM: 28.                     The length of the data *note n: 611. is not a positive integer (i.e. *note n: 611. is zero).
                                             
                                             
     *note GSL_EINVAL: 2b.                   The length of the data *note n: 611. and the length used to compute the given *note wavetable: 611. do not
                                             match.
                                             

Here is an example program which computes the FFT of a short pulse in a
sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>

     #define REAL(z,i) ((z)[2*(i)])
     #define IMAG(z,i) ((z)[2*(i)+1])

     int
     main (void)
     {
       int i;
       const int n = 630;
       double data[2*n];

       gsl_fft_complex_wavetable * wavetable;
       gsl_fft_complex_workspace * workspace;

       for (i = 0; i < n; i++)
         {
           REAL(data,i) = 0.0;
           IMAG(data,i) = 0.0;
         }

       data[0] = 1.0;

       for (i = 1; i <= 10; i++)
         {
           REAL(data,i) = REAL(data,n-i) = 1.0;
         }

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e %e\n", i, REAL(data,i),
                                     IMAG(data,i));
         }
       printf ("\n");

       wavetable = gsl_fft_complex_wavetable_alloc (n);
       workspace = gsl_fft_complex_workspace_alloc (n);

       for (i = 0; i < (int) wavetable->nf; i++)
         {
            printf ("# factor %d: %zu\n", i,
                    wavetable->factor[i]);
         }

       gsl_fft_complex_forward (data, 1, n,
                                wavetable, workspace);

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e %e\n", i, REAL(data,i),
                                     IMAG(data,i));
         }

       gsl_fft_complex_wavetable_free (wavetable);
       gsl_fft_complex_workspace_free (workspace);
       return 0;
     }

Note that we have assumed that the program is using the default ‘gsl’
error handler (which calls ‘abort()’ for any errors).  If you are not
using a safe error handler you would need to check the return status of
all the ‘gsl’ routines.


File: gsl-ref.info,  Node: Overview of real data FFTs,  Next: Radix-2 FFT routines for real data,  Prev: Mixed-radix FFT routines for complex data,  Up: Fast Fourier Transforms FFTs

16.5 Overview of real data FFTs
===============================

The functions for real data are similar to those for complex data.
However, there is an important difference between forward and inverse
transforms.  The Fourier transform of a real sequence is not real.  It
is a complex sequence with a special symmetry:

     z_k = z_{n-k}^*

A sequence with this symmetry is called `conjugate-complex' or
`half-complex'.  This different structure requires different storage
layouts for the forward transform (from real to half-complex) and
inverse transform (from half-complex back to real).  As a consequence
the routines are divided into two sets: functions in ‘gsl_fft_real’
which operate on real sequences and functions in ‘gsl_fft_halfcomplex’
which operate on half-complex sequences.

Functions in ‘gsl_fft_real’ compute the frequency coefficients of a real
sequence.  The half-complex coefficients c of a real sequence x are
given by Fourier analysis,

     c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)

Functions in ‘gsl_fft_halfcomplex’ compute inverse or backwards
transforms.  They reconstruct real sequences by Fourier synthesis from
their half-complex frequency coefficients, c,

     x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)

The symmetry of the half-complex sequence implies that only half of the
complex numbers in the output need to be stored.  The remaining half can
be reconstructed using the half-complex symmetry condition.  This works
for all lengths, even and odd—when the length is even the middle value
where k=n/2 is also real.  Thus only ‘n’ real numbers are required to
store the half-complex sequence, and the transform of a real sequence
can be stored in the same size array as the original data.

The precise storage arrangements depend on the algorithm, and are
different for radix-2 and mixed-radix routines.  The radix-2 function
operates in-place, which constrains the locations where each element can
be stored.  The restriction forces real and imaginary parts to be stored
far apart.  The mixed-radix algorithm does not have this restriction,
and it stores the real and imaginary parts of a given term in
neighboring locations (which is desirable for better locality of memory
accesses).


File: gsl-ref.info,  Node: Radix-2 FFT routines for real data,  Next: Mixed-radix FFT routines for real data,  Prev: Overview of real data FFTs,  Up: Fast Fourier Transforms FFTs

16.6 Radix-2 FFT routines for real data
=======================================

This section describes radix-2 FFT algorithms for real data.  They use
the Cooley-Tukey algorithm to compute in-place FFTs for lengths which
are a power of 2.

The radix-2 FFT functions for real data are declared in the header files
‘gsl_fft_real.h’

 -- Function: int gsl_fft_real_radix2_transform (double data[], size_t
          stride, size_t n)

     This function computes an in-place radix-2 FFT of length *note n:
     627. and stride *note stride: 627. on the real array *note data:
     627.  The output is a half-complex sequence, which is stored
     in-place.  The arrangement of the half-complex terms uses the
     following scheme: for k < n/2 the real part of the k-th term is
     stored in location k, and the corresponding imaginary part is
     stored in location n-k.  Terms with k > n/2 can be reconstructed
     using the symmetry z_k = z^*_{n-k}.  The terms for k=0 and k=n/2
     are both purely real, and count as a special case.  Their real
     parts are stored in locations 0 and n/2 respectively, while their
     imaginary parts which are zero are not stored.

     The following table shows the correspondence between the output
     *note data: 627. and the equivalent results obtained by considering
     the input data as a complex sequence with zero imaginary part
     (assuming *note stride: 627. = 1}):

          complex[0].real    =    data[0]
          complex[0].imag    =    0
          complex[1].real    =    data[1]
          complex[1].imag    =    data[n-1]
          ...............         ................
          complex[k].real    =    data[k]
          complex[k].imag    =    data[n-k]
          ...............         ................
          complex[n/2].real  =    data[n/2]
          complex[n/2].imag  =    0
          ...............         ................
          complex[k'].real   =    data[k]        k' = n - k
          complex[k'].imag   =   -data[n-k]
          ...............         ................
          complex[n-1].real  =    data[1]
          complex[n-1].imag  =   -data[n-1]

     Note that the output data can be converted into the full complex
     sequence using the function *note
     gsl_fft_halfcomplex_radix2_unpack(): 628. described below.

The radix-2 FFT functions for halfcomplex data are declared in the
header file ‘gsl_fft_halfcomplex.h’.

 -- Function: int gsl_fft_halfcomplex_radix2_inverse (double data[],
          size_t stride, size_t n)
 -- Function: int gsl_fft_halfcomplex_radix2_backward (double data[],
          size_t stride, size_t n)

     These functions compute the inverse or backwards in-place radix-2
     FFT of length *note n: 62a. and stride *note stride: 62a. on the
     half-complex sequence *note data: 62a. stored according the output
     scheme used by ‘gsl_fft_real_radix2()’.  The result is a real array
     stored in natural order.

 -- Function: int gsl_fft_halfcomplex_radix2_unpack (const double
          halfcomplex_coefficient[], gsl_complex_packed_array
          complex_coefficient, size_t stride, size_t n)

     This function converts *note halfcomplex_coefficient: 628, an array
     of half-complex coefficients as returned by *note
     gsl_fft_real_radix2_transform(): 627, into an ordinary complex
     array, *note complex_coefficient: 628.  It fills in the complex
     array using the symmetry z_k = z_{n-k}^* to reconstruct the
     redundant elements.  The algorithm for the conversion is:

          complex_coefficient[0].real = halfcomplex_coefficient[0];
          complex_coefficient[0].imag = 0.0;

          for (i = 1; i < n - i; i++)
            {
              double hc_real = halfcomplex_coefficient[i*stride];
              double hc_imag = halfcomplex_coefficient[(n-i)*stride];
              complex_coefficient[i*stride].real = hc_real;
              complex_coefficient[i*stride].imag = hc_imag;
              complex_coefficient[(n - i)*stride].real = hc_real;
              complex_coefficient[(n - i)*stride].imag = -hc_imag;
            }

          if (i == n - i)
            {
              complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride];
              complex_coefficient[i*stride].imag = 0.0;
            }


File: gsl-ref.info,  Node: Mixed-radix FFT routines for real data,  Next: References and Further Reading<11>,  Prev: Radix-2 FFT routines for real data,  Up: Fast Fourier Transforms FFTs

16.7 Mixed-radix FFT routines for real data
===========================================

This section describes mixed-radix FFT algorithms for real data.  The
mixed-radix functions work for FFTs of any length.  They are a
reimplementation of the real-FFT routines in the Fortran FFTPACK library
by Paul Swarztrauber.  The theory behind the algorithm is explained in
the article “Fast Mixed-Radix Real Fourier Transforms” by Clive
Temperton.  The routines here use the same indexing scheme and basic
algorithms as FFTPACK.

The functions use the FFTPACK storage convention for half-complex
sequences.  In this convention the half-complex transform of a real
sequence is stored with frequencies in increasing order, starting at
zero, with the real and imaginary parts of each frequency in neighboring
locations.  When a value is known to be real the imaginary part is not
stored.  The imaginary part of the zero-frequency component is never
stored.  It is known to be zero (since the zero frequency component is
simply the sum of the input data (all real)).  For a sequence of even
length the imaginary part of the frequency n/2 is not stored either,
since the symmetry z_k = z_{n-k}^* implies that this is purely real too.

The storage scheme is best shown by some examples.  The table below
shows the output for an odd-length sequence, n=5.  The two columns give
the correspondence between the 5 values in the half-complex sequence
returned by *note gsl_fft_real_transform(): 62c, ‘halfcomplex[]’ and the
values ‘complex[]’ that would be returned if the same real input
sequence were passed to *note gsl_fft_complex_backward(): 624. as a
complex sequence (with imaginary parts set to ‘0’):

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[3]
     complex[3].imag  = -halfcomplex[4]
     complex[4].real  =  halfcomplex[1]
     complex[4].imag  = -halfcomplex[2]

The upper elements of the ‘complex’ array, ‘complex[3]’ and ‘complex[4]’
are filled in using the symmetry condition.  The imaginary part of the
zero-frequency term ‘complex[0].imag’ is known to be zero by the
symmetry.

The next table shows the output for an even-length sequence, n=6.  In
the even case there are two values which are purely real:

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[5]
     complex[3].imag  =  0
     complex[4].real  =  halfcomplex[3]
     complex[4].imag  = -halfcomplex[4]
     complex[5].real  =  halfcomplex[1]
     complex[5].imag  = -halfcomplex[2]

The upper elements of the ‘complex’ array, ‘complex[4]’ and ‘complex[5]’
are filled in using the symmetry condition.  Both ‘complex[0].imag’ and
‘complex[3].imag’ are known to be zero.

All these functions are declared in the header files ‘gsl_fft_real.h’
and ‘gsl_fft_halfcomplex.h’.

 -- Type: gsl_fft_real_wavetable
 -- Type: gsl_fft_halfcomplex_wavetable

     These data structures contain lookup tables for an FFT of a fixed
     size.

 -- Function: *note gsl_fft_real_wavetable: 62d.
          *gsl_fft_real_wavetable_alloc (size_t n)
 -- Function: *note gsl_fft_halfcomplex_wavetable: 62e.
          *gsl_fft_halfcomplex_wavetable_alloc (size_t n)

     These functions prepare trigonometric lookup tables for an FFT of
     size n real elements.  The functions return a pointer to the newly
     allocated struct if no errors were detected, and a null pointer in
     the case of error.  The length *note n: 630. is factorized into a
     product of subtransforms, and the factors and their trigonometric
     coefficients are stored in the wavetable.  The trigonometric
     coefficients are computed using direct calls to ‘sin’ and ‘cos’,
     for accuracy.  Recursion relations could be used to compute the
     lookup table faster, but if an application performs many FFTs of
     the same length then computing the wavetable is a one-off overhead
     which does not affect the final throughput.

     The wavetable structure can be used repeatedly for any transform of
     the same length.  The table is not modified by calls to any of the
     other FFT functions.  The appropriate type of wavetable must be
     used for forward real or inverse half-complex transforms.

 -- Function: void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable
          *wavetable)
 -- Function: void gsl_fft_halfcomplex_wavetable_free
          (gsl_fft_halfcomplex_wavetable *wavetable)

     These functions free the memory associated with the wavetable *note
     wavetable: 632.  The wavetable can be freed if no further FFTs of
     the same length will be needed.

The mixed radix algorithms require additional working space to hold the
intermediate steps of the transform,

 -- Type: gsl_fft_real_workspace

     This workspace contains parameters needed to compute a real FFT.

 -- Function: *note gsl_fft_real_workspace: 633.
          *gsl_fft_real_workspace_alloc (size_t n)

     This function allocates a workspace for a real transform of length
     *note n: 634.  The same workspace can be used for both forward real
     and inverse halfcomplex transforms.

 -- Function: void gsl_fft_real_workspace_free (gsl_fft_real_workspace
          *workspace)

     This function frees the memory associated with the workspace *note
     workspace: 635.  The workspace can be freed if no further FFTs of
     the same length will be needed.

The following functions compute the transforms of real and half-complex
data,

 -- Function: int gsl_fft_real_transform (double data[], size_t stride,
          size_t n, const gsl_fft_real_wavetable *wavetable,
          gsl_fft_real_workspace *work)
 -- Function: int gsl_fft_halfcomplex_transform (double data[], size_t
          stride, size_t n, const gsl_fft_halfcomplex_wavetable
          *wavetable, gsl_fft_real_workspace *work)

     These functions compute the FFT of *note data: 636, a real or
     half-complex array of length *note n: 636, using a mixed radix
     decimation-in-frequency algorithm.  For *note
     gsl_fft_real_transform(): 62c. *note data: 636. is an array of
     time-ordered real data.  For *note gsl_fft_halfcomplex_transform():
     636. *note data: 636. contains Fourier coefficients in the
     half-complex ordering described above.  There is no restriction on
     the length *note n: 636.  Efficient modules are provided for
     subtransforms of length 2, 3, 4 and 5.  Any remaining factors are
     computed with a slow, O(n^2), general-n module.  The caller must
     supply a *note wavetable: 636. containing trigonometric lookup
     tables and a workspace *note work: 636.

 -- Function: int gsl_fft_real_unpack (const double real_coefficient[],
          gsl_complex_packed_array complex_coefficient, size_t stride,
          size_t n)

     This function converts a single real array, *note real_coefficient:
     637. into an equivalent complex array, *note complex_coefficient:
     637, (with imaginary part set to zero), suitable for
     ‘gsl_fft_complex’ routines.  The algorithm for the conversion is
     simply:

          for (i = 0; i < n; i++)
            {
              complex_coefficient[i*stride].real = real_coefficient[i*stride];
              complex_coefficient[i*stride].imag = 0.0;
            }

 -- Function: int gsl_fft_halfcomplex_unpack (const double
          halfcomplex_coefficient[], gsl_complex_packed_array
          complex_coefficient, size_t stride, size_t n)

     This function converts *note halfcomplex_coefficient: 638, an array
     of half-complex coefficients as returned by *note
     gsl_fft_real_transform(): 62c, into an ordinary complex array,
     *note complex_coefficient: 638.  It fills in the complex array
     using the symmetry z_k = z_{n-k}^* to reconstruct the redundant
     elements.  The algorithm for the conversion is:

          complex_coefficient[0].real = halfcomplex_coefficient[0];
          complex_coefficient[0].imag = 0.0;

          for (i = 1; i < n - i; i++)
            {
              double hc_real = halfcomplex_coefficient[(2 * i - 1)*stride];
              double hc_imag = halfcomplex_coefficient[(2 * i)*stride];
              complex_coefficient[i*stride].real = hc_real;
              complex_coefficient[i*stride].imag = hc_imag;
              complex_coefficient[(n - i)*stride].real = hc_real;
              complex_coefficient[(n - i)*stride].imag = -hc_imag;
            }

          if (i == n - i)
            {
              complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride];
              complex_coefficient[i*stride].imag = 0.0;
            }

Here is an example program using *note gsl_fft_real_transform(): 62c.
and ‘gsl_fft_halfcomplex_inverse()’.  It generates a real signal in the
shape of a square pulse.  The pulse is Fourier transformed to frequency
space, and all but the lowest ten frequency components are removed from
the array of Fourier coefficients returned by *note
gsl_fft_real_transform(): 62c.

The remaining Fourier coefficients are transformed back to the
time-domain, to give a filtered version of the square pulse.  Since
Fourier coefficients are stored using the half-complex symmetry both
positive and negative frequencies are removed and the final filtered
signal is also real.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_real.h>
     #include <gsl/gsl_fft_halfcomplex.h>

     int
     main (void)
     {
       int i, n = 100;
       double data[n];

       gsl_fft_real_wavetable * real;
       gsl_fft_halfcomplex_wavetable * hc;
       gsl_fft_real_workspace * work;

       for (i = 0; i < n; i++)
         {
           data[i] = 0.0;
         }

       for (i = n / 3; i < 2 * n / 3; i++)
         {
           data[i] = 1.0;
         }

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }
       printf ("\n");

       work = gsl_fft_real_workspace_alloc (n);
       real = gsl_fft_real_wavetable_alloc (n);

       gsl_fft_real_transform (data, 1, n,
                               real, work);

       gsl_fft_real_wavetable_free (real);

       for (i = 11; i < n; i++)
         {
           data[i] = 0;
         }

       hc = gsl_fft_halfcomplex_wavetable_alloc (n);

       gsl_fft_halfcomplex_inverse (data, 1, n,
                                    hc, work);
       gsl_fft_halfcomplex_wavetable_free (hc);

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }

       gsl_fft_real_workspace_free (work);
       return 0;
     }

The program output is shown in Fig.  %s.

[gsl-ref-figures/fft-real-mixedradix]
Figure: Low-pass filtered version of a real pulse, output from the
example program.


File: gsl-ref.info,  Node: References and Further Reading<11>,  Prev: Mixed-radix FFT routines for real data,  Up: Fast Fourier Transforms FFTs

16.8 References and Further Reading
===================================

A good starting point for learning more about the FFT is the following
review article,

   * P. Duhamel and M. Vetterli.  Fast Fourier transforms: A tutorial
     review and a state of the art.  Signal Processing, 19:259–299,
     1990.

To find out about the algorithms used in the GSL routines you may want
to consult the document “GSL FFT Algorithms” (it is included in GSL, as
‘doc/fftalgorithms.tex’).  This has general information on FFTs and
explicit derivations of the implementation for each routine.  There are
also references to the relevant literature.  For convenience some of the
more important references are reproduced below.

There are several introductory books on the FFT with example programs,
such as “The Fast Fourier Transform” by Brigham and “DFT/FFT and
Convolution Algorithms” by Burrus and Parks,

   * 
       5. Oran Brigham.  “The Fast Fourier Transform”.  Prentice Hall,
          1974.

   * C. S. Burrus and T. W. Parks.  “DFT/FFT and Convolution
     Algorithms”, Wiley, 1984.

Both these introductory books cover the radix-2 FFT in some detail.  The
mixed-radix algorithm at the heart of the FFTPACK routines is reviewed
in Clive Temperton’s paper,

   * Clive Temperton, Self-sorting mixed-radix fast Fourier transforms,
     Journal of Computational Physics, 52(1):1–23, 1983.

The derivation of FFTs for real-valued data is explained in the
following two articles,

   * Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C.
     Sidney Burrus.  Real-valued fast Fourier transform algorithms.
     “IEEE Transactions on Acoustics, Speech, and Signal Processing”,
     ASSP-35(6):849–863, 1987.

   * Clive Temperton.  Fast mixed-radix real Fourier transforms.
     “Journal of Computational Physics”, 52:340–350, 1983.

In 1979 the IEEE published a compendium of carefully-reviewed Fortran
FFT programs in “Programs for Digital Signal Processing”.  It is a
useful reference for implementations of many different FFT algorithms,

   * Digital Signal Processing Committee and IEEE Acoustics, Speech, and
     Signal Processing Committee, editors.  Programs for Digital Signal
     Processing.  IEEE Press, 1979.

For large-scale FFT work we recommend the use of the dedicated FFTW
library by Frigo and Johnson.  The FFTW library is self-optimizing—it
automatically tunes itself for each hardware platform in order to
achieve maximum performance.  It is available under the GNU GPL.

   * FFTW Website, ‘http://www.fftw.org/’

The source code for FFTPACK is available from
‘http://www.netlib.org/fftpack/’


File: gsl-ref.info,  Node: Numerical Integration,  Next: Random Number Generation,  Prev: Fast Fourier Transforms FFTs,  Up: Top

17 Numerical Integration
************************

This chapter describes routines for performing numerical integration
(quadrature) of a function in one dimension.  There are routines for
adaptive and non-adaptive integration of general functions, with
specialised routines for specific cases.  These include integration over
infinite and semi-infinite ranges, singular integrals, including
logarithmic singularities, computation of Cauchy principal values and
oscillatory integrals.  The library reimplements the algorithms used in
QUADPACK, a numerical integration package written by Piessens, de
Doncker-Kapenga, Ueberhuber and Kahaner.  Fortran code for QUADPACK is
available on Netlib.  Also included are non-adaptive, fixed-order
Gauss-Legendre integration routines with high precision coefficients, as
well as fixed-order quadrature rules for a variety of weighting
functions from IQPACK.

The functions described in this chapter are declared in the header file
‘gsl_integration.h’.

* Menu:

* Introduction: Introduction<2>.
* QNG non-adaptive Gauss-Kronrod integration::
* QAG adaptive integration::
* QAGS adaptive integration with singularities::
* QAGP adaptive integration with known singular points::
* QAGI adaptive integration on infinite intervals::
* QAWC adaptive integration for Cauchy principal values::
* QAWS adaptive integration for singular functions::
* QAWO adaptive integration for oscillatory functions::
* QAWF adaptive integration for Fourier integrals::
* CQUAD doubly-adaptive integration::
* Romberg integration::
* Gauss-Legendre integration::
* Fixed point quadratures::
* Error codes::
* Examples: Examples<11>.
* References and Further Reading: References and Further Reading<12>.


File: gsl-ref.info,  Node: Introduction<2>,  Next: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration

17.1 Introduction
=================

Each algorithm computes an approximation to a definite integral of the
form,

     I = \int_a^b f(x) w(x) dx

where w(x) is a weight function (for general integrands w(x) = 1).  The
user provides absolute and relative error bounds (epsabs, epsrel) which
specify the following accuracy requirement,

     |RESULT - I| <= max(epsabs, epsrel |I|)

where RESULT is the numerical approximation obtained by the algorithm.
The algorithms attempt to estimate the absolute error ABSERR = |RESULT -
I| in such a way that the following inequality holds,

     |RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)

In short, the routines return the first approximation which has an
absolute error smaller than epsabs or a relative error smaller than
epsrel.

Note that this is an `either-or' constraint, not simultaneous.  To
compute to a specified absolute error, set epsrel to zero.  To compute
to a specified relative error, set epsabs to zero.  The routines will
fail to converge if the error bounds are too stringent, but always
return the best approximation obtained up to that stage.

The algorithms in QUADPACK use a naming convention based on the
following letters:

     Q - quadrature routine

     N - non-adaptive integrator
     A - adaptive integrator

     G - general integrand (user-defined)
     W - weight function with integrand

     S - singularities can be more readily integrated
     P - points of special difficulty can be supplied
     I - infinite range of integration
     O - oscillatory weight function, cos or sin
     F - Fourier integral
     C - Cauchy principal value

The algorithms are built on pairs of quadrature rules, a higher order
rule and a lower order rule.  The higher order rule is used to compute
the best approximation to an integral over a small range.  The
difference between the results of the higher order rule and the lower
order rule gives an estimate of the error in the approximation.

* Menu:

* Integrands without weight functions::
* Integrands with weight functions::
* Integrands with singular weight functions::


File: gsl-ref.info,  Node: Integrands without weight functions,  Next: Integrands with weight functions,  Up: Introduction<2>

17.1.1 Integrands without weight functions
------------------------------------------

The algorithms for general functions (without a weight function) are
based on Gauss-Kronrod rules.

A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of
order m.  This is extended with additional points between each of the
abscissae to give a higher order Kronrod rule of order 2m + 1.  The
Kronrod rule is efficient because it reuses existing function
evaluations from the Gaussian rule.

The higher order Kronrod rule is used as the best approximation to the
integral, and the difference between the two rules is used as an
estimate of the error in the approximation.


File: gsl-ref.info,  Node: Integrands with weight functions,  Next: Integrands with singular weight functions,  Prev: Integrands without weight functions,  Up: Introduction<2>

17.1.2 Integrands with weight functions
---------------------------------------

For integrands with weight functions the algorithms use Clenshaw-Curtis
quadrature rules.

A Clenshaw-Curtis rule begins with an n-th order Chebyshev polynomial
approximation to the integrand.  This polynomial can be integrated
exactly to give an approximation to the integral of the original
function.  The Chebyshev expansion can be extended to higher orders to
improve the approximation and provide an estimate of the error.


File: gsl-ref.info,  Node: Integrands with singular weight functions,  Prev: Integrands with weight functions,  Up: Introduction<2>

17.1.3 Integrands with singular weight functions
------------------------------------------------

The presence of singularities (or other behavior) in the integrand can
cause slow convergence in the Chebyshev approximation.  The modified
Clenshaw-Curtis rules used in QUADPACK separate out several common
weight functions which cause slow convergence.

These weight functions are integrated analytically against the Chebyshev
polynomials to precompute `modified Chebyshev moments'.  Combining the
moments with the Chebyshev approximation to the function gives the
desired integral.  The use of analytic integration for the singular part
of the function allows exact cancellations and substantially improves
the overall convergence behavior of the integration.


File: gsl-ref.info,  Node: QNG non-adaptive Gauss-Kronrod integration,  Next: QAG adaptive integration,  Prev: Introduction<2>,  Up: Numerical Integration

17.2 QNG non-adaptive Gauss-Kronrod integration
===============================================

The QNG algorithm is a non-adaptive procedure which uses fixed
Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum
of 87 points.  It is provided for fast integration of smooth functions.

 -- Function: int gsl_integration_qng (const gsl_function *f, double a,
          double b, double epsabs, double epsrel, double *result, double
          *abserr, size_t *neval)

     This function applies the Gauss-Kronrod 10-point, 21-point,
     43-point and 87-point integration rules in succession until an
     estimate of the integral of f over (a,b) is achieved within the
     desired absolute and relative error limits, *note epsabs: 642. and
     *note epsrel: 642.  The function returns the final approximation,
     *note result: 642, an estimate of the absolute error, *note abserr:
     642. and the number of function evaluations used, *note neval: 642.
     The Gauss-Kronrod rules are designed in such a way that each rule
     uses all the results of its predecessors, in order to minimize the
     total number of function evaluations.


File: gsl-ref.info,  Node: QAG adaptive integration,  Next: QAGS adaptive integration with singularities,  Prev: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration

17.3 QAG adaptive integration
=============================

The QAG algorithm is a simple adaptive integration procedure.  The
integration region is divided into subintervals, and on each iteration
the subinterval with the largest estimated error is bisected.  This
reduces the overall error rapidly, as the subintervals become
concentrated around local difficulties in the integrand.  These
subintervals are managed by the following struct,

 -- Type: gsl_integration_workspace

     This workspace handles the memory for the subinterval ranges,
     results and error estimates.

 -- Function: *note gsl_integration_workspace: 644.
          *gsl_integration_workspace_alloc (size_t n)

     This function allocates a workspace sufficient to hold *note n:
     645. double precision intervals, their integration results and
     error estimates.  One workspace may be used multiple times as all
     necessary reinitialization is performed automatically by the
     integration routines.

 -- Function: void gsl_integration_workspace_free
          (gsl_integration_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 646.

 -- Function: int gsl_integration_qag (const gsl_function *f, double a,
          double b, double epsabs, double epsrel, size_t limit, int key,
          gsl_integration_workspace *workspace, double *result, double
          *abserr)

     This function applies an integration rule adaptively until an
     estimate of the integral of f over (a,b) is achieved within the
     desired absolute and relative error limits, *note epsabs: 647. and
     *note epsrel: 647.  The function returns the final approximation,
     *note result: 647, and an estimate of the absolute error, *note
     abserr: 647.  The integration rule is determined by the value of
     *note key: 647, which should be chosen from the following symbolic
     names,

     Symbolic Name                  Key
                                    
     ---------------------------------------
                                    
     ‘GSL_INTEG_GAUSS15’            1
                                    
                                    
     ‘GSL_INTEG_GAUSS21’            2
                                    
                                    
     ‘GSL_INTEG_GAUSS31’            3
                                    
                                    
     ‘GSL_INTEG_GAUSS41’            4
                                    
                                    
     ‘GSL_INTEG_GAUSS51’            5
                                    
                                    
     ‘GSL_INTEG_GAUSS61’            6
                                    

     corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod
     rules.  The higher-order rules give better accuracy for smooth
     functions, while lower-order rules save time when the function
     contains local difficulties, such as discontinuities.

     On each iteration the adaptive integration strategy bisects the
     interval with the largest error estimate.  The subintervals and
     their results are stored in the memory provided by *note workspace:
     647.  The maximum number of subintervals is given by *note limit:
     647, which may not exceed the allocated size of the workspace.


File: gsl-ref.info,  Node: QAGS adaptive integration with singularities,  Next: QAGP adaptive integration with known singular points,  Prev: QAG adaptive integration,  Up: Numerical Integration

17.4 QAGS adaptive integration with singularities
=================================================

The presence of an integrable singularity in the integration region
causes an adaptive routine to concentrate new subintervals around the
singularity.  As the subintervals decrease in size the successive
approximations to the integral converge in a limiting fashion.  This
approach to the limit can be accelerated using an extrapolation
procedure.  The QAGS algorithm combines adaptive bisection with the Wynn
epsilon-algorithm to speed up the integration of many types of
integrable singularities.

 -- Function: int gsl_integration_qags (const gsl_function *f, double a,
          double b, double epsabs, double epsrel, size_t limit,
          gsl_integration_workspace *workspace, double *result, double
          *abserr)

     This function applies the Gauss-Kronrod 21-point integration rule
     adaptively until an estimate of the integral of f over (a,b) is
     achieved within the desired absolute and relative error limits,
     *note epsabs: 649. and *note epsrel: 649.  The results are
     extrapolated using the epsilon-algorithm, which accelerates the
     convergence of the integral in the presence of discontinuities and
     integrable singularities.  The function returns the final
     approximation from the extrapolation, *note result: 649, and an
     estimate of the absolute error, *note abserr: 649.  The
     subintervals and their results are stored in the memory provided by
     *note workspace: 649.  The maximum number of subintervals is given
     by *note limit: 649, which may not exceed the allocated size of the
     workspace.


File: gsl-ref.info,  Node: QAGP adaptive integration with known singular points,  Next: QAGI adaptive integration on infinite intervals,  Prev: QAGS adaptive integration with singularities,  Up: Numerical Integration

17.5 QAGP adaptive integration with known singular points
=========================================================

 -- Function: int gsl_integration_qagp (const gsl_function *f, double
          *pts, size_t npts, double epsabs, double epsrel, size_t limit,
          gsl_integration_workspace *workspace, double *result, double
          *abserr)

     This function applies the adaptive integration algorithm QAGS
     taking account of the user-supplied locations of singular points.
     The array *note pts: 64b. of length *note npts: 64b. should contain
     the endpoints of the integration ranges defined by the integration
     region and locations of the singularities.  For example, to
     integrate over the region (a,b) with break-points at x_1, x_2, x_3
     (where a < x_1 < x_2 < x_3 < b) the following *note pts: 64b. array
     should be used:

          pts[0] = a
          pts[1] = x_1
          pts[2] = x_2
          pts[3] = x_3
          pts[4] = b

     with *note npts: 64b. = 5.

     If you know the locations of the singular points in the integration
     region then this routine will be faster than *note
     gsl_integration_qags(): 649.


File: gsl-ref.info,  Node: QAGI adaptive integration on infinite intervals,  Next: QAWC adaptive integration for Cauchy principal values,  Prev: QAGP adaptive integration with known singular points,  Up: Numerical Integration

17.6 QAGI adaptive integration on infinite intervals
====================================================

 -- Function: int gsl_integration_qagi (gsl_function *f, double epsabs,
          double epsrel, size_t limit, gsl_integration_workspace
          *workspace, double *result, double *abserr)

     This function computes the integral of the function *note f: 64d.
     over the infinite interval (-\infty,+\infty).  The integral is
     mapped onto the semi-open interval (0,1] using the transformation x
     = (1-t)/t,

          \int_{-\infty}^{+\infty} dx f(x) = \int_0^1 dt (f((1-t)/t) + f(-(1-t)/t))/t^2.

     It is then integrated using the QAGS algorithm.  The normal
     21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule,
     because the transformation can generate an integrable singularity
     at the origin.  In this case a lower-order rule is more efficient.

 -- Function: int gsl_integration_qagiu (gsl_function *f, double a,
          double epsabs, double epsrel, size_t limit,
          gsl_integration_workspace *workspace, double *result, double
          *abserr)

     This function computes the integral of the function *note f: 64e.
     over the semi-infinite interval (a,+\infty).  The integral is
     mapped onto the semi-open interval (0,1] using the transformation x
     = a + (1-t)/t,

          \int_{a}^{+\infty} dx f(x) = \int_0^1 dt f(a + (1-t)/t)/t^2

     and then integrated using the QAGS algorithm.

 -- Function: int gsl_integration_qagil (gsl_function *f, double b,
          double epsabs, double epsrel, size_t limit,
          gsl_integration_workspace *workspace, double *result, double
          *abserr)

     This function computes the integral of the function *note f: 64f.
     over the semi-infinite interval (-\infty,b).  The integral is
     mapped onto the semi-open interval (0,1] using the transformation x
     = b - (1-t)/t,

          \int_{-\infty}^{b} dx f(x) = \int_0^1 dt f(b - (1-t)/t)/t^2

     and then integrated using the QAGS algorithm.


File: gsl-ref.info,  Node: QAWC adaptive integration for Cauchy principal values,  Next: QAWS adaptive integration for singular functions,  Prev: QAGI adaptive integration on infinite intervals,  Up: Numerical Integration

17.7 QAWC adaptive integration for Cauchy principal values
==========================================================

 -- Function: int gsl_integration_qawc (gsl_function *f, double a,
          double b, double c, double epsabs, double epsrel, size_t
          limit, gsl_integration_workspace *workspace, double *result,
          double *abserr)

     This function computes the Cauchy principal value of the integral
     of f over (a,b), with a singularity at *note c: 651,

          I = \int_a^b dx f(x) / (x - c)

     The adaptive bisection algorithm of QAG is used, with modifications
     to ensure that subdivisions do not occur at the singular point x =
     c.  When a subinterval contains the point x = c or is close to it
     then a special 25-point modified Clenshaw-Curtis rule is used to
     control the singularity.  Further away from the singularity the
     algorithm uses an ordinary 15-point Gauss-Kronrod integration rule.


File: gsl-ref.info,  Node: QAWS adaptive integration for singular functions,  Next: QAWO adaptive integration for oscillatory functions,  Prev: QAWC adaptive integration for Cauchy principal values,  Up: Numerical Integration

17.8 QAWS adaptive integration for singular functions
=====================================================

The QAWS algorithm is designed for integrands with algebraic-logarithmic
singularities at the end-points of an integration region.  In order to
work efficiently the algorithm requires a precomputed table of Chebyshev
moments.

 -- Type: gsl_integration_qaws_table

     This structure contains precomputed quantities for the QAWS
     algorithm.

 -- Function: *note gsl_integration_qaws_table: 653.
          *gsl_integration_qaws_table_alloc (double alpha, double beta,
          int mu, int nu)

     This function allocates space for a *note
     gsl_integration_qaws_table: 653. struct describing a singular
     weight function w(x) with the parameters (\alpha, \beta, \mu, \nu),

          w(x) = (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x)

     where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1.  The
     weight function can take four different forms depending on the
     values of \mu and \nu,

     Weight function w(x)                                             (\mu,\nu)
                                                                      
     ---------------------------------------------------------------------------------------
                                                                      
     (x - a)^\alpha (b - x)^\beta                                     (0,0)
                                                                      
                                                                      
     (x - a)^\alpha (b - x)^\beta \log{(x-a)}                         (1,0)
                                                                      
                                                                      
     (x - a)^\alpha (b - x)^\beta \log{(b-x)}                         (0,1)
                                                                      
                                                                      
     (x - a)^\alpha (b - x)^\beta \log{(x-a)} \log{(b-x)}             (1,1)
                                                                      

     The singular points (a,b) do not have to be specified until the
     integral is computed, where they are the endpoints of the
     integration range.

     The function returns a pointer to the newly allocated table *note
     gsl_integration_qaws_table: 653. if no errors were detected, and 0
     in the case of error.

 -- Function: int gsl_integration_qaws_table_set
          (gsl_integration_qaws_table *t, double alpha, double beta, int
          mu, int nu)

     This function modifies the parameters (\alpha, \beta, \mu, \nu) of
     an existing *note gsl_integration_qaws_table: 653. struct *note t:
     655.

 -- Function: void gsl_integration_qaws_table_free
          (gsl_integration_qaws_table *t)

     This function frees all the memory associated with the *note
     gsl_integration_qaws_table: 653. struct *note t: 656.

 -- Function: int gsl_integration_qaws (gsl_function *f, const double a,
          const double b, gsl_integration_qaws_table *t, const double
          epsabs, const double epsrel, const size_t limit,
          gsl_integration_workspace *workspace, double *result, double
          *abserr)

     This function computes the integral of the function f(x) over the
     interval (a,b) with the singular weight function (x-a)^\alpha
     (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x).  The parameters of the
     weight function (\alpha, \beta, \mu, \nu) are taken from the table
     *note t: 657.  The integral is,

          I = \int_a^b dx f(x) (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x).

     The adaptive bisection algorithm of QAG is used.  When a
     subinterval contains one of the endpoints then a special 25-point
     modified Clenshaw-Curtis rule is used to control the singularities.
     For subintervals which do not include the endpoints an ordinary
     15-point Gauss-Kronrod integration rule is used.


File: gsl-ref.info,  Node: QAWO adaptive integration for oscillatory functions,  Next: QAWF adaptive integration for Fourier integrals,  Prev: QAWS adaptive integration for singular functions,  Up: Numerical Integration

17.9 QAWO adaptive integration for oscillatory functions
========================================================

The QAWO algorithm is designed for integrands with an oscillatory
factor, \sin(\omega x) or \cos(\omega x).  In order to work efficiently
the algorithm requires a table of Chebyshev moments which must be
pre-computed with calls to the functions below.

 -- Function: gsl_integration_qawo_table
          *gsl_integration_qawo_table_alloc (double omega, double L,
          enum gsl_integration_qawo_enum sine, size_t n)

     This function allocates space for a ‘gsl_integration_qawo_table’
     struct and its associated workspace describing a sine or cosine
     weight function w(x) with the parameters (\omega, L),

          w(x) = sin(omega x) 
          w(x) = cos(omega x) 

     The parameter *note L: 659. must be the length of the interval over
     which the function will be integrated L = b - a.  The choice of
     sine or cosine is made with the parameter *note sine: 659. which
     should be chosen from one of the two following symbolic values:

      -- Macro: GSL_INTEG_COSINE

      -- Macro: GSL_INTEG_SINE

     The ‘gsl_integration_qawo_table’ is a table of the trigonometric
     coefficients required in the integration process.  The parameter
     *note n: 659. determines the number of levels of coefficients that
     are computed.  Each level corresponds to one bisection of the
     interval L, so that *note n: 659. levels are sufficient for
     subintervals down to the length L/2^n.  The integration routine
     *note gsl_integration_qawo(): 65c. returns the error ‘GSL_ETABLE’
     if the number of levels is insufficient for the requested accuracy.

 -- Function: int gsl_integration_qawo_table_set
          (gsl_integration_qawo_table *t, double omega, double L, enum
          gsl_integration_qawo_enum sine)

     This function changes the parameters *note omega: 65d, *note L:
     65d. and *note sine: 65d. of the existing workspace *note t: 65d.

 -- Function: int gsl_integration_qawo_table_set_length
          (gsl_integration_qawo_table *t, double L)

     This function allows the length parameter *note L: 65e. of the
     workspace *note t: 65e. to be changed.

 -- Function: void gsl_integration_qawo_table_free
          (gsl_integration_qawo_table *t)

     This function frees all the memory associated with the workspace
     *note t: 65f.

 -- Function: int gsl_integration_qawo (gsl_function *f, const double a,
          const double epsabs, const double epsrel, const size_t limit,
          gsl_integration_workspace *workspace,
          gsl_integration_qawo_table *wf, double *result, double
          *abserr)

     This function uses an adaptive algorithm to compute the integral of
     f over (a,b) with the weight function \sin(\omega x) or \cos(\omega
     x) defined by the table *note wf: 65c,

          I = int_a^b dx f(x) sin(omega x) 
          I = int_a^b dx f(x) cos(omega x) 

     The results are extrapolated using the epsilon-algorithm to
     accelerate the convergence of the integral.  The function returns
     the final approximation from the extrapolation, *note result: 65c,
     and an estimate of the absolute error, *note abserr: 65c.  The
     subintervals and their results are stored in the memory provided by
     *note workspace: 65c.  The maximum number of subintervals is given
     by *note limit: 65c, which may not exceed the allocated size of the
     workspace.

     Those subintervals with “large” widths d where d\omega > 4 are
     computed using a 25-point Clenshaw-Curtis integration rule, which
     handles the oscillatory behavior.  Subintervals with a “small”
     widths where d\omega < 4 are computed using a 15-point
     Gauss-Kronrod integration.


File: gsl-ref.info,  Node: QAWF adaptive integration for Fourier integrals,  Next: CQUAD doubly-adaptive integration,  Prev: QAWO adaptive integration for oscillatory functions,  Up: Numerical Integration

17.10 QAWF adaptive integration for Fourier integrals
=====================================================

 -- Function: int gsl_integration_qawf (gsl_function *f, const double a,
          const double epsabs, const size_t limit,
          gsl_integration_workspace *workspace,
          gsl_integration_workspace *cycle_workspace,
          gsl_integration_qawo_table *wf, double *result, double
          *abserr)

     This function attempts to compute a Fourier integral of the
     function *note f: 661. over the semi-infinite interval [a,+\infty)

          I = \int_a^{+\infty} dx f(x) sin(omega x)
          I = \int_a^{+\infty} dx f(x) cos(omega x)

     The parameter \omega and choice of \sin or \cos is taken from the
     table *note wf: 661. (the length ‘L’ can take any value, since it
     is overridden by this function to a value appropriate for the
     Fourier integration).  The integral is computed using the QAWO
     algorithm over each of the subintervals,

          C_1 = [a, a + c]
          C_2 = [a + c, a + 2 c]
          ... = ...
          C_k = [a + (k-1) c, a + k c]

     where c = (2 floor(|\omega|) + 1) \pi/|\omega|.  The width c is
     chosen to cover an odd number of periods so that the contributions
     from the intervals alternate in sign and are monotonically
     decreasing when *note f: 661. is positive and monotonically
     decreasing.  The sum of this sequence of contributions is
     accelerated using the epsilon-algorithm.

     This function works to an overall absolute tolerance of *note
     abserr: 661.  The following strategy is used: on each interval C_k
     the algorithm tries to achieve the tolerance

          TOL_k = u_k abserr

     where u_k = (1 - p)p^{k-1} and p = 9/10.  The sum of the geometric
     series of contributions from each interval gives an overall
     tolerance of *note abserr: 661.

     If the integration of a subinterval leads to difficulties then the
     accuracy requirement for subsequent intervals is relaxed,

          TOL_k = u_k max(abserr, max_{i<k}(E_i))

     where E_k is the estimated error on the interval C_k.

     The subintervals and their results are stored in the memory
     provided by *note workspace: 661.  The maximum number of
     subintervals is given by *note limit: 661, which may not exceed the
     allocated size of the workspace.  The integration over each
     subinterval uses the memory provided by *note cycle_workspace: 661.
     as workspace for the QAWO algorithm.


File: gsl-ref.info,  Node: CQUAD doubly-adaptive integration,  Next: Romberg integration,  Prev: QAWF adaptive integration for Fourier integrals,  Up: Numerical Integration

17.11 CQUAD doubly-adaptive integration
=======================================

CQUAD is a new doubly-adaptive general-purpose quadrature routine which
can handle most types of singularities, non-numerical function values
such as ‘Inf’ or ‘NaN’, as well as some divergent integrals.  It
generally requires more function evaluations than the integration
routines in QUADPACK, yet fails less often for difficult integrands.

The underlying algorithm uses a doubly-adaptive scheme in which
Clenshaw-Curtis quadrature rules of increasing degree are used to
compute the integral in each interval.  The L_2-norm of the difference
between the underlying interpolatory polynomials of two successive rules
is used as an error estimate.  The interval is subdivided if the
difference between two successive rules is too large or a rule of
maximum degree has been reached.

 -- Function: gsl_integration_cquad_workspace
          *gsl_integration_cquad_workspace_alloc (size_t n)

     This function allocates a workspace sufficient to hold the data for
     *note n: 663. intervals.  The number *note n: 663. is not the
     maximum number of intervals that will be evaluated.  If the
     workspace is full, intervals with smaller error estimates will be
     discarded.  A minimum of 3 intervals is required and for most
     functions, a workspace of size 100 is sufficient.

 -- Function: void gsl_integration_cquad_workspace_free
          (gsl_integration_cquad_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 664.

 -- Function: int gsl_integration_cquad (const gsl_function *f, double
          a, double b, double epsabs, double epsrel,
          gsl_integration_cquad_workspace *workspace, double *result,
          double *abserr, size_t *nevals)

     This function computes the integral of f over (a,b) within the
     desired absolute and relative error limits, *note epsabs: 665. and
     *note epsrel: 665. using the CQUAD algorithm.  The function returns
     the final approximation, *note result: 665, an estimate of the
     absolute error, *note abserr: 665, and the number of function
     evaluations required, *note nevals: 665.

     The CQUAD algorithm divides the integration region into
     subintervals, and in each iteration, the subinterval with the
     largest estimated error is processed.  The algorithm uses
     Clenshaw-Curtis quadrature rules of degree 4, 8, 16 and 32 over 5,
     9, 17 and 33 nodes respectively.  Each interval is initialized with
     the lowest-degree rule.  When an interval is processed, the
     next-higher degree rule is evaluated and an error estimate is
     computed based on the L_2-norm of the difference between the
     underlying interpolating polynomials of both rules.  If the
     highest-degree rule has already been used, or the interpolatory
     polynomials differ significantly, the interval is bisected.

     The subintervals and their results are stored in the memory
     provided by *note workspace: 665.  If the error estimate or the
     number of function evaluations is not needed, the pointers *note
     abserr: 665. and *note nevals: 665. can be set to ‘NULL’.


File: gsl-ref.info,  Node: Romberg integration,  Next: Gauss-Legendre integration,  Prev: CQUAD doubly-adaptive integration,  Up: Numerical Integration

17.12 Romberg integration
=========================

The Romberg integration method estimates the definite integral

     I = \int_a^b f(x) dx

by applying Richardson extrapolation on the trapezoidal rule, using
equally spaced points with spacing

     h_k = (b - a) 2^{-k}

for k = 1, \dots, n.  For each k, Richardson extrapolation is used k-1
times on previous approximations to improve the order of accuracy as
much as possible.  Romberg integration typically works well (and
converges quickly) for smooth integrands with no singularities in the
interval or at the end points.

 -- Function: gsl_integration_romberg_workspace
          *gsl_integration_romberg_alloc (const size_t n)

     This function allocates a workspace for Romberg integration,
     specifying a maximum of n iterations, or divisions of the interval.
     Since the number of divisions is 2^n + 1, n can be kept relatively
     small (i.e.  10 or 20).  It is capped at a maximum value of 30 to
     prevent overflow.  The size of the workspace is O(2n).

 -- Function: void gsl_integration_romberg_free
          (gsl_integration_romberg_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 668.

 -- Function: int gsl_integration_romberg (const gsl_function *f, const
          double a, const double b, const double epsabs, const double
          epsrel, double *result, size_t *neval,
          gsl_integration_romberg_workspace *w)

     This function integrates f(x), specified by *note f: 669, from
     *note a: 669. to *note b: 669, storing the answer in *note result:
     669.  At each step in the iteration, convergence is tested by
     checking:

          | I_k - I_{k-1} | \le \textrm{max} \left( epsabs, epsrel \times |I_k| \right)

     where I_k is the current approximation and I_{k-1} is the
     approximation of the previous iteration.  If the method does not
     converge within the previously specified n iterations, the function
     stores the best current estimate in *note result: 669. and returns
     ‘GSL_EMAXITER’.  If the method converges, the function returns
     ‘GSL_SUCCESS’.  The total number of function evaluations is
     returned in *note neval: 669.


File: gsl-ref.info,  Node: Gauss-Legendre integration,  Next: Fixed point quadratures,  Prev: Romberg integration,  Up: Numerical Integration

17.13 Gauss-Legendre integration
================================

The fixed-order Gauss-Legendre integration routines are provided for
fast integration of smooth functions with known polynomial order.  The
n-point Gauss-Legendre rule is exact for polynomials of order 2n - 1 or
less.  For example, these rules are useful when integrating basis
functions to form mass matrices for the Galerkin method.  Unlike other
numerical integration routines within the library, these routines do not
accept absolute or relative error bounds.

 -- Function: gsl_integration_glfixed_table
          *gsl_integration_glfixed_table_alloc (size_t n)

     This function determines the Gauss-Legendre abscissae and weights
     necessary for an n-point fixed order integration scheme.  If
     possible, high precision precomputed coefficients are used.  If
     precomputed weights are not available, lower precision coefficients
     are computed on the fly.

 -- Function: double gsl_integration_glfixed (const gsl_function *f,
          double a, double b, const gsl_integration_glfixed_table *t)

     This function applies the Gauss-Legendre integration rule contained
     in table *note t: 66c. and returns the result.

 -- Function: int gsl_integration_glfixed_point (double a, double b,
          size_t i, double *xi, double *wi, const
          gsl_integration_glfixed_table *t)

     For *note i: 66d. in [0, \dots, n - 1], this function obtains the
     *note i: 66d.-th Gauss-Legendre point *note xi: 66d. and weight
     *note wi: 66d. on the interval [*note a: 66d, *note b: 66d.].  The
     points and weights are ordered by increasing point value.  A
     function f may be integrated on [*note a: 66d, *note b: 66d.] by
     summing wi * f(xi) over *note i: 66d.

 -- Function: void gsl_integration_glfixed_table_free
          (gsl_integration_glfixed_table *t)

     This function frees the memory associated with the table *note t:
     66e.


File: gsl-ref.info,  Node: Fixed point quadratures,  Next: Error codes,  Prev: Gauss-Legendre integration,  Up: Numerical Integration

17.14 Fixed point quadratures
=============================

The routines in this section approximate an integral by the sum

     \int_a^b w(x) f(x) dx = \sum_{i=1}^n w_i f(x_i)

where f(x) is the function to be integrated and w(x) is a weighting
function.  The n weights w_i and nodes x_i are carefully chosen so that
the result is exact when f(x) is a polynomial of degree 2n - 1 or less.
Once the user chooses the order n and weighting function w(x), the
weights w_i and nodes x_i can be precomputed and used to efficiently
evaluate integrals for any number of functions f(x).

This method works best when f(x) is well approximated by a polynomial on
the interval (a,b), and so is not suitable for functions with
singularities.  Since the user specifies ahead of time how many
quadrature nodes will be used, these routines do not accept absolute or
relative error bounds.  The table below lists the weighting functions
currently supported.

Name                 Interval                     Weighting function w(x)                         Constraints
                                                                                                  
--------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                  
Legendre             (a,b)                        1                                               b > a
                                                                                                  
                                                                                                  
Chebyshev Type 1     (a,b)                        1 / \sqrt{(b - x) (x - a)}                      b > a
                                                                                                  
                                                                                                  
Gegenbauer           (a,b)                        ((b - x) (x - a))^{\alpha}                      \alpha > -1, b > a
                                                                                                  
                                                                                                  
Jacobi               (a,b)                        (b - x)^{\alpha} (x - a)^{\beta}                \alpha,\beta > -1, b > a
                                                                                                  
                                                                                                  
Laguerre             (a,\infty)                   (x-a)^{\alpha} \exp{( -b (x - a) )}             \alpha > -1, b > 0
                                                                                                  
                                                                                                  
Hermite              (-\infty,\infty)             |x-a|^{\alpha} \exp{( -b (x-a)^2 )}             \alpha > -1, b > 0
                                                                                                  
                                                                                                  
Exponential          (a,b)                        |x - (a + b)/2|^{\alpha}                        \alpha > -1, b > a
                                                                                                  
                                                                                                  
Rational             (a,\infty)                   (x - a)^{\alpha} (x + b)^{\beta}                \alpha > -1, \alpha + \beta + 2n < 0, a + b > 0
                                                                                                  
                                                                                                  
Chebyshev Type 2     (a,b)                        \sqrt{(b - x) (x - a)}                          b > a
                                                                                                  

The fixed point quadrature routines use the following workspace to store
the nodes and weights, as well as additional variables for intermediate
calculations:

 -- Type: gsl_integration_fixed_workspace

     This workspace is used for fixed point quadrature rules and looks
     like this:

          typedef struct
          {
            size_t n;        /* number of nodes/weights */
            double *weights; /* quadrature weights */
            double *x;       /* quadrature nodes */
            double *diag;    /* diagonal of Jacobi matrix */
            double *subdiag; /* subdiagonal of Jacobi matrix */
            const gsl_integration_fixed_type * type;
          } gsl_integration_fixed_workspace;

 -- Function: *note gsl_integration_fixed_workspace: 671.
          *gsl_integration_fixed_alloc (const gsl_integration_fixed_type
          *T, const size_t n, const double a, const double b, const
          double alpha, const double beta)

     This function allocates a workspace for computing integrals with
     interpolating quadratures using *note n: 672. quadrature nodes.
     The parameters *note a: 672, *note b: 672, *note alpha: 672, and
     *note beta: 672. specify the integration interval and/or weighting
     function for the various quadrature types.  See the *note table:
     670. above for constraints on these parameters.  The size of the
     workspace is O(4n).

      -- Type: gsl_integration_fixed_type

          The type of quadrature used is specified by *note T: 672.
          which can be set to the following choices:

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_legendre

               This specifies Legendre quadrature integration.  The
               parameters *note alpha: 672. and *note beta: 672. are
               ignored for this type.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_chebyshev

               This specifies Chebyshev type 1 quadrature integration.
               The parameters *note alpha: 672. and *note beta: 672. are
               ignored for this type.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_gegenbauer

               This specifies Gegenbauer quadrature integration.  The
               parameter *note beta: 672. is ignored for this type.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_jacobi

               This specifies Jacobi quadrature integration.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_laguerre

               This specifies Laguerre quadrature integration.  The
               parameter *note beta: 672. is ignored for this type.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_hermite

               This specifies Hermite quadrature integration.  The
               parameter *note beta: 672. is ignored for this type.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_exponential

               This specifies exponential quadrature integration.  The
               parameter *note beta: 672. is ignored for this type.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_rational

               This specifies rational quadrature integration.

           -- Variable: *note gsl_integration_fixed_type: 673.
                    *gsl_integration_fixed_chebyshev2

               This specifies Chebyshev type 2 quadrature integration.
               The parameters *note alpha: 672. and *note beta: 672. are
               ignored for this type.

 -- Function: void gsl_integration_fixed_free
          (gsl_integration_fixed_workspace *w)

     This function frees the memory assocated with the workspace *note
     w: 67d.

 -- Function: size_t gsl_integration_fixed_n (const
          gsl_integration_fixed_workspace *w)

     This function returns the number of quadrature nodes and weights.

 -- Function: double *gsl_integration_fixed_nodes (const
          gsl_integration_fixed_workspace *w)

     This function returns a pointer to an array of size ‘n’ containing
     the quadrature nodes x_i.

 -- Function: double *gsl_integration_fixed_weights (const
          gsl_integration_fixed_workspace *w)

     This function returns a pointer to an array of size ‘n’ containing
     the quadrature weights w_i.

 -- Function: int gsl_integration_fixed (const gsl_function *func,
          double *result, const gsl_integration_fixed_workspace *w)

     This function integrates the function f(x) provided in *note func:
     681. using previously computed fixed quadrature rules.  The
     integral is approximated as

          \sum_{i=1}^n w_i f(x_i)

     where w_i are the quadrature weights and x_i are the quadrature
     nodes computed previously by *note gsl_integration_fixed_alloc():
     672.  The sum is stored in *note result: 681. on output.


File: gsl-ref.info,  Node: Error codes,  Next: Examples<11>,  Prev: Fixed point quadratures,  Up: Numerical Integration

17.15 Error codes
=================

In addition to the standard error codes for invalid arguments the
functions can return the following values,

‘GSL_EMAXITER’            the maximum number of subdivisions was exceeded.
                          
                          
‘GSL_EROUND’              cannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.
                          
                          
‘GSL_ESING’               a non-integrable singularity or other bad integrand behavior was found in the integration interval.
                          
                          
‘GSL_EDIVERGE’            the integral is divergent, or too slowly convergent to be integrated numerically.
                          
                          
*note GSL_EDOM: 28.       error in the values of the input arguments
                          


File: gsl-ref.info,  Node: Examples<11>,  Next: References and Further Reading<12>,  Prev: Error codes,  Up: Numerical Integration

17.16 Examples
==============

* Menu:

* Adaptive integration example::
* Fixed-point quadrature example::


File: gsl-ref.info,  Node: Adaptive integration example,  Next: Fixed-point quadrature example,  Up: Examples<11>

17.16.1 Adaptive integration example
------------------------------------

The integrator ‘QAGS’ will handle a large class of definite integrals.
For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,

     \int_0^1 x^{-1/2} \log(x) dx = -4

The program below computes this integral to a relative accuracy bound of
‘1e-7’.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_integration.h>

     double f (double x, void * params) {
       double alpha = *(double *) params;
       double f = log(alpha*x) / sqrt(x);
       return f;
     }

     int
     main (void)
     {
       gsl_integration_workspace * w
         = gsl_integration_workspace_alloc (1000);

       double result, error;
       double expected = -4.0;
       double alpha = 1.0;

       gsl_function F;
       F.function = &f;
       F.params = &alpha;

       gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                             w, &result, &error);

       printf ("result          = % .18f\n", result);
       printf ("exact result    = % .18f\n", expected);
       printf ("estimated error = % .18f\n", error);
       printf ("actual error    = % .18f\n", result - expected);
       printf ("intervals       = %zu\n", w->size);

       gsl_integration_workspace_free (w);

       return 0;
     }

The results below show that the desired accuracy is achieved after 8
subdivisions.

     result          = -4.000000000000085265
     exact result    = -4.000000000000000000
     estimated error =  0.000000000000135447
     actual error    = -0.000000000000085265
     intervals       = 8

In fact, the extrapolation procedure used by ‘QAGS’ produces an accuracy
of almost twice as many digits.  The error estimate returned by the
extrapolation procedure is larger than the actual error, giving a margin
of safety of one order of magnitude.


File: gsl-ref.info,  Node: Fixed-point quadrature example,  Prev: Adaptive integration example,  Up: Examples<11>

17.16.2 Fixed-point quadrature example
--------------------------------------

In this example, we use a fixed-point quadrature rule to integrate the
integral

     \int_{-\infty}^{\infty} e^{-x^2} \left( x^m + 1 \right) dx = \left\{ \begin{array}{cc} \sqrt{\pi} + \Gamma{\left( \frac{m+1}{2} \right)}, & m \textrm{ even} \\ \sqrt{\pi}, & m \textrm{ odd} \end{array} \right.

for integer m.  Consulting our *note table: 670. of fixed point
quadratures, we see that this integral can be evaluated with a Hermite
quadrature rule, setting \alpha = 0, a = 0, b = 1.  Since we are
integrating a polynomial of degree m, we need to choose the number of
nodes n \ge (m+1)/2 to achieve the best results.

First we will try integrating for m = 10, n = 5, which does not satisfy
our criteria above:

     $ ./integration2 10 5

The output is,

     m               = 10
     intervals       = 5
     result          =  47.468529694563351029
     exact result    =  54.115231635459025483
     actual error    = -6.646701940895674454

So, we find a large error.  Now we try integrating for m = 10, n = 6
which does satisfy the criteria above:

     $ ./integration2 10 6

The output is,

     m               = 10
     intervals       = 6
     result          =  54.115231635459096537
     exact result    =  54.115231635459025483
     actual error    =  0.000000000000071054

The program is given below.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_integration.h>
     #include <gsl/gsl_sf_gamma.h>

     double
     f(double x, void * params)
     {
       int m = *(int *) params;
       double f = gsl_pow_int(x, m) + 1.0;
       return f;
     }

     int
     main (int argc, char *argv[])
     {
       gsl_integration_fixed_workspace * w;
       const gsl_integration_fixed_type * T = gsl_integration_fixed_hermite;
       int m = 10;
       int n = 6;
       double expected, result;
       gsl_function F;

       if (argc > 1)
         m = atoi(argv[1]);

       if (argc > 2)
         n = atoi(argv[2]);

       w = gsl_integration_fixed_alloc(T, n, 0.0, 1.0, 0.0, 0.0);

       F.function = &f;
       F.params = &m;

       gsl_integration_fixed(&F, &result, w);

       if (m % 2 == 0)
         expected = M_SQRTPI + gsl_sf_gamma(0.5*(1.0 + m));
       else
         expected = M_SQRTPI;

       printf ("m               = %d\n", m);
       printf ("intervals       = %zu\n", gsl_integration_fixed_n(w));
       printf ("result          = % .18f\n", result);
       printf ("exact result    = % .18f\n", expected);
       printf ("actual error    = % .18f\n", result - expected);

       gsl_integration_fixed_free (w);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<12>,  Prev: Examples<11>,  Up: Numerical Integration

17.17 References and Further Reading
====================================

The following book is the definitive reference for QUADPACK, and was
written by the original authors.  It provides descriptions of the
algorithms, program listings, test programs and examples.  It also
includes useful advice on numerical integration and many references to
the numerical integration literature used in developing QUADPACK.

   * R. Piessens, E. de Doncker-Kapenga, C.W. Ueberhuber, D.K. Kahaner.
     QUADPACK A subroutine package for automatic integration Springer
     Verlag, 1983.

The CQUAD integration algorithm is described in the following paper:

   * P. Gonnet, “Increasing the Reliability of Adaptive Quadrature Using
     Explicit Interpolants”, ACM Transactions on Mathematical Software,
     Volume 37 (2010), Issue 3, Article 26.

The fixed-point quadrature routines are based on IQPACK, described in
the following papers:

   * S. Elhay, J. Kautsky, Algorithm 655: IQPACK, FORTRAN Subroutines
     for the Weights of Interpolatory Quadrature, ACM Transactions on
     Mathematical Software, Volume 13, Number 4, December 1987, pages
     399-415.

   * J. Kautsky, S. Elhay, Calculation of the Weights of Interpolatory
     Quadratures, Numerische Mathematik, Volume 40, Number 3, October
     1982, pages 407-422.


File: gsl-ref.info,  Node: Random Number Generation,  Next: Quasi-Random Sequences,  Prev: Numerical Integration,  Up: Top

18 Random Number Generation
***************************

The library provides a large collection of random number generators
which can be accessed through a uniform interface.  Environment
variables allow you to select different generators and seeds at runtime,
so that you can easily switch between generators without needing to
recompile your program.  Each instance of a generator keeps track of its
own state, allowing the generators to be used in multi-threaded
programs.  Additional functions are available for transforming uniform
random numbers into samples from continuous or discrete probability
distributions such as the Gaussian, log-normal or Poisson distributions.

These functions are declared in the header file ‘gsl_rng.h’.

* Menu:

* General comments on random numbers::
* The Random Number Generator Interface::
* Random number generator initialization::
* Sampling from a random number generator::
* Auxiliary random number generator functions::
* Random number environment variables::
* Copying random number generator state::
* Reading and writing random number generator state::
* Random number generator algorithms::
* Unix random number generators::
* Other random number generators::
* Performance::
* Examples: Examples<12>.
* References and Further Reading: References and Further Reading<13>.
* Acknowledgements::


File: gsl-ref.info,  Node: General comments on random numbers,  Next: The Random Number Generator Interface,  Up: Random Number Generation

18.1 General comments on random numbers
=======================================

In 1988, Park and Miller wrote a paper entitled “Random number
generators: good ones are hard to find.” [Commun.: ACM, 31, 1192–1201].
Fortunately, some excellent random number generators are available,
though poor ones are still in common use.  You may be happy with the
system-supplied random number generator on your computer, but you should
be aware that as computers get faster, requirements on random number
generators increase.  Nowadays, a simulation that calls a random number
generator millions of times can often finish before you can make it down
the hall to the coffee machine and back.

A very nice review of random number generators was written by Pierre
L’Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks,
ed.  (Wiley, 1997).  The chapter is available in postscript from
L’Ecuyer’s ftp site (see references).  Knuth’s volume on Seminumerical
Algorithms (originally published in 1968) devotes 170 pages to random
number generators, and has recently been updated in its 3rd edition
(1997).  It is brilliant, a classic.  If you don’t own it, you should
stop reading right now, run to the nearest bookstore, and buy it.

A good random number generator will satisfy both theoretical and
statistical properties.  Theoretical properties are often hard to obtain
(they require real math!), but one prefers a random number generator
with a long period, low serial correlation, and a tendency `not' to
“fall mainly on the planes.” Statistical tests are performed with
numerical simulations.  Generally, a random number generator is used to
estimate some quantity for which the theory of probability provides an
exact answer.  Comparison to this exact answer provides a measure of
“randomness”.


File: gsl-ref.info,  Node: The Random Number Generator Interface,  Next: Random number generator initialization,  Prev: General comments on random numbers,  Up: Random Number Generation

18.2 The Random Number Generator Interface
==========================================

It is important to remember that a random number generator is not a
“real” function like sine or cosine.  Unlike real functions, successive
calls to a random number generator yield different return values.  Of
course that is just what you want for a random number generator, but to
achieve this effect, the generator must keep track of some kind of
“state” variable.  Sometimes this state is just an integer (sometimes
just the value of the previously generated random number), but often it
is more complicated than that and may involve a whole array of numbers,
possibly with some indices thrown in.  To use the random number
generators, you do not need to know the details of what comprises the
state, and besides that varies from algorithm to algorithm.

 -- Type: gsl_rng_type
 -- Type: gsl_rng

     The random number generator library uses two special structs, *note
     gsl_rng_type: 68b. which holds static information about each type
     of generator and *note gsl_rng: 68c. which describes an instance of
     a generator created from a given *note gsl_rng_type: 68b.

The functions described in this section are declared in the header file
‘gsl_rng.h’.


File: gsl-ref.info,  Node: Random number generator initialization,  Next: Sampling from a random number generator,  Prev: The Random Number Generator Interface,  Up: Random Number Generation

18.3 Random number generator initialization
===========================================

 -- Function: *note gsl_rng: 68c. *gsl_rng_alloc (const gsl_rng_type *T)

     This function returns a pointer to a newly-created instance of a
     random number generator of type *note T: 68e.  For example, the
     following code creates an instance of the Tausworthe generator:

          gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);

     If there is insufficient memory to create the generator then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

     The generator is automatically initialized with the default seed,
     *note gsl_rng_default_seed: 68f.  This is zero by default but can
     be changed either directly or by using the environment variable
     *note GSL_RNG_SEED: 690.

     The details of the available generator types are described later in
     this chapter.

 -- Function: void gsl_rng_set (const gsl_rng *r, unsigned long int s)

     This function initializes (or “seeds”) the random number generator.
     If the generator is seeded with the same value of *note s: 691. on
     two different runs, the same stream of random numbers will be
     generated by successive calls to the routines below.  If different
     values of s \geq 1 are supplied, then the generated streams of
     random numbers should be completely different.  If the seed *note
     s: 691. is zero then the standard seed from the original
     implementation is used instead.  For example, the original Fortran
     source code for the ‘ranlux’ generator used a seed of 314159265,
     and so choosing *note s: 691. equal to zero reproduces this when
     using *note gsl_rng_ranlux: 692.

     When using multiple seeds with the same generator, choose seed
     values greater than zero to avoid collisions with the default
     setting.

     Note that the most generators only accept 32-bit seeds, with higher
     values being reduced modulo 2^{32}.  For generators with smaller
     ranges the maximum seed value will typically be lower.

 -- Function: void gsl_rng_free (gsl_rng *r)

     This function frees all the memory associated with the generator
     *note r: 693.


File: gsl-ref.info,  Node: Sampling from a random number generator,  Next: Auxiliary random number generator functions,  Prev: Random number generator initialization,  Up: Random Number Generation

18.4 Sampling from a random number generator
============================================

The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers.  Inline
versions of these functions are used when ‘HAVE_INLINE’ is defined.  To
obtain non-uniform distributions, see *note Random Number Distributions:
695.

 -- Function: unsigned long int gsl_rng_get (const gsl_rng *r)

     This function returns a random integer from the generator *note r:
     696.  The minimum and maximum values depend on the algorithm used,
     but all integers in the range [‘min’, ‘max’] are equally likely.
     The values of ‘min’ and ‘max’ can be determined using the auxiliary
     functions *note gsl_rng_max(): 697. and *note gsl_rng_min(): 698.

 -- Function: double gsl_rng_uniform (const gsl_rng *r)

     This function returns a double precision floating point number
     uniformly distributed in the range [0,1).  The range includes 0.0
     but excludes 1.0.  The value is typically obtained by dividing the
     result of ‘gsl_rng_get(r)’ by ‘gsl_rng_max(r) + 1.0’ in double
     precision.  Some generators compute this ratio internally so that
     they can provide floating point numbers with more than 32 bits of
     randomness (the maximum number of bits that can be portably
     represented in a single ‘unsigned long int’).

 -- Function: double gsl_rng_uniform_pos (const gsl_rng *r)

     This function returns a positive double precision floating point
     number uniformly distributed in the range (0,1), excluding both 0.0
     and 1.0.  The number is obtained by sampling the generator with the
     algorithm of *note gsl_rng_uniform(): 699. until a non-zero value
     is obtained.  You can use this function if you need to avoid a
     singularity at 0.0.

 -- Function: unsigned long int gsl_rng_uniform_int (const gsl_rng *r,
          unsigned long int n)

     This function returns a random integer from 0 to n-1 inclusive by
     scaling down and/or discarding samples from the generator *note r:
     69b.  All integers in the range [0,n-1] are produced with equal
     probability.  For generators with a non-zero minimum value an
     offset is applied so that zero is returned with the correct
     probability.

     Note that this function is designed for sampling from ranges
     smaller than the range of the underlying generator.  The parameter
     *note n: 69b. must be less than or equal to the range of the
     generator *note r: 69b.  If *note n: 69b. is larger than the range
     of the generator then the function calls the error handler with an
     error code of *note GSL_EINVAL: 2b. and returns zero.

     In particular, this function is not intended for generating the
     full range of unsigned integer values [0,2^{32}-1].  Instead choose
     a generator with the maximal integer range and zero minimum value,
     such as *note gsl_rng_ranlxd1: 69c, *note gsl_rng_mt19937: 69d. or
     *note gsl_rng_taus: 69e, and sample it directly using *note
     gsl_rng_get(): 696.  The range of each generator can be found using
     the auxiliary functions described in the next section.


File: gsl-ref.info,  Node: Auxiliary random number generator functions,  Next: Random number environment variables,  Prev: Sampling from a random number generator,  Up: Random Number Generation

18.5 Auxiliary random number generator functions
================================================

The following functions provide information about an existing generator.
You should use them in preference to hard-coding the generator
parameters into your own code.

 -- Function: const char *gsl_rng_name (const gsl_rng *r)

     This function returns a pointer to the name of the generator.  For
     example:

          printf ("r is a '%s' generator\n", gsl_rng_name (r));

     would print something like:

          r is a 'taus' generator

 -- Function: unsigned long int gsl_rng_max (const gsl_rng *r)

     This function returns the largest value that *note gsl_rng_get():
     696. can return.

 -- Function: unsigned long int gsl_rng_min (const gsl_rng *r)

     This function returns the smallest value that *note gsl_rng_get():
     696. can return.  Usually this value is zero.  There are some
     generators with algorithms that cannot return zero, and for these
     generators the minimum value is 1.

 -- Function: void *gsl_rng_state (const gsl_rng *r)
 -- Function: size_t gsl_rng_size (const gsl_rng *r)

     These functions return a pointer to the state of generator *note r:
     6a2. and its size.  You can use this information to access the
     state directly.  For example, the following code will write the
     state of a generator to a stream:

          void * state = gsl_rng_state (r);
          size_t n = gsl_rng_size (r);
          fwrite (state, n, 1, stream);

 -- Function: const *note gsl_rng_type: 68b. **gsl_rng_types_setup
          (void)

     This function returns a pointer to an array of all the available
     generator types, terminated by a null pointer.  The function should
     be called once at the start of the program, if needed.  The
     following code fragment shows how to iterate over the array of
     generator types to print the names of the available algorithms:

          const gsl_rng_type **t, **t0;

          t0 = gsl_rng_types_setup ();

          printf ("Available generators:\n");

          for (t = t0; *t != 0; t++)
            {
              printf ("%s\n", (*t)->name);
            }


File: gsl-ref.info,  Node: Random number environment variables,  Next: Copying random number generator state,  Prev: Auxiliary random number generator functions,  Up: Random Number Generation

18.6 Random number environment variables
========================================

The library allows you to choose a default generator and seed from the
environment variables *note GSL_RNG_TYPE: 6a5. and *note GSL_RNG_SEED:
690. and the function *note gsl_rng_env_setup(): 6a6.  This makes it
easy try out different generators and seeds without having to recompile
your program.

 -- Macro: GSL_RNG_TYPE

     This environment variable specifies the default random number
     generator.  It should be the name of a generator, such as ‘taus’ or
     ‘mt19937’.

 -- Macro: GSL_RNG_SEED

     This environment variable specifies the default seed for the random
     number generator

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_default

     This global library variable specifies the default random number
     generator, and can be initialized from *note GSL_RNG_TYPE: 6a5.
     using *note gsl_rng_env_setup(): 6a6.  It is defined as follows:

          extern const gsl_rng_type *gsl_rng_default

 -- Variable: unsigned long int gsl_rng_default_seed

     This global library variable specifies the seed for the default
     random number generator, and can be initialized from *note
     GSL_RNG_SEED: 690. using *note gsl_rng_env_setup(): 6a6.  It is set
     to zero by default and is defined as follows:

          extern unsigned long int gsl_rng_default_seed

 -- Function: const *note gsl_rng_type: 68b. *gsl_rng_env_setup (void)

     This function reads the environment variables *note GSL_RNG_TYPE:
     6a5. and *note GSL_RNG_SEED: 690. and uses their values to set the
     corresponding library variables *note gsl_rng_default: 6a7. and
     *note gsl_rng_default_seed: 68f.

     The value of *note GSL_RNG_SEED: 690. is converted to an ‘unsigned
     long int’ using the C library function ‘strtoul()’.

     If you don’t specify a generator for *note GSL_RNG_TYPE: 6a5. then
     *note gsl_rng_mt19937: 69d. is used as the default.  The initial
     value of *note gsl_rng_default_seed: 68f. is zero.

Here is a short program which shows how to create a global generator
using the environment variables *note GSL_RNG_TYPE: 6a5. and *note
GSL_RNG_SEED: 690,

     #include <stdio.h>
     #include <gsl/gsl_rng.h>

     gsl_rng * r;  /* global generator */

     int
     main (void)
     {
       const gsl_rng_type * T;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       printf ("generator type: %s\n", gsl_rng_name (r));
       printf ("seed = %lu\n", gsl_rng_default_seed);
       printf ("first value = %lu\n", gsl_rng_get (r));

       gsl_rng_free (r);
       return 0;
     }

Running the program without any environment variables uses the initial
defaults, an ‘mt19937’ generator with a seed of 0,

     generator type: mt19937
     seed = 0
     first value = 4293858116

By setting the two variables on the command line we can change the
default generator and the seed:

     $ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
     GSL_RNG_TYPE=taus
     GSL_RNG_SEED=123
     generator type: taus
     seed = 123
     first value = 2720986350


File: gsl-ref.info,  Node: Copying random number generator state,  Next: Reading and writing random number generator state,  Prev: Random number environment variables,  Up: Random Number Generation

18.7 Copying random number generator state
==========================================

The above methods do not expose the random number state which changes
from call to call.  It is often useful to be able to save and restore
the state.  To permit these practices, a few somewhat more advanced
functions are supplied.  These include:

 -- Function: int gsl_rng_memcpy (gsl_rng *dest, const gsl_rng *src)

     This function copies the random number generator *note src: 6a9.
     into the pre-existing generator *note dest: 6a9, making *note dest:
     6a9. into an exact copy of *note src: 6a9.  The two generators must
     be of the same type.

 -- Function: *note gsl_rng: 68c. *gsl_rng_clone (const gsl_rng *r)

     This function returns a pointer to a newly created generator which
     is an exact copy of the generator *note r: 6aa.


File: gsl-ref.info,  Node: Reading and writing random number generator state,  Next: Random number generator algorithms,  Prev: Copying random number generator state,  Up: Random Number Generation

18.8 Reading and writing random number generator state
======================================================

The library provides functions for reading and writing the random number
state to a file as binary data.

 -- Function: int gsl_rng_fwrite (FILE *stream, const gsl_rng *r)

     This function writes the random number state of the random number
     generator *note r: 6ac. to the stream *note stream: 6ac. in binary
     format.  The return value is 0 for success and ‘GSL_EFAILED’ if
     there was a problem writing to the file.  Since the data is written
     in the native binary format it may not be portable between
     different architectures.

 -- Function: int gsl_rng_fread (FILE *stream, gsl_rng *r)

     This function reads the random number state into the random number
     generator *note r: 6ad. from the open stream *note stream: 6ad. in
     binary format.  The random number generator *note r: 6ad. must be
     preinitialized with the correct random number generator type since
     type information is not saved.  The return value is 0 for success
     and ‘GSL_EFAILED’ if there was a problem reading from the file.
     The data is assumed to have been written in the native binary
     format on the same architecture.


File: gsl-ref.info,  Node: Random number generator algorithms,  Next: Unix random number generators,  Prev: Reading and writing random number generator state,  Up: Random Number Generation

18.9 Random number generator algorithms
=======================================

The functions described above make no reference to the actual algorithm
used.  This is deliberate so that you can switch algorithms without
having to change any of your application source code.  The library
provides a large number of generators of different types, including
simulation quality generators, generators provided for compatibility
with other libraries and historical generators from the past.

The following generators are recommended for use in simulation.  They
have extremely long periods, low correlation and pass most statistical
tests.  For the most reliable source of uncorrelated numbers, the
second-generation RANLUX generators have the strongest proof of
randomness.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_mt19937

     The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
     variant of the twisted generalized feedback shift-register
     algorithm, and is known as the “Mersenne Twister” generator.  It
     has a Mersenne prime period of 2^{19937} - 1 (about 10^{6000}) and
     is equi-distributed in 623 dimensions.  It has passed the DIEHARD
     statistical tests.  It uses 624 words of state per generator and is
     comparable in speed to the other generators.  The original
     generator used a default seed of 4357 and choosing ‘s’ equal to
     zero in *note gsl_rng_set(): 691. reproduces this.  Later versions
     switched to 5489 as the default seed, you can choose this
     explicitly via *note gsl_rng_set(): 691. instead if you require it.

     For more information see,

        * Makoto Matsumoto and Takuji Nishimura, “Mersenne Twister: A
          623-dimensionally equidistributed uniform pseudorandom number
          generator”.  ACM Transactions on Modeling and Computer
          Simulation, Vol.: 8, No.: 1 (Jan.  1998), Pages 3–30

     The generator *note gsl_rng_mt19937: 69d. uses the second revision
     of the seeding procedure published by the two authors above in
     2002.  The original seeding procedures could cause spurious
     artifacts for some seed values.  They are still available through
     the alternative generators ‘gsl_rng_mt19937_1999’ and
     ‘gsl_rng_mt19937_1998’.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranlxs0
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranlxs1
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranlxs2

     The generator ‘ranlxs0’ is a second-generation version of the
     RANLUX algorithm of Luscher, which produces “luxury random
     numbers”.  This generator provides single precision output (24
     bits) at three luxury levels ‘ranlxs0’, ‘ranlxs1’ and ‘ranlxs2’, in
     increasing order of strength.  It uses double-precision floating
     point arithmetic internally and can be significantly faster than
     the integer version of ‘ranlux’, particularly on 64-bit
     architectures.  The period of the generator is about 10^{171}.  The
     algorithm has mathematically proven properties and can provide
     truly decorrelated numbers at a known level of randomness.  The
     higher luxury levels provide increased decorrelation between
     samples as an additional safety margin.

     Note that the range of allowed seeds for this generator is
     [0,2^{31}-1].  Higher seed values are wrapped modulo 2^{31}.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranlxd1
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranlxd2

     These generators produce double precision output (48 bits) from the
     RANLXS generator.  The library provides two luxury levels ‘ranlxd1’
     and ‘ranlxd2’, in increasing order of strength.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranlux
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranlux389

     The ‘ranlux’ generator is an implementation of the original
     algorithm developed by Luscher.  It uses a
     lagged-fibonacci-with-skipping algorithm to produce “luxury random
     numbers”.  It is a 24-bit generator, originally designed for
     single-precision IEEE floating point numbers.  This implementation
     is based on integer arithmetic, while the second-generation
     versions RANLXS and RANLXD described above provide floating-point
     implementations which will be faster on many platforms.  The period
     of the generator is about 10^{171}.  The algorithm has
     mathematically proven properties and it can provide truly
     decorrelated numbers at a known level of randomness.  The default
     level of decorrelation recommended by Luscher is provided by *note
     gsl_rng_ranlux: 692, while *note gsl_rng_ranlux389: 6b3. gives the
     highest level of randomness, with all 24 bits decorrelated.  Both
     types of generator use 24 words of state per generator.

     For more information see,

        * M. Luscher, “A portable high-quality random number generator
          for lattice field theory calculations”, Computer Physics
          Communications, 79 (1994) 100–110.

        * F. James, “RANLUX: A Fortran implementation of the
          high-quality pseudo-random number generator of Luscher”,
          Computer Physics Communications, 79 (1994) 111–114

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_cmrg

     This is a combined multiple recursive generator by L’Ecuyer.  Its
     sequence is,

          z_n = (x_n - y_n) \mod m_1

     where the two underlying generators x_n and y_n are,

          x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) mod m_1
          y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) mod m_2

     with coefficients a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098,
     b_2 = 0, b_3 = -539608, and moduli m_1 = 2^{31} - 1 = 2147483647
     and m_2 = 2145483479.

     The period of this generator is \hbox{lcm}(m_1^3-1, m_2^3-1), which
     is approximately 2^{185} (about 10^{56}).  It uses 6 words of state
     per generator.  For more information see,

        * P. L’Ecuyer, “Combined Multiple Recursive Random Number
          Generators”, Operations Research, 44, 5 (1996), 816–822.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_mrg

     This is a fifth-order multiple recursive generator by L’Ecuyer,
     Blouin and Coutre.  Its sequence is,

          x_n = (a_1 x_{n-1} + a_5 x_{n-5}) \mod m

     with a_1 = 107374182, a_2 = a_3 = a_4 = 0, a_5 = 104480 and m =
     2^{31}-1.

     The period of this generator is about 10^{46}.  It uses 5 words of
     state per generator.  More information can be found in the
     following paper,

        * P. L’Ecuyer, F. Blouin, and R. Coutre, “A search for good
          multiple recursive random number generators”, ACM Transactions
          on Modeling and Computer Simulation 3, 87–98 (1993).

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_taus
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_taus2

     This is a maximally equidistributed combined Tausworthe generator
     by L’Ecuyer.  The sequence is,

          x_n = (s1_n ^^ s2_n ^^ s3_n)

     where,

          s1_{n+1} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19))
          s2_{n+1} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25))
          s3_{n+1} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))

     computed modulo 2^{32}.  In the formulas above \oplus denotes
     `exclusive-or'.  Note that the algorithm relies on the properties
     of 32-bit unsigned integers and has been implemented using a
     bitmask of ‘0xFFFFFFFF’ to make it work on 64 bit machines.

     The period of this generator is 2^{88} (about 10^{26}).  It uses 3
     words of state per generator.  For more information see,

        * P. L’Ecuyer, “Maximally Equidistributed Combined Tausworthe
          Generators”, Mathematics of Computation, 65, 213 (1996),
          203–213.

     The generator *note gsl_rng_taus2: 6b6. uses the same algorithm as
     *note gsl_rng_taus: 69e. but with an improved seeding procedure
     described in the paper,

        * P. L’Ecuyer, “Tables of Maximally Equidistributed Combined
          LFSR Generators”, Mathematics of Computation, 68, 225 (1999),
          261–269

     The generator *note gsl_rng_taus2: 6b6. should now be used in
     preference to *note gsl_rng_taus: 69e.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_gfsr4

     The ‘gfsr4’ generator is like a lagged-fibonacci generator, and
     produces each number as an ‘xor’’d sum of four previous values.

          r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}

     Ziff (ref below) notes that “it is now widely known” that two-tap
     registers (such as R250, which is described below) have serious
     flaws, the most obvious one being the three-point correlation that
     comes from the definition of the generator.  Nice mathematical
     properties can be derived for GFSR’s, and numerics bears out the
     claim that 4-tap GFSR’s with appropriately chosen offsets are as
     random as can be measured, using the author’s test.

     This implementation uses the values suggested the example on p392
     of Ziff’s article: A=471, B=1586, C=6988, D=9689.

     If the offsets are appropriately chosen (such as the one ones in
     this implementation), then the sequence is said to be maximal; that
     means that the period is 2^D - 1, where D is the longest lag.  (It
     is one less than 2^D because it is not permitted to have all zeros
     in the ‘ra[]’ array.)  For this implementation with D=9689 that
     works out to about 10^{2917}.

     Note that the implementation of this generator using a 32-bit
     integer amounts to 32 parallel implementations of one-bit
     generators.  One consequence of this is that the period of this
     32-bit generator is the same as for the one-bit generator.
     Moreover, this independence means that all 32-bit patterns are
     equally likely, and in particular that 0 is an allowed random
     value.  (We are grateful to Heiko Bauke for clarifying for us these
     properties of GFSR random number generators.)

     For more information see,

        * Robert M. Ziff, “Four-tap shift-register-sequence
          random-number generators”, Computers in Physics, 12(4),
          Jul/Aug 1998, pp 385–392.


File: gsl-ref.info,  Node: Unix random number generators,  Next: Other random number generators,  Prev: Random number generator algorithms,  Up: Random Number Generation

18.10 Unix random number generators
===================================

The standard Unix random number generators ‘rand’, ‘random’ and ‘rand48’
are provided as part of GSL. Although these generators are widely
available individually often they aren’t all available on the same
platform.  This makes it difficult to write portable code using them and
so we have included the complete set of Unix generators in GSL for
convenience.  Note that these generators don’t produce high-quality
randomness and aren’t suitable for work requiring accurate statistics.
However, if you won’t be measuring statistical quantities and just want
to introduce some variation into your program then these generators are
quite acceptable.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_rand

     This is the BSD ‘rand’ generator.  Its sequence is

          x_{n+1} = (a x_n + c) \mod m

     with a = 1103515245, c = 12345 and m = 2^{31}.  The seed specifies
     the initial value, x_1.  The period of this generator is 2^{31},
     and it uses 1 word of storage per generator.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_random_bsd
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_random_libc5
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_random_glibc2

     These generators implement the ‘random’ family of functions, a set
     of linear feedback shift register generators originally used in BSD
     Unix.  There are several versions of ‘random’ in use today: the
     original BSD version (e.g.  on SunOS4), a libc5 version (found on
     older GNU/Linux systems) and a glibc2 version.  Each version uses a
     different seeding procedure, and thus produces different sequences.

     The original BSD routines accepted a variable length buffer for the
     generator state, with longer buffers providing higher-quality
     randomness.  The ‘random’ function implemented algorithms for
     buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm
     with the largest length that would fit into the user-supplied
     buffer was used.  To support these algorithms additional generators
     are available with the following names:

          gsl_rng_random8_bsd
          gsl_rng_random32_bsd
          gsl_rng_random64_bsd
          gsl_rng_random128_bsd
          gsl_rng_random256_bsd

     where the numeric suffix indicates the buffer length.  The original
     BSD ‘random’ function used a 128-byte default buffer and so *note
     gsl_rng_random_bsd: 6ba. has been made equivalent to
     ‘gsl_rng_random128_bsd’.  Corresponding versions of the ‘libc5’ and
     ‘glibc2’ generators are also available, with the names
     ‘gsl_rng_random8_libc5’, ‘gsl_rng_random8_glibc2’, etc.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_rand48

     This is the Unix ‘rand48’ generator.  Its sequence is

          x_{n+1} = (a x_n + c) \mod m

     defined on 48-bit unsigned integers with a = 25214903917, c = 11
     and m = 2^{48}.  The seed specifies the upper 32 bits of the
     initial value, x_1, with the lower 16 bits set to ‘0x330E’.  The
     function *note gsl_rng_get(): 696. returns the upper 32 bits from
     each term of the sequence.  This does not have a direct parallel in
     the original ‘rand48’ functions, but forcing the result to type
     ‘long int’ reproduces the output of ‘mrand48’.  The function *note
     gsl_rng_uniform(): 699. uses the full 48 bits of internal state to
     return the double precision number x_n/m, which is equivalent to
     the function ‘drand48’.  Note that some versions of the GNU C
     Library contained a bug in ‘mrand48’ function which caused it to
     produce different results (only the lower 16-bits of the return
     value were set).


File: gsl-ref.info,  Node: Other random number generators,  Next: Performance,  Prev: Unix random number generators,  Up: Random Number Generation

18.11 Other random number generators
====================================

The generators in this section are provided for compatibility with
existing libraries.  If you are converting an existing program to use
GSL then you can select these generators to check your new
implementation against the original one, using the same random number
generator.  After verifying that your new program reproduces the
original results you can then switch to a higher-quality generator.

Note that most of the generators in this section are based on single
linear congruence relations, which are the least sophisticated type of
generator.  In particular, linear congruences have poor properties when
used with a non-prime modulus, as several of these routines do (e.g.
with a power of two modulus, 2^{31} or 2^{32}).  This leads to
periodicity in the least significant bits of each number, with only the
higher bits having any randomness.  Thus if you want to produce a random
bitstream it is best to avoid using the least significant bits.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranf

     This is the CRAY random number generator ‘RANF’.  Its sequence is

          x_{n+1} = (a x_n) \mod m

     defined on 48-bit unsigned integers with a = 44485709377909 and m =
     2^{48}.  The seed specifies the lower 32 bits of the initial value,
     x_1, with the lowest bit set to prevent the seed taking an even
     value.  The upper 16 bits of x_1 are set to 0.  A consequence of
     this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc.:
     produce the same sequences.

     The generator compatible with the CRAY MATHLIB routine RANF. It
     produces double precision floating point numbers which should be
     identical to those from the original RANF.

     There is a subtlety in the implementation of the seeding.  The
     initial state is reversed through one step, by multiplying by the
     modular inverse of a mod m.  This is done for compatibility with
     the original CRAY implementation.

     Note that you can only seed the generator with integers up to
     2^{32}, while the original CRAY implementation uses non-portable
     wide integers which can cover all 2^{48} states of the generator.

     The function *note gsl_rng_get(): 696. returns the upper 32 bits
     from each term of the sequence.  The function *note
     gsl_rng_uniform(): 699. uses the full 48 bits to return the double
     precision number x_n/m.

     The period of this generator is 2^{46}.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_ranmar

     This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman
     and Tsang.  It is a 24-bit generator, originally designed for
     single-precision IEEE floating point numbers.  It was included in
     the CERNLIB high-energy physics library.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_r250

     This is the shift-register generator of Kirkpatrick and Stoll.  The
     sequence is based on the recurrence

          x_n = x_{n-103} ^^ x_{n-250}

     where \oplus denotes `exclusive-or', defined on 32-bit words.  The
     period of this generator is about 2^{250} and it uses 250 words of
     state per generator.

     For more information see,

        * S. Kirkpatrick and E. Stoll, “A very fast shift-register
          sequence random number generator”, Journal of Computational
          Physics, 40, 517–526 (1981)

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_tt800

     This is an earlier version of the twisted generalized feedback
     shift-register generator, and has been superseded by the
     development of MT19937.  However, it is still an acceptable
     generator in its own right.  It has a period of 2^{800} and uses 33
     words of storage per generator.

     For more information see,

        * Makoto Matsumoto and Yoshiharu Kurita, “Twisted GFSR
          Generators II”, ACM Transactions on Modelling and Computer
          Simulation, Vol.: 4, No.: 3, 1994, pages 254–266.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_vax

     This is the VAX generator ‘MTH$RANDOM’.  Its sequence is,

          x_{n+1} = (a x_n + c) \mod m

     with a = 69069, c = 1 and m = 2^{32}.  The seed specifies the
     initial value, x_1.  The period of this generator is 2^{32} and it
     uses 1 word of storage per generator.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_transputer

     This is the random number generator from the INMOS Transputer
     Development system.  Its sequence is,

          x_{n+1} = (a x_n) \mod m

     with a = 1664525 and m = 2^{32}.  The seed specifies the initial
     value, x_1.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_randu

     This is the IBM ‘RANDU’ generator.  Its sequence is

          x_{n+1} = (a x_n) \mod m

     with a = 65539 and m = 2^{31}.  The seed specifies the initial
     value, x_1.  The period of this generator was only 2^{29}.  It has
     become a textbook example of a poor generator.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_minstd

     This is Park and Miller’s “minimal standard” MINSTD generator, a
     simple linear congruence which takes care to avoid the major
     pitfalls of such algorithms.  Its sequence is,

          x_{n+1} = (a x_n) \mod m

     with a = 16807 and m = 2^{31} - 1 = 2147483647.  The seed specifies
     the initial value, x_1.  The period of this generator is about
     2^{31}.

     This generator was used in the IMSL Library (subroutine RNUN) and
     in MATLAB (the RAND function) in the past.  It is also sometimes
     known by the acronym “GGL” (I’m not sure what that stands for).

     For more information see,

        * Park and Miller, “Random Number Generators: Good ones are hard
          to find”, Communications of the ACM, October 1988, Volume 31,
          No 10, pages 1192–1201.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_uni
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_uni32

     This is a reimplementation of the 16-bit SLATEC random number
     generator RUNIF. A generalization of the generator to 32 bits is
     provided by *note gsl_rng_uni32: 6c8.  The original source code is
     available from NETLIB.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_slatec

     This is the SLATEC random number generator RAND. It is ancient.
     The original source code is available from NETLIB.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_zuf

     This is the ZUFALL lagged Fibonacci series generator of Peterson.
     Its sequence is,

          t = u_{n-273} + u_{n-607}
          u_n  = t - floor(t)

     The original source code is available from NETLIB. For more
     information see,

        * W. Petersen, “Lagged Fibonacci Random Number Generators for
          the NEC SX-3”, International Journal of High Speed Computing
          (1994).

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_knuthran2

     This is a second-order multiple recursive generator described by
     Knuth in Seminumerical Algorithms, 3rd Ed., page 108.  Its sequence
     is,

          x_n = (a_1 x_{n-1} + a_2 x_{n-2}) \mod m

     with a_1 = 271828183, a_2 = 314159269, and m = 2^{31}-1.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_knuthran2002
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_knuthran

     This is a second-order multiple recursive generator described by
     Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6.  Knuth
     provides its C code.  The updated routine *note
     gsl_rng_knuthran2002: 6cc. is from the revised 9th printing and
     corrects some weaknesses in the earlier version, which is
     implemented as *note gsl_rng_knuthran: 6cd.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_borosh13
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_fishman18
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_fishman20
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_lecuyer21
 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_waterman14

     These multiplicative generators are taken from Knuth’s
     Seminumerical Algorithms, 3rd Ed., pages 106–108.  Their sequence
     is,

          x_{n+1} = (a x_n) \mod m

     where the seed specifies the initial value, x_1.  The parameters a
     and m are as follows, Borosh-Niederreiter: a = 1812433253, m =
     2^{32}, Fishman18: a = 62089911, m = 2^{31}-1, Fishman20: a =
     48271, m = 2^{31}-1, L’Ecuyer: a = 40692, m = 2^{31}-249, Waterman:
     a = 1566083941, m = 2^{32}.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_fishman2x

     This is the L’Ecuyer–Fishman random number generator.  It is taken
     from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108.  Its
     sequence is,

          z_{n+1} = (x_n - y_n) \mod m

     with m = 2^{31}-1.  x_n and y_n are given by the ‘fishman20’ and
     ‘lecuyer21’ algorithms.  The seed specifies the initial value, x_1.

 -- Variable: *note gsl_rng_type: 68b. *gsl_rng_coveyou

     This is the Coveyou random number generator.  It is taken from
     Knuth’s Seminumerical Algorithms, 3rd Ed., Section 3.2.2.  Its
     sequence is,

          x_{n+1} = (x_n (x_n + 1)) \mod m

     with m = 2^{32}.  The seed specifies the initial value, x_1.


File: gsl-ref.info,  Node: Performance,  Next: Examples<12>,  Prev: Other random number generators,  Up: Random Number Generation

18.12 Performance
=================

The following table shows the relative performance of a selection the
available random number generators.  The fastest simulation quality
generators are ‘taus’, ‘gfsr4’ and ‘mt19937’.  The generators which
offer the best mathematically-proven quality are those based on the
RANLUX algorithm:

     1754 k ints/sec,    870 k doubles/sec, taus
     1613 k ints/sec,    855 k doubles/sec, gfsr4
     1370 k ints/sec,    769 k doubles/sec, mt19937
      565 k ints/sec,    571 k doubles/sec, ranlxs0
      400 k ints/sec,    405 k doubles/sec, ranlxs1
      490 k ints/sec,    389 k doubles/sec, mrg
      407 k ints/sec,    297 k doubles/sec, ranlux
      243 k ints/sec,    254 k doubles/sec, ranlxd1
      251 k ints/sec,    253 k doubles/sec, ranlxs2
      238 k ints/sec,    215 k doubles/sec, cmrg
      247 k ints/sec,    198 k doubles/sec, ranlux389
      141 k ints/sec,    140 k doubles/sec, ranlxd2


File: gsl-ref.info,  Node: Examples<12>,  Next: References and Further Reading<13>,  Prev: Performance,  Up: Random Number Generation

18.13 Examples
==============

The following program demonstrates the use of a random number generator
to produce uniform random numbers in the range [0.0, 1.0),

     #include <stdio.h>
     #include <gsl/gsl_rng.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       int i, n = 10;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < n; i++)
         {
           double u = gsl_rng_uniform (r);
           printf ("%.5f\n", u);
         }

       gsl_rng_free (r);

       return 0;
     }

Here is the output of the program,

     0.99974
     0.16291
     0.28262
     0.94720
     0.23166
     0.48497
     0.95748
     0.74431
     0.54004
     0.73995

The numbers depend on the seed used by the generator.  The default seed
can be changed with the *note GSL_RNG_SEED: 690. environment variable to
produce a different stream of numbers.  The generator itself can be
changed using the environment variable *note GSL_RNG_TYPE: 6a5.  Here is
the output of the program using a seed value of 123 and the
multiple-recursive generator ‘mrg’:

     $ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out

     0.33050
     0.86631
     0.32982
     0.67620
     0.53391
     0.06457
     0.16847
     0.70229
     0.04371
     0.86374


File: gsl-ref.info,  Node: References and Further Reading<13>,  Next: Acknowledgements,  Prev: Examples<12>,  Up: Random Number Generation

18.14 References and Further Reading
====================================

The subject of random number generation and testing is reviewed
extensively in Knuth’s `Seminumerical Algorithms'.

   * Donald E. Knuth, The Art of Computer Programming: Seminumerical
     Algorithms (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

Further information is available in the review paper written by Pierre
L’Ecuyer,

   * P. L’Ecuyer, “Random Number Generation”, Chapter 4 of the Handbook
     on Simulation, Jerry Banks Ed., Wiley, 1998, 93–137.

   * ‘http://www.iro.umontreal.ca/~lecuyer/papers.html’ in the file
     ‘handsim.ps’.

The source code for the DIEHARD random number generator tests is also
available online,

   * DIEHARD source code, G. Marsaglia,
     ‘http://stat.fsu.edu/pub/diehard/’

A comprehensive set of random number generator tests is available from
NIST,

   * NIST Special Publication 800-22, “A Statistical Test Suite for the
     Validation of Random Number Generators and Pseudo Random Number
     Generators for Cryptographic Applications”.

   * ‘http://csrc.nist.gov/rng/’


File: gsl-ref.info,  Node: Acknowledgements,  Prev: References and Further Reading<13>,  Up: Random Number Generation

18.15 Acknowledgements
======================

Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for
making the source code to their generators (MT19937, MM&TN; TT800,
MM&YK) available under the GNU General Public License.  Thanks to Martin
Luscher for providing notes and source code for the RANLXS and RANLXD
generators.


File: gsl-ref.info,  Node: Quasi-Random Sequences,  Next: Random Number Distributions,  Prev: Random Number Generation,  Up: Top

19 Quasi-Random Sequences
*************************

This chapter describes functions for generating quasi-random sequences
in arbitrary dimensions.  A quasi-random sequence progressively covers a
d-dimensional space with a set of points that are uniformly distributed.
Quasi-random sequences are also known as low-discrepancy sequences.  The
quasi-random sequence generators use an interface that is similar to the
interface for random number generators, except that seeding is not
required—each generator produces a single sequence.

The functions described in this section are declared in the header file
‘gsl_qrng.h’.

* Menu:

* Quasi-random number generator initialization::
* Sampling from a quasi-random number generator::
* Auxiliary quasi-random number generator functions::
* Saving and restoring quasi-random number generator state::
* Quasi-random number generator algorithms::
* Examples: Examples<13>.
* References::


File: gsl-ref.info,  Node: Quasi-random number generator initialization,  Next: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences

19.1 Quasi-random number generator initialization
=================================================

 -- Type: gsl_qrng

     This is a workspace for computing quasi-random sequences.

 -- Function: *note gsl_qrng: 6dc. *gsl_qrng_alloc (const gsl_qrng_type
          *T, unsigned int d)

     This function returns a pointer to a newly-created instance of a
     quasi-random sequence generator of type *note T: 6dd. and dimension
     *note d: 6dd.  If there is insufficient memory to create the
     generator then the function returns a null pointer and the error
     handler is invoked with an error code of *note GSL_ENOMEM: 2a.

 -- Function: void gsl_qrng_free (gsl_qrng *q)

     This function frees all the memory associated with the generator
     *note q: 6de.

 -- Function: void gsl_qrng_init (gsl_qrng *q)

     This function reinitializes the generator *note q: 6df. to its
     starting point.  Note that quasi-random sequences do not use a seed
     and always produce the same set of values.


File: gsl-ref.info,  Node: Sampling from a quasi-random number generator,  Next: Auxiliary quasi-random number generator functions,  Prev: Quasi-random number generator initialization,  Up: Quasi-Random Sequences

19.2 Sampling from a quasi-random number generator
==================================================

 -- Function: int gsl_qrng_get (const gsl_qrng *q, double x[])

     This function stores the next point from the sequence generator
     *note q: 6e1. in the array *note x: 6e1.  The space available for
     *note x: 6e1. must match the dimension of the generator.  The point
     *note x: 6e1. will lie in the range 0 < x_i < 1 for each x_i.  An
     inline version of this function is used when ‘HAVE_INLINE’ is
     defined.


File: gsl-ref.info,  Node: Auxiliary quasi-random number generator functions,  Next: Saving and restoring quasi-random number generator state,  Prev: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences

19.3 Auxiliary quasi-random number generator functions
======================================================

 -- Function: const char *gsl_qrng_name (const gsl_qrng *q)

     This function returns a pointer to the name of the generator.

 -- Function: size_t gsl_qrng_size (const gsl_qrng *q)
 -- Function: void *gsl_qrng_state (const gsl_qrng *q)

     These functions return a pointer to the state of generator ‘r’ and
     its size.  You can use this information to access the state
     directly.  For example, the following code will write the state of
     a generator to a stream:

          void * state = gsl_qrng_state (q);
          size_t n = gsl_qrng_size (q);
          fwrite (state, n, 1, stream);


File: gsl-ref.info,  Node: Saving and restoring quasi-random number generator state,  Next: Quasi-random number generator algorithms,  Prev: Auxiliary quasi-random number generator functions,  Up: Quasi-Random Sequences

19.4 Saving and restoring quasi-random number generator state
=============================================================

 -- Function: int gsl_qrng_memcpy (gsl_qrng *dest, const gsl_qrng *src)

     This function copies the quasi-random sequence generator *note src:
     6e7. into the pre-existing generator *note dest: 6e7, making *note
     dest: 6e7. into an exact copy of *note src: 6e7.  The two
     generators must be of the same type.

 -- Function: *note gsl_qrng: 6dc. *gsl_qrng_clone (const gsl_qrng *q)

     This function returns a pointer to a newly created generator which
     is an exact copy of the generator *note q: 6e8.


File: gsl-ref.info,  Node: Quasi-random number generator algorithms,  Next: Examples<13>,  Prev: Saving and restoring quasi-random number generator state,  Up: Quasi-Random Sequences

19.5 Quasi-random number generator algorithms
=============================================

The following quasi-random sequence algorithms are available,

 -- Type: gsl_qrng_type

      -- Variable: *note gsl_qrng_type: 6ea. *gsl_qrng_niederreiter_2

          This generator uses the algorithm described in Bratley, Fox,
          Niederreiter, ACM Trans.  Model.  Comp.  Sim.  2, 195 (1992).
          It is valid up to 12 dimensions.

      -- Variable: *note gsl_qrng_type: 6ea. *gsl_qrng_sobol

          This generator uses the Sobol sequence described in Antonov,
          Saleev, USSR Comput.  Maths.  Math.  Phys.  19, 252 (1980).
          It is valid up to 40 dimensions.

      -- Variable: *note gsl_qrng_type: 6ea. *gsl_qrng_halton
      -- Variable: *note gsl_qrng_type: 6ea. *gsl_qrng_reversehalton

          These generators use the Halton and reverse Halton sequences
          described in J.H. Halton, Numerische Mathematik, 2, 84-90
          (1960) and B. Vandewoestyne and R. Cools Computational and
          Applied Mathematics, 189, 1&2, 341-361 (2006).  They are valid
          up to 1229 dimensions.


File: gsl-ref.info,  Node: Examples<13>,  Next: References,  Prev: Quasi-random number generator algorithms,  Up: Quasi-Random Sequences

19.6 Examples
=============

The following program prints the first 1024 points of the 2-dimensional
Sobol sequence.

     #include <stdio.h>
     #include <gsl/gsl_qrng.h>

     int
     main (void)
     {
       int i;
       gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);

       for (i = 0; i < 1024; i++)
         {
           double v[2];
           gsl_qrng_get (q, v);
           printf ("%.5f %.5f\n", v[0], v[1]);
         }

       gsl_qrng_free (q);
       return 0;
     }

Here is the output from the program:

     $ ./a.out
     0.50000 0.50000
     0.75000 0.25000
     0.25000 0.75000
     0.37500 0.37500
     0.87500 0.87500
     0.62500 0.12500
     0.12500 0.62500
     ....

It can be seen that successive points progressively fill-in the spaces
between previous points.

Fig.  %s shows the distribution in the x-y plane of the first 1024
points from the Sobol sequence,

[gsl-ref-figures/qrng]
Figure: Distribution of the first 1024 points from the quasi-random
Sobol sequence


File: gsl-ref.info,  Node: References,  Prev: Examples<13>,  Up: Quasi-Random Sequences

19.7 References
===============

The implementations of the quasi-random sequence routines are based on
the algorithms described in the following paper,

   * P. Bratley and B.L. Fox and H. Niederreiter, “Algorithm 738:
     Programs to Generate Niederreiter’s Low-discrepancy Sequences”, ACM
     Transactions on Mathematical Software, Vol.: 20, No.: 4, December,
     1994, p.: 494–495.


File: gsl-ref.info,  Node: Random Number Distributions,  Next: Statistics,  Prev: Quasi-Random Sequences,  Up: Top

20 Random Number Distributions
******************************

This chapter describes functions for generating random variates and
computing their probability distributions.  Samples from the
distributions described in this chapter can be obtained using any of the
random number generators in the library as an underlying source of
randomness.

In the simplest cases a non-uniform distribution can be obtained
analytically from the uniform distribution of a random number generator
by applying an appropriate transformation.  This method uses one call to
the random number generator.  More complicated distributions are created
by the `acceptance-rejection' method, which compares the desired
distribution against a distribution which is similar and known
analytically.  This usually requires several samples from the generator.

The library also provides cumulative distribution functions and inverse
cumulative distribution functions, sometimes referred to as quantile
functions.  The cumulative distribution functions and their inverses are
computed separately for the upper and lower tails of the distribution,
allowing full accuracy to be retained for small results.

The functions for random variates and probability density functions
described in this section are declared in ‘gsl_randist.h’.  The
corresponding cumulative distribution functions are declared in
‘gsl_cdf.h’.

Note that the discrete random variate functions always return a value of
type ‘unsigned int’, and on most platforms this has a maximum value of

     2^32-1 ~=~ 4.29e9

They should only be called with a safe range of parameters (where there
is a negligible probability of a variate exceeding this limit) to
prevent incorrect results due to overflow.

* Menu:

* Introduction: Introduction<3>.
* The Gaussian Distribution::
* The Gaussian Tail Distribution::
* The Bivariate Gaussian Distribution::
* The Multivariate Gaussian Distribution::
* The Exponential Distribution::
* The Laplace Distribution::
* The Exponential Power Distribution::
* The Cauchy Distribution::
* The Rayleigh Distribution::
* The Rayleigh Tail Distribution::
* The Landau Distribution::
* The Levy alpha-Stable Distributions::
* The Levy skew alpha-Stable Distribution::
* The Gamma Distribution::
* The Flat (Uniform) Distribution: The Flat Uniform Distribution.
* The Lognormal Distribution::
* The Chi-squared Distribution::
* The F-distribution::
* The t-distribution::
* The Beta Distribution::
* The Logistic Distribution::
* The Pareto Distribution::
* Spherical Vector Distributions::
* The Weibull Distribution::
* The Type-1 Gumbel Distribution::
* The Type-2 Gumbel Distribution::
* The Dirichlet Distribution::
* General Discrete Distributions::
* The Poisson Distribution::
* The Bernoulli Distribution::
* The Binomial Distribution::
* The Multinomial Distribution::
* The Negative Binomial Distribution::
* The Pascal Distribution::
* The Geometric Distribution::
* The Hypergeometric Distribution::
* The Logarithmic Distribution::
* The Wishart Distribution::
* Shuffling and Sampling::
* Examples: Examples<14>.
* References and Further Reading: References and Further Reading<14>.


File: gsl-ref.info,  Node: Introduction<3>,  Next: The Gaussian Distribution,  Up: Random Number Distributions

20.1 Introduction
=================

Continuous random number distributions are defined by a probability
density function, p(x), such that the probability of x occurring in the
infinitesimal range x to x + dx is p(x) dx.

The cumulative distribution function for the lower tail P(x) is defined
by the integral,

     P(x) = \int_{-\infty}^{x} dx' p(x')

and gives the probability of a variate taking a value less than x.

The cumulative distribution function for the upper tail Q(x) is defined
by the integral,

     Q(x) = \int_{x}^{+\infty} dx' p(x')

and gives the probability of a variate taking a value greater than x.

The upper and lower cumulative distribution functions are related by
P(x) + Q(x) = 1 and satisfy 0 \le P(x) \le 1, 0 \le Q(x) \le 1.

The inverse cumulative distributions, x = P^{-1}(P) and x = Q^{-1}(Q)
give the values of x which correspond to a specific value of P or Q.
They can be used to find confidence limits from probability values.

For discrete distributions the probability of sampling the integer value
k is given by p(k), where \sum_k p(k) = 1.  The cumulative distribution
for the lower tail P(k) of a discrete distribution is defined as,

     P(k) = \sum_{i \le k} p(i)

where the sum is over the allowed range of the distribution less than or
equal to k.

The cumulative distribution for the upper tail of a discrete
distribution Q(k) is defined as

     Q(k) = \sum_{i > k} p(i)

giving the sum of probabilities for all values greater than k.  These
two definitions satisfy the identity P(k)+Q(k)=1.

If the range of the distribution is 1 to n inclusive then P(n) = 1, Q(n)
= 0 while P(1) = p(1), Q(1) = 1 - p(1).


File: gsl-ref.info,  Node: The Gaussian Distribution,  Next: The Gaussian Tail Distribution,  Prev: Introduction<3>,  Up: Random Number Distributions

20.2 The Gaussian Distribution
==============================

 -- Function: double gsl_ran_gaussian (const gsl_rng *r, double sigma)

     This function returns a Gaussian random variate, with mean zero and
     standard deviation *note sigma: 6f6.  The probability distribution
     for Gaussian random variates is,

          p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx

     for x in the range -\infty to +\infty.  Use the transformation z =
     \mu + x on the numbers returned by *note gsl_ran_gaussian(): 6f6.
     to obtain a Gaussian distribution with mean \mu.  This function
     uses the Box-Muller algorithm which requires two calls to the
     random number generator *note r: 6f6.

 -- Function: double gsl_ran_gaussian_pdf (double x, double sigma)

     This function computes the probability density p(x) at *note x:
     6f7. for a Gaussian distribution with standard deviation *note
     sigma: 6f7, using the formula given above.

[gsl-ref-figures/rand-gaussian]
 -- Function: double gsl_ran_gaussian_ziggurat (const gsl_rng *r, double
          sigma)
 -- Function: double gsl_ran_gaussian_ratio_method (const gsl_rng *r,
          double sigma)

     This function computes a Gaussian random variate using the
     alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva
     ratio methods.  The Ziggurat algorithm is the fastest available
     algorithm in most cases.

 -- Function: double gsl_ran_ugaussian (const gsl_rng *r)
 -- Function: double gsl_ran_ugaussian_pdf (double x)
 -- Function: double gsl_ran_ugaussian_ratio_method (const gsl_rng *r)

     These functions compute results for the unit Gaussian distribution.
     They are equivalent to the functions above with a standard
     deviation of one, ‘sigma’ = 1.

 -- Function: double gsl_cdf_gaussian_P (double x, double sigma)
 -- Function: double gsl_cdf_gaussian_Q (double x, double sigma)
 -- Function: double gsl_cdf_gaussian_Pinv (double P, double sigma)
 -- Function: double gsl_cdf_gaussian_Qinv (double Q, double sigma)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Gaussian distribution with standard
     deviation *note sigma: 700.

 -- Function: double gsl_cdf_ugaussian_P (double x)
 -- Function: double gsl_cdf_ugaussian_Q (double x)
 -- Function: double gsl_cdf_ugaussian_Pinv (double P)
 -- Function: double gsl_cdf_ugaussian_Qinv (double Q)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the unit Gaussian distribution.


File: gsl-ref.info,  Node: The Gaussian Tail Distribution,  Next: The Bivariate Gaussian Distribution,  Prev: The Gaussian Distribution,  Up: Random Number Distributions

20.3 The Gaussian Tail Distribution
===================================

 -- Function: double gsl_ran_gaussian_tail (const gsl_rng *r, double a,
          double sigma)

     This function provides random variates from the upper tail of a
     Gaussian distribution with standard deviation *note sigma: 706.
     The values returned are larger than the lower limit *note a: 706,
     which must be positive.  The method is based on Marsaglia’s famous
     rectangle-wedge-tail algorithm (Ann.  Math.  Stat.  32, 894–899
     (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586
     (exercise 11).

     The probability distribution for Gaussian tail random variates is,

          p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2 / 2\sigma^2) dx

     for x > a where N(a;\sigma) is the normalization constant,

          N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).

 -- Function: double gsl_ran_gaussian_tail_pdf (double x, double a,
          double sigma)

     This function computes the probability density p(x) at *note x:
     707. for a Gaussian tail distribution with standard deviation *note
     sigma: 707. and lower limit *note a: 707, using the formula given
     above.

[gsl-ref-figures/rand-gaussian-tail]
 -- Function: double gsl_ran_ugaussian_tail (const gsl_rng *r, double a)
 -- Function: double gsl_ran_ugaussian_tail_pdf (double x, double a)

     These functions compute results for the tail of a unit Gaussian
     distribution.  They are equivalent to the functions above with a
     standard deviation of one, ‘sigma’ = 1.


File: gsl-ref.info,  Node: The Bivariate Gaussian Distribution,  Next: The Multivariate Gaussian Distribution,  Prev: The Gaussian Tail Distribution,  Up: Random Number Distributions

20.4 The Bivariate Gaussian Distribution
========================================

 -- Function: void gsl_ran_bivariate_gaussian (const gsl_rng *r, double
          sigma_x, double sigma_y, double rho, double *x, double *y)

     This function generates a pair of correlated Gaussian variates,
     with mean zero, correlation coefficient *note rho: 70b. and
     standard deviations *note sigma_x: 70b. and *note sigma_y: 70b. in
     the x and y directions.  The probability distribution for bivariate
     Gaussian random variates is,

          p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy

     for x,y in the range -\infty to +\infty.  The correlation
     coefficient *note rho: 70b. should lie between 1 and -1.

 -- Function: double gsl_ran_bivariate_gaussian_pdf (double x, double y,
          double sigma_x, double sigma_y, double rho)

     This function computes the probability density p(x,y) at (*note x:
     70c, *note y: 70c.) for a bivariate Gaussian distribution with
     standard deviations *note sigma_x: 70c, *note sigma_y: 70c. and
     correlation coefficient *note rho: 70c, using the formula given
     above.

[gsl-ref-figures/rand-bivariate-gaussian]

File: gsl-ref.info,  Node: The Multivariate Gaussian Distribution,  Next: The Exponential Distribution,  Prev: The Bivariate Gaussian Distribution,  Up: Random Number Distributions

20.5 The Multivariate Gaussian Distribution
===========================================

 -- Function: int gsl_ran_multivariate_gaussian (const gsl_rng *r, const
          gsl_vector *mu, const gsl_matrix *L, gsl_vector *result)

     This function generates a random vector satisfying the
     k-dimensional multivariate Gaussian distribution with mean \mu and
     variance-covariance matrix \Sigma.  On input, the k-vector \mu is
     given in *note mu: 70e, and the Cholesky factor of the k-by-k
     matrix \Sigma = L L^T is given in the lower triangle of *note L:
     70e, as output from *note gsl_linalg_cholesky_decomp(): 553.  The
     random vector is stored in *note result: 70e. on output.  The
     probability distribution for multivariate Gaussian random variates
     is

          p(x_1,...,x_k) dx_1 ... dx_k = 1 / ( \sqrt{(2 \pi)^k |\Sigma| ) \exp (-1/2 (x - \mu)^T \Sigma^{-1} (x - \mu)) dx_1 ... dx_k

 -- Function: int gsl_ran_multivariate_gaussian_pdf (const gsl_vector
          *x, const gsl_vector *mu, const gsl_matrix *L, double *result,
          gsl_vector *work)
 -- Function: int gsl_ran_multivariate_gaussian_log_pdf (const
          gsl_vector *x, const gsl_vector *mu, const gsl_matrix *L,
          double *result, gsl_vector *work)

     These functions compute p(x) or \log{p(x)} at the point *note x:
     710, using mean vector *note mu: 710. and variance-covariance
     matrix specified by its Cholesky factor *note L: 710. using the
     formula above.  Additional workspace of length k is required in
     *note work: 710.

 -- Function: int gsl_ran_multivariate_gaussian_mean (const gsl_matrix
          *X, gsl_vector *mu_hat)

     Given a set of n samples X_j from a k-dimensional multivariate
     Gaussian distribution, this function computes the maximum
     likelihood estimate of the mean of the distribution, given by

          \Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j

     The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix *note
     X: 711, and the maximum likelihood estimate of the mean is stored
     in *note mu_hat: 711. on output.

 -- Function: int gsl_ran_multivariate_gaussian_vcov (const gsl_matrix
          *X, gsl_matrix *sigma_hat)

     Given a set of n samples X_j from a k-dimensional multivariate
     Gaussian distribution, this function computes the maximum
     likelihood estimate of the variance-covariance matrix of the
     distribution, given by

          \Hat{\Sigma} = (1 / n) \sum_{j=1}^n ( X_j - \Hat{\mu} ) ( X_j - \Hat{\mu} )^T

     The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix *note
     X: 712. and the maximum likelihood estimate of the
     variance-covariance matrix is stored in *note sigma_hat: 712. on
     output.


File: gsl-ref.info,  Node: The Exponential Distribution,  Next: The Laplace Distribution,  Prev: The Multivariate Gaussian Distribution,  Up: Random Number Distributions

20.6 The Exponential Distribution
=================================

 -- Function: double gsl_ran_exponential (const gsl_rng *r, double mu)

     This function returns a random variate from the exponential
     distribution with mean *note mu: 714.  The distribution is,

          p(x) dx = {1 \over \mu} \exp(-x/\mu) dx

     for x \ge 0.

 -- Function: double gsl_ran_exponential_pdf (double x, double mu)

     This function computes the probability density p(x) at *note x:
     715. for an exponential distribution with mean *note mu: 715, using
     the formula given above.

[gsl-ref-figures/rand-exponential]
 -- Function: double gsl_cdf_exponential_P (double x, double mu)
 -- Function: double gsl_cdf_exponential_Q (double x, double mu)
 -- Function: double gsl_cdf_exponential_Pinv (double P, double mu)
 -- Function: double gsl_cdf_exponential_Qinv (double Q, double mu)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the exponential distribution with mean
     *note mu: 719.


File: gsl-ref.info,  Node: The Laplace Distribution,  Next: The Exponential Power Distribution,  Prev: The Exponential Distribution,  Up: Random Number Distributions

20.7 The Laplace Distribution
=============================

 -- Function: double gsl_ran_laplace (const gsl_rng *r, double a)

     This function returns a random variate from the Laplace
     distribution with width *note a: 71b.  The distribution is,

          p(x) dx = {1 \over 2 a} \exp(-|x/a|) dx

     for -\infty < x < \infty.

 -- Function: double gsl_ran_laplace_pdf (double x, double a)

     This function computes the probability density p(x) at *note x:
     71c. for a Laplace distribution with width *note a: 71c, using the
     formula given above.

[gsl-ref-figures/rand-laplace]
 -- Function: double gsl_cdf_laplace_P (double x, double a)
 -- Function: double gsl_cdf_laplace_Q (double x, double a)
 -- Function: double gsl_cdf_laplace_Pinv (double P, double a)
 -- Function: double gsl_cdf_laplace_Qinv (double Q, double a)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Laplace distribution with width
     *note a: 720.


File: gsl-ref.info,  Node: The Exponential Power Distribution,  Next: The Cauchy Distribution,  Prev: The Laplace Distribution,  Up: Random Number Distributions

20.8 The Exponential Power Distribution
=======================================

 -- Function: double gsl_ran_exppow (const gsl_rng *r, double a, double
          b)

     This function returns a random variate from the exponential power
     distribution with scale parameter *note a: 722. and exponent *note
     b: 722.  The distribution is,

          p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx

     for x \ge 0.  For b = 1 this reduces to the Laplace distribution.
     For b = 2 it has the same form as a Gaussian distribution, but with
     a = \sqrt{2} \sigma.

 -- Function: double gsl_ran_exppow_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     723. for an exponential power distribution with scale parameter
     *note a: 723. and exponent *note b: 723, using the formula given
     above.

[gsl-ref-figures/rand-exppow]
 -- Function: double gsl_cdf_exppow_P (double x, double a, double b)
 -- Function: double gsl_cdf_exppow_Q (double x, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) for the exponential power distribution with parameters *note
     a: 725. and *note b: 725.


File: gsl-ref.info,  Node: The Cauchy Distribution,  Next: The Rayleigh Distribution,  Prev: The Exponential Power Distribution,  Up: Random Number Distributions

20.9 The Cauchy Distribution
============================

 -- Function: double gsl_ran_cauchy (const gsl_rng *r, double a)

     This function returns a random variate from the Cauchy distribution
     with scale parameter *note a: 727.  The probability distribution
     for Cauchy random variates is,

          p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx

     for x in the range -\infty to +\infty.  The Cauchy distribution is
     also known as the Lorentz distribution.

 -- Function: double gsl_ran_cauchy_pdf (double x, double a)

     This function computes the probability density p(x) at *note x:
     728. for a Cauchy distribution with scale parameter *note a: 728,
     using the formula given above.

[gsl-ref-figures/rand-cauchy]
 -- Function: double gsl_cdf_cauchy_P (double x, double a)
 -- Function: double gsl_cdf_cauchy_Q (double x, double a)
 -- Function: double gsl_cdf_cauchy_Pinv (double P, double a)
 -- Function: double gsl_cdf_cauchy_Qinv (double Q, double a)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Cauchy distribution with scale
     parameter *note a: 72c.


File: gsl-ref.info,  Node: The Rayleigh Distribution,  Next: The Rayleigh Tail Distribution,  Prev: The Cauchy Distribution,  Up: Random Number Distributions

20.10 The Rayleigh Distribution
===============================

 -- Function: double gsl_ran_rayleigh (const gsl_rng *r, double sigma)

     This function returns a random variate from the Rayleigh
     distribution with scale parameter *note sigma: 72e.  The
     distribution is,

          p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx

     for x > 0.

 -- Function: double gsl_ran_rayleigh_pdf (double x, double sigma)

     This function computes the probability density p(x) at *note x:
     72f. for a Rayleigh distribution with scale parameter *note sigma:
     72f, using the formula given above.

[gsl-ref-figures/rand-rayleigh]
 -- Function: double gsl_cdf_rayleigh_P (double x, double sigma)
 -- Function: double gsl_cdf_rayleigh_Q (double x, double sigma)
 -- Function: double gsl_cdf_rayleigh_Pinv (double P, double sigma)
 -- Function: double gsl_cdf_rayleigh_Qinv (double Q, double sigma)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Rayleigh distribution with scale
     parameter *note sigma: 733.


File: gsl-ref.info,  Node: The Rayleigh Tail Distribution,  Next: The Landau Distribution,  Prev: The Rayleigh Distribution,  Up: Random Number Distributions

20.11 The Rayleigh Tail Distribution
====================================

 -- Function: double gsl_ran_rayleigh_tail (const gsl_rng *r, double a,
          double sigma)

     This function returns a random variate from the tail of the
     Rayleigh distribution with scale parameter *note sigma: 735. and a
     lower limit of *note a: 735.  The distribution is,

          p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx

     for x > a.

 -- Function: double gsl_ran_rayleigh_tail_pdf (double x, double a,
          double sigma)

     This function computes the probability density p(x) at *note x:
     736. for a Rayleigh tail distribution with scale parameter *note
     sigma: 736. and lower limit *note a: 736, using the formula given
     above.

[gsl-ref-figures/rand-rayleigh-tail]

File: gsl-ref.info,  Node: The Landau Distribution,  Next: The Levy alpha-Stable Distributions,  Prev: The Rayleigh Tail Distribution,  Up: Random Number Distributions

20.12 The Landau Distribution
=============================

 -- Function: double gsl_ran_landau (const gsl_rng *r)

     This function returns a random variate from the Landau
     distribution.  The probability distribution for Landau random
     variates is defined analytically by the complex integral,

          p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)

     For numerical purposes it is more convenient to use the following
     equivalent form of the integral,

          p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).

 -- Function: double gsl_ran_landau_pdf (double x)

     This function computes the probability density p(x) at *note x:
     739. for the Landau distribution using an approximation to the
     formula given above.

[gsl-ref-figures/rand-landau]

File: gsl-ref.info,  Node: The Levy alpha-Stable Distributions,  Next: The Levy skew alpha-Stable Distribution,  Prev: The Landau Distribution,  Up: Random Number Distributions

20.13 The Levy alpha-Stable Distributions
=========================================

 -- Function: double gsl_ran_levy (const gsl_rng *r, double c, double
          alpha)

     This function returns a random variate from the Levy symmetric
     stable distribution with scale *note c: 73b. and exponent *note
     alpha: 73b.  The symmetric stable probability distribution is
     defined by a Fourier transform,

          p(x) = 1 / (2 \pi) \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)

     There is no explicit solution for the form of p(x) and the library
     does not define a corresponding ‘pdf’ function.  For \alpha = 1 the
     distribution reduces to the Cauchy distribution.  For \alpha = 2 it
     is a Gaussian distribution with \sigma = \sqrt{2} c.  For \alpha <
     1 the tails of the distribution become extremely wide.

     The algorithm only works for 0 < \alpha \le 2.

[gsl-ref-figures/rand-levy]

File: gsl-ref.info,  Node: The Levy skew alpha-Stable Distribution,  Next: The Gamma Distribution,  Prev: The Levy alpha-Stable Distributions,  Up: Random Number Distributions

20.14 The Levy skew alpha-Stable Distribution
=============================================

 -- Function: double gsl_ran_levy_skew (const gsl_rng *r, double c,
          double alpha, double beta)

     This function returns a random variate from the Levy skew stable
     distribution with scale *note c: 73d, exponent *note alpha: 73d.
     and skewness parameter *note beta: 73d.  The skewness parameter
     must lie in the range [-1,1].  The Levy skew stable probability
     distribution is defined by a Fourier transform,

          p(x) = 1 / (2 \pi) \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))

     When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
     -(2/\pi)\log|t|.  There is no explicit solution for the form of
     p(x) and the library does not define a corresponding ‘pdf’
     function.  For \alpha = 2 the distribution reduces to a Gaussian
     distribution with \sigma = \sqrt{2} c and the skewness parameter
     has no effect.  For \alpha < 1 the tails of the distribution become
     extremely wide.  The symmetric distribution corresponds to \beta =
     0.

     The algorithm only works for 0 < \alpha \le 2.

The Levy alpha-stable distributions have the property that if N
alpha-stable variates are drawn from the distribution p(c, \alpha,
\beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be distributed
as an alpha-stable variate, p(N^{1/\alpha} c, \alpha, \beta).

[gsl-ref-figures/rand-levyskew]

File: gsl-ref.info,  Node: The Gamma Distribution,  Next: The Flat Uniform Distribution,  Prev: The Levy skew alpha-Stable Distribution,  Up: Random Number Distributions

20.15 The Gamma Distribution
============================

 -- Function: double gsl_ran_gamma (const gsl_rng *r, double a, double
          b)

     This function returns a random variate from the gamma distribution.
     The distribution function is,

          p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx

     for x > 0.

     The gamma distribution with an integer parameter *note a: 73f. is
     known as the Erlang distribution.

     The variates are computed using the Marsaglia-Tsang fast gamma
     method.  This function for this method was previously called
     ‘gsl_ran_gamma_mt()’ and can still be accessed using this name.

 -- Function: double gsl_ran_gamma_knuth (const gsl_rng *r, double a,
          double b)

     This function returns a gamma variate using the algorithms from
     Knuth (vol 2).

 -- Function: double gsl_ran_gamma_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     741. for a gamma distribution with parameters *note a: 741. and
     *note b: 741, using the formula given above.

[gsl-ref-figures/rand-gamma]
 -- Function: double gsl_cdf_gamma_P (double x, double a, double b)
 -- Function: double gsl_cdf_gamma_Q (double x, double a, double b)
 -- Function: double gsl_cdf_gamma_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_gamma_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the gamma distribution with parameters
     *note a: 745. and *note b: 745.


File: gsl-ref.info,  Node: The Flat Uniform Distribution,  Next: The Lognormal Distribution,  Prev: The Gamma Distribution,  Up: Random Number Distributions

20.16 The Flat (Uniform) Distribution
=====================================

 -- Function: double gsl_ran_flat (const gsl_rng *r, double a, double b)

     This function returns a random variate from the flat (uniform)
     distribution from *note a: 747. to *note b: 747.  The distribution
     is,

          p(x) dx = {1 \over (b-a)} dx

     if a \le x < b and 0 otherwise.

 -- Function: double gsl_ran_flat_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     748. for a uniform distribution from *note a: 748. to *note b: 748,
     using the formula given above.

[gsl-ref-figures/rand-flat]
 -- Function: double gsl_cdf_flat_P (double x, double a, double b)
 -- Function: double gsl_cdf_flat_Q (double x, double a, double b)
 -- Function: double gsl_cdf_flat_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_flat_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for a uniform distribution from *note a:
     74c. to *note b: 74c.


File: gsl-ref.info,  Node: The Lognormal Distribution,  Next: The Chi-squared Distribution,  Prev: The Flat Uniform Distribution,  Up: Random Number Distributions

20.17 The Lognormal Distribution
================================

 -- Function: double gsl_ran_lognormal (const gsl_rng *r, double zeta,
          double sigma)

     This function returns a random variate from the lognormal
     distribution.  The distribution function is,

          p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2}} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx

     for x > 0.

 -- Function: double gsl_ran_lognormal_pdf (double x, double zeta,
          double sigma)

     This function computes the probability density p(x) at *note x:
     74f. for a lognormal distribution with parameters *note zeta: 74f.
     and *note sigma: 74f, using the formula given above.

[gsl-ref-figures/rand-lognormal]
 -- Function: double gsl_cdf_lognormal_P (double x, double zeta, double
          sigma)
 -- Function: double gsl_cdf_lognormal_Q (double x, double zeta, double
          sigma)
 -- Function: double gsl_cdf_lognormal_Pinv (double P, double zeta,
          double sigma)
 -- Function: double gsl_cdf_lognormal_Qinv (double Q, double zeta,
          double sigma)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the lognormal distribution with
     parameters *note zeta: 753. and *note sigma: 753.


File: gsl-ref.info,  Node: The Chi-squared Distribution,  Next: The F-distribution,  Prev: The Lognormal Distribution,  Up: Random Number Distributions

20.18 The Chi-squared Distribution
==================================

The chi-squared distribution arises in statistics.  If Y_i are n
independent Gaussian random variates with unit variance then the
sum-of-squares,

     X_i = \sum_i Y_i^2

has a chi-squared distribution with n degrees of freedom.

 -- Function: double gsl_ran_chisq (const gsl_rng *r, double nu)

     This function returns a random variate from the chi-squared
     distribution with *note nu: 755. degrees of freedom.  The
     distribution function is,

          p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx

     for x \ge 0.

 -- Function: double gsl_ran_chisq_pdf (double x, double nu)

     This function computes the probability density p(x) at *note x:
     756. for a chi-squared distribution with *note nu: 756. degrees of
     freedom, using the formula given above.

[gsl-ref-figures/rand-chisq]
 -- Function: double gsl_cdf_chisq_P (double x, double nu)
 -- Function: double gsl_cdf_chisq_Q (double x, double nu)
 -- Function: double gsl_cdf_chisq_Pinv (double P, double nu)
 -- Function: double gsl_cdf_chisq_Qinv (double Q, double nu)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the chi-squared distribution with *note
     nu: 75a. degrees of freedom.


File: gsl-ref.info,  Node: The F-distribution,  Next: The t-distribution,  Prev: The Chi-squared Distribution,  Up: Random Number Distributions

20.19 The F-distribution
========================

The F-distribution arises in statistics.  If Y_1 and Y_2 are chi-squared
deviates with \nu_1 and \nu_2 degrees of freedom then the ratio,

     X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }

has an F-distribution F(x;\nu_1,\nu_2).

 -- Function: double gsl_ran_fdist (const gsl_rng *r, double nu1, double
          nu2)

     This function returns a random variate from the F-distribution with
     degrees of freedom *note nu1: 75c. and *note nu2: 75c.  The
     distribution function is,

          p(x) dx =
             { \Gamma((\nu_1 + \nu_2)/2)
                  \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) }
             \nu_1^{\nu_1/2} \nu_2^{\nu_2/2}
             x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}

     for x \ge 0.

 -- Function: double gsl_ran_fdist_pdf (double x, double nu1, double
          nu2)

     This function computes the probability density p(x) at *note x:
     75d. for an F-distribution with *note nu1: 75d. and *note nu2: 75d.
     degrees of freedom, using the formula given above.

[gsl-ref-figures/rand-fdist]
 -- Function: double gsl_cdf_fdist_P (double x, double nu1, double nu2)
 -- Function: double gsl_cdf_fdist_Q (double x, double nu1, double nu2)
 -- Function: double gsl_cdf_fdist_Pinv (double P, double nu1, double
          nu2)
 -- Function: double gsl_cdf_fdist_Qinv (double Q, double nu1, double
          nu2)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the F-distribution with *note nu1: 761.
     and *note nu2: 761. degrees of freedom.


File: gsl-ref.info,  Node: The t-distribution,  Next: The Beta Distribution,  Prev: The F-distribution,  Up: Random Number Distributions

20.20 The t-distribution
========================

The t-distribution arises in statistics.  If Y_1 has a normal
distribution and Y_2 has a chi-squared distribution with \nu degrees of
freedom then the ratio,

     X = { Y_1 \over \sqrt{Y_2 / \nu} }

has a t-distribution t(x;\nu) with \nu degrees of freedom.

 -- Function: double gsl_ran_tdist (const gsl_rng *r, double nu)

     This function returns a random variate from the t-distribution.
     The distribution function is,

          p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)} (1 + x^2/\nu)^{-(\nu + 1)/2} dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_tdist_pdf (double x, double nu)

     This function computes the probability density p(x) at *note x:
     764. for a t-distribution with *note nu: 764. degrees of freedom,
     using the formula given above.

[gsl-ref-figures/rand-tdist]
 -- Function: double gsl_cdf_tdist_P (double x, double nu)
 -- Function: double gsl_cdf_tdist_Q (double x, double nu)
 -- Function: double gsl_cdf_tdist_Pinv (double P, double nu)
 -- Function: double gsl_cdf_tdist_Qinv (double Q, double nu)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the t-distribution with *note nu: 768.
     degrees of freedom.


File: gsl-ref.info,  Node: The Beta Distribution,  Next: The Logistic Distribution,  Prev: The t-distribution,  Up: Random Number Distributions

20.21 The Beta Distribution
===========================

 -- Function: double gsl_ran_beta (const gsl_rng *r, double a, double b)

     This function returns a random variate from the beta distribution.
     The distribution function is,

          p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx

     for 0 \le x \le 1.

 -- Function: double gsl_ran_beta_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     76b. for a beta distribution with parameters *note a: 76b. and
     *note b: 76b, using the formula given above.

[gsl-ref-figures/rand-beta]
 -- Function: double gsl_cdf_beta_P (double x, double a, double b)
 -- Function: double gsl_cdf_beta_Q (double x, double a, double b)
 -- Function: double gsl_cdf_beta_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_beta_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the beta distribution with parameters
     *note a: 76f. and *note b: 76f.


File: gsl-ref.info,  Node: The Logistic Distribution,  Next: The Pareto Distribution,  Prev: The Beta Distribution,  Up: Random Number Distributions

20.22 The Logistic Distribution
===============================

 -- Function: double gsl_ran_logistic (const gsl_rng *r, double a)

     This function returns a random variate from the logistic
     distribution.  The distribution function is,

          p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_logistic_pdf (double x, double a)

     This function computes the probability density p(x) at *note x:
     772. for a logistic distribution with scale parameter *note a: 772,
     using the formula given above.

[gsl-ref-figures/rand-logistic]
 -- Function: double gsl_cdf_logistic_P (double x, double a)
 -- Function: double gsl_cdf_logistic_Q (double x, double a)
 -- Function: double gsl_cdf_logistic_Pinv (double P, double a)
 -- Function: double gsl_cdf_logistic_Qinv (double Q, double a)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the logistic distribution with scale
     parameter *note a: 776.


File: gsl-ref.info,  Node: The Pareto Distribution,  Next: Spherical Vector Distributions,  Prev: The Logistic Distribution,  Up: Random Number Distributions

20.23 The Pareto Distribution
=============================

 -- Function: double gsl_ran_pareto (const gsl_rng *r, double a, double
          b)

     This function returns a random variate from the Pareto distribution
     of order *note a: 778.  The distribution function is,

          p(x) dx = (a/b) / (x/b)^{a+1} dx

     for x \ge b.

 -- Function: double gsl_ran_pareto_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     779. for a Pareto distribution with exponent *note a: 779. and
     scale *note b: 779, using the formula given above.

[gsl-ref-figures/rand-pareto]
 -- Function: double gsl_cdf_pareto_P (double x, double a, double b)
 -- Function: double gsl_cdf_pareto_Q (double x, double a, double b)
 -- Function: double gsl_cdf_pareto_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_pareto_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Pareto distribution with exponent
     *note a: 77d. and scale *note b: 77d.


File: gsl-ref.info,  Node: Spherical Vector Distributions,  Next: The Weibull Distribution,  Prev: The Pareto Distribution,  Up: Random Number Distributions

20.24 Spherical Vector Distributions
====================================

The spherical distributions generate random vectors, located on a
spherical surface.  They can be used as random directions, for example
in the steps of a random walk.

 -- Function: void gsl_ran_dir_2d (const gsl_rng *r, double *x, double
          *y)
 -- Function: void gsl_ran_dir_2d_trig_method (const gsl_rng *r, double
          *x, double *y)

     This function returns a random direction vector v = (*note x: 780,
     *note y: 780.) in two dimensions.  The vector is normalized such
     that |v|^2 = x^2 + y^2 = 1.  The obvious way to do this is to take
     a uniform random number between 0 and 2\pi and let *note x: 780.
     and *note y: 780. be the sine and cosine respectively.  Two trig
     functions would have been expensive in the old days, but with
     modern hardware implementations, this is sometimes the fastest way
     to go.  This is the case for the Pentium (but not the case for the
     Sun Sparcstation).  One can avoid the trig evaluations by choosing
     *note x: 780. and *note y: 780. in the interior of a unit circle
     (choose them at random from the interior of the enclosing square,
     and then reject those that are outside the unit circle), and then
     dividing by \sqrt{x^2 + y^2}.  A much cleverer approach, attributed
     to von Neumann (See Knuth, v2, 3rd ed, p140, exercise 23), requires
     neither trig nor a square root.  In this approach, ‘u’ and ‘v’ are
     chosen at random from the interior of a unit circle, and then
     x=(u^2-v^2)/(u^2+v^2) and y=2uv/(u^2+v^2).

 -- Function: void gsl_ran_dir_3d (const gsl_rng *r, double *x, double
          *y, double *z)

     This function returns a random direction vector v = (*note x: 781,
     *note y: 781, *note z: 781.) in three dimensions.  The vector is
     normalized such that |v|^2 = x^2 + y^2 + z^2 = 1.  The method
     employed is due to Robert E. Knop (CACM 13, 326 (1970)), and
     explained in Knuth, v2, 3rd ed, p136.  It uses the surprising fact
     that the distribution projected along any axis is actually uniform
     (this is only true for 3 dimensions).

 -- Function: void gsl_ran_dir_nd (const gsl_rng *r, size_t n, double
          *x)

     This function returns a random direction vector v =
     (x_1,x_2,\ldots,x_n) in *note n: 782. dimensions.  The vector is
     normalized such that |v|^2 = x_1^2 + x_2^2 + \cdots + x_n^2 = 1.
     The method uses the fact that a multivariate Gaussian distribution
     is spherically symmetric.  Each component is generated to have a
     Gaussian distribution, and then the components are normalized.  The
     method is described by Knuth, v2, 3rd ed, p135–136, and attributed
     to G. W. Brown, Modern Mathematics for the Engineer (1956).


File: gsl-ref.info,  Node: The Weibull Distribution,  Next: The Type-1 Gumbel Distribution,  Prev: Spherical Vector Distributions,  Up: Random Number Distributions

20.25 The Weibull Distribution
==============================

 -- Function: double gsl_ran_weibull (const gsl_rng *r, double a, double
          b)

     This function returns a random variate from the Weibull
     distribution.  The distribution function is,

          p(x) dx = {b \over a^b} x^{b-1} \exp(-(x/a)^b) dx

     for x \ge 0.

 -- Function: double gsl_ran_weibull_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     785. for a Weibull distribution with scale *note a: 785. and
     exponent *note b: 785, using the formula given above.

[gsl-ref-figures/rand-weibull]
 -- Function: double gsl_cdf_weibull_P (double x, double a, double b)
 -- Function: double gsl_cdf_weibull_Q (double x, double a, double b)
 -- Function: double gsl_cdf_weibull_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_weibull_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Weibull distribution with scale
     *note a: 789. and exponent *note b: 789.


File: gsl-ref.info,  Node: The Type-1 Gumbel Distribution,  Next: The Type-2 Gumbel Distribution,  Prev: The Weibull Distribution,  Up: Random Number Distributions

20.26 The Type-1 Gumbel Distribution
====================================

 -- Function: double gsl_ran_gumbel1 (const gsl_rng *r, double a, double
          b)

     This function returns a random variate from the Type-1 Gumbel
     distribution.  The Type-1 Gumbel distribution function is,

          p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx

     for -\infty < x < \infty.

 -- Function: double gsl_ran_gumbel1_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     78c. for a Type-1 Gumbel distribution with parameters *note a: 78c.
     and *note b: 78c, using the formula given above.

[gsl-ref-figures/rand-gumbel1]
 -- Function: double gsl_cdf_gumbel1_P (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel1_Q (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel1_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_gumbel1_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Type-1 Gumbel distribution with
     parameters *note a: 790. and *note b: 790.


File: gsl-ref.info,  Node: The Type-2 Gumbel Distribution,  Next: The Dirichlet Distribution,  Prev: The Type-1 Gumbel Distribution,  Up: Random Number Distributions

20.27 The Type-2 Gumbel Distribution
====================================

 -- Function: double gsl_ran_gumbel2 (const gsl_rng *r, double a, double
          b)

     This function returns a random variate from the Type-2 Gumbel
     distribution.  The Type-2 Gumbel distribution function is,

          p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx

     for 0 < x < \infty.

 -- Function: double gsl_ran_gumbel2_pdf (double x, double a, double b)

     This function computes the probability density p(x) at *note x:
     793. for a Type-2 Gumbel distribution with parameters *note a: 793.
     and *note b: 793, using the formula given above.

[gsl-ref-figures/rand-gumbel2]
 -- Function: double gsl_cdf_gumbel2_P (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel2_Q (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel2_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_gumbel2_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Type-2 Gumbel distribution with
     parameters *note a: 797. and *note b: 797.


File: gsl-ref.info,  Node: The Dirichlet Distribution,  Next: General Discrete Distributions,  Prev: The Type-2 Gumbel Distribution,  Up: Random Number Distributions

20.28 The Dirichlet Distribution
================================

 -- Function: void gsl_ran_dirichlet (const gsl_rng *r, size_t K, const
          double alpha[], double theta[])

     This function returns an array of *note K: 799. random variates
     from a Dirichlet distribution of order *note K: 799.-1.  The
     distribution function is

          p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =
            (1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K

     for \theta_i \ge 0 and \alpha_i > 0.  The delta function ensures
     that \sum \theta_i = 1.  The normalization factor Z is

          Z = {\prod_{i=1}^K \Gamma(\alpha_i) \over \Gamma( \sum_{i=1}^K \alpha_i)}

     The random variates are generated by sampling *note K: 799. values
     from gamma distributions with parameters a=\alpha_i$, $b=1, and
     renormalizing.  See A.M. Law, W.D. Kelton, `Simulation Modeling and
     Analysis' (1991).

 -- Function: double gsl_ran_dirichlet_pdf (size_t K, const double
          alpha[], const double theta[])

     This function computes the probability density p(\theta_1, \ldots ,
     \theta_K) at ‘theta[K]’ for a Dirichlet distribution with
     parameters ‘alpha[K]’, using the formula given above.

 -- Function: double gsl_ran_dirichlet_lnpdf (size_t K, const double
          alpha[], const double theta[])

     This function computes the logarithm of the probability density
     p(\theta_1, \ldots , \theta_K) for a Dirichlet distribution with
     parameters ‘alpha[K]’.


File: gsl-ref.info,  Node: General Discrete Distributions,  Next: The Poisson Distribution,  Prev: The Dirichlet Distribution,  Up: Random Number Distributions

20.29 General Discrete Distributions
====================================

Given K discrete events with different probabilities P[k], produce a
random value k consistent with its probability.

The obvious way to do this is to preprocess the probability list by
generating a cumulative probability array with K + 1 elements:

     C[0] = 0
     C[k+1] = C[k] + P[k]

Note that this construction produces C[K] = 1.  Now choose a uniform
deviate u between 0 and 1, and find the value of k such that C[k] \le u
< C[k+1].  Although this in principle requires of order \log K steps per
random number generation, they are fast steps, and if you use something
like \lfloor uK \rfloor as a starting point, you can often do pretty
well.

But faster methods have been devised.  Again, the idea is to preprocess
the probability list, and save the result in some form of lookup table;
then the individual calls for a random discrete event can go rapidly.
An approach invented by G. Marsaglia (Generating discrete random
variables in a computer, Comm ACM 6, 37–38 (1963)) is very clever, and
readers interested in examples of good algorithm design are directed to
this short and well-written paper.  Unfortunately, for large K,
Marsaglia’s lookup table can be quite large.

A much better approach is due to Alastair J. Walker (An efficient method
for generating discrete random variables with general distributions, ACM
Trans on Mathematical Software 3, 253–256 (1977); see also Knuth, v2,
3rd ed, p120–121,139).  This requires two lookup tables, one floating
point and one integer, but both only of size K. After preprocessing, the
random numbers are generated in O(1) time, even for large K. The
preprocessing suggested by Walker requires O(K^2) effort, but that is
not actually necessary, and the implementation provided here only takes
O(K) effort.  In general, more preprocessing leads to faster generation
of the individual random numbers, but a diminishing return is reached
pretty early.  Knuth points out that the optimal preprocessing is
combinatorially difficult for large K.

This method can be used to speed up some of the discrete random number
generators below, such as the binomial distribution.  To use it for
something like the Poisson Distribution, a modification would have to be
made, since it only takes a finite set of K outcomes.

 -- Type: gsl_ran_discrete_t

     This structure contains the lookup table for the discrete random
     number generator.

 -- Function: *note gsl_ran_discrete_t: 79d. *gsl_ran_discrete_preproc
          (size_t K, const double *P)

     This function returns a pointer to a structure that contains the
     lookup table for the discrete random number generator.  The array
     *note P: 79e. contains the probabilities of the discrete events;
     these array elements must all be positive, but they needn’t add up
     to one (so you can think of them more generally as “weights”)—the
     preprocessor will normalize appropriately.  This return value is
     used as an argument for the *note gsl_ran_discrete(): 79f. function
     below.

 -- Function: size_t gsl_ran_discrete (const gsl_rng *r, const
          gsl_ran_discrete_t *g)

     After the preprocessor, above, has been called, you use this
     function to get the discrete random numbers.

 -- Function: double gsl_ran_discrete_pdf (size_t k, const
          gsl_ran_discrete_t *g)

     Returns the probability P[k] of observing the variable *note k:
     7a0.  Since P[k] is not stored as part of the lookup table, it must
     be recomputed; this computation takes O(K), so if ‘K’ is large and
     you care about the original array P[k] used to create the lookup
     table, then you should just keep this original array P[k] around.

 -- Function: void gsl_ran_discrete_free (gsl_ran_discrete_t *g)

     De-allocates the lookup table pointed to by *note g: 7a1.


File: gsl-ref.info,  Node: The Poisson Distribution,  Next: The Bernoulli Distribution,  Prev: General Discrete Distributions,  Up: Random Number Distributions

20.30 The Poisson Distribution
==============================

 -- Function: unsigned int gsl_ran_poisson (const gsl_rng *r, double mu)

     This function returns a random integer from the Poisson
     distribution with mean *note mu: 7a3.  The probability distribution
     for Poisson variates is,

          p(k) = {\mu^k \over k!} \exp(-\mu)

     for k \ge 0.

 -- Function: double gsl_ran_poisson_pdf (unsigned int k, double mu)

     This function computes the probability p(k) of obtaining *note k:
     7a4. from a Poisson distribution with mean *note mu: 7a4, using the
     formula given above.

[gsl-ref-figures/rand-poisson]
 -- Function: double gsl_cdf_poisson_P (unsigned int k, double mu)
 -- Function: double gsl_cdf_poisson_Q (unsigned int k, double mu)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the Poisson distribution with parameter *note mu: 7a6.


File: gsl-ref.info,  Node: The Bernoulli Distribution,  Next: The Binomial Distribution,  Prev: The Poisson Distribution,  Up: Random Number Distributions

20.31 The Bernoulli Distribution
================================

 -- Function: unsigned int gsl_ran_bernoulli (const gsl_rng *r, double
          p)

     This function returns either 0 or 1, the result of a Bernoulli
     trial with probability *note p: 7a8.  The probability distribution
     for a Bernoulli trial is,

          p(0) = 1 - p
          p(1) = p

 -- Function: double gsl_ran_bernoulli_pdf (unsigned int k, double p)

     This function computes the probability p(k) of obtaining *note k:
     7a9. from a Bernoulli distribution with probability parameter *note
     p: 7a9, using the formula given above.

[gsl-ref-figures/rand-bernoulli]

File: gsl-ref.info,  Node: The Binomial Distribution,  Next: The Multinomial Distribution,  Prev: The Bernoulli Distribution,  Up: Random Number Distributions

20.32 The Binomial Distribution
===============================

 -- Function: unsigned int gsl_ran_binomial (const gsl_rng *r, double p,
          unsigned int n)

     This function returns a random integer from the binomial
     distribution, the number of successes in *note n: 7ab. independent
     trials with probability *note p: 7ab.  The probability distribution
     for binomial variates is,

          p(k) = {n! \over k! (n-k)!} p^k (1-p)^{n-k}

     for 0 \le k \le n.

 -- Function: double gsl_ran_binomial_pdf (unsigned int k, double p,
          unsigned int n)

     This function computes the probability p(k) of obtaining *note k:
     7ac. from a binomial distribution with parameters *note p: 7ac. and
     *note n: 7ac, using the formula given above.

[gsl-ref-figures/rand-binomial]
 -- Function: double gsl_cdf_binomial_P (unsigned int k, double p,
          unsigned int n)
 -- Function: double gsl_cdf_binomial_Q (unsigned int k, double p,
          unsigned int n)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the binomial distribution with parameters *note p: 7ae.
     and *note n: 7ae.


File: gsl-ref.info,  Node: The Multinomial Distribution,  Next: The Negative Binomial Distribution,  Prev: The Binomial Distribution,  Up: Random Number Distributions

20.33 The Multinomial Distribution
==================================

 -- Function: void gsl_ran_multinomial (const gsl_rng *r, size_t K,
          unsigned int N, const double p[], unsigned int n[])

     This function computes a random sample *note n: 7b0. from the
     multinomial distribution formed by *note N: 7b0. trials from an
     underlying distribution ‘p[K]’.  The distribution function for
     *note n: 7b0. is,

          P(n_1, n_2, ..., n_K) =
            (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K

     where (n_1, n_2, \ldots, n_K) are nonnegative integers with
     \sum_{k=1}^{K} n_k = N, and (p_1, p_2, \ldots, p_K) is a
     probability distribution with \sum p_i = 1.  If the array ‘p[K]’ is
     not normalized then its entries will be treated as weights and
     normalized appropriately.  The arrays *note n: 7b0. and *note p:
     7b0. must both be of length *note K: 7b0.

     Random variates are generated using the conditional binomial method
     (see C.S. Davis, `The computer generation of multinomial random
     variates', Comp.  Stat.  Data Anal.  16 (1993) 205–217 for
     details).

 -- Function: double gsl_ran_multinomial_pdf (size_t K, const double
          p[], const unsigned int n[])

     This function computes the probability P(n_1, n_2, \ldots, n_K) of
     sampling ‘n[K]’ from a multinomial distribution with parameters
     ‘p[K]’, using the formula given above.

 -- Function: double gsl_ran_multinomial_lnpdf (size_t K, const double
          p[], const unsigned int n[])

     This function returns the logarithm of the probability for the
     multinomial distribution P(n_1, n_2, \ldots, n_K) with parameters
     ‘p[K]’.


File: gsl-ref.info,  Node: The Negative Binomial Distribution,  Next: The Pascal Distribution,  Prev: The Multinomial Distribution,  Up: Random Number Distributions

20.34 The Negative Binomial Distribution
========================================

 -- Function: unsigned int gsl_ran_negative_binomial (const gsl_rng *r,
          double p, double n)

     This function returns a random integer from the negative binomial
     distribution, the number of failures occurring before *note n: 7b4.
     successes in independent trials with probability *note p: 7b4. of
     success.  The probability distribution for negative binomial
     variates is,

          p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k

     Note that n is not required to be an integer.

 -- Function: double gsl_ran_negative_binomial_pdf (unsigned int k,
          double p, double n)

     This function computes the probability p(k) of obtaining *note k:
     7b5. from a negative binomial distribution with parameters *note p:
     7b5. and *note n: 7b5, using the formula given above.

[gsl-ref-figures/rand-nbinomial]
 -- Function: double gsl_cdf_negative_binomial_P (unsigned int k, double
          p, double n)
 -- Function: double gsl_cdf_negative_binomial_Q (unsigned int k, double
          p, double n)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the negative binomial distribution with parameters *note
     p: 7b7. and *note n: 7b7.


File: gsl-ref.info,  Node: The Pascal Distribution,  Next: The Geometric Distribution,  Prev: The Negative Binomial Distribution,  Up: Random Number Distributions

20.35 The Pascal Distribution
=============================

 -- Function: unsigned int gsl_ran_pascal (const gsl_rng *r, double p,
          unsigned int n)

     This function returns a random integer from the Pascal
     distribution.  The Pascal distribution is simply a negative
     binomial distribution with an integer value of n.

          p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k

     for k \ge 0.

 -- Function: double gsl_ran_pascal_pdf (unsigned int k, double p,
          unsigned int n)

     This function computes the probability p(k) of obtaining *note k:
     7ba. from a Pascal distribution with parameters *note p: 7ba. and
     *note n: 7ba, using the formula given above.

[gsl-ref-figures/rand-pascal]
 -- Function: double gsl_cdf_pascal_P (unsigned int k, double p,
          unsigned int n)
 -- Function: double gsl_cdf_pascal_Q (unsigned int k, double p,
          unsigned int n)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the Pascal distribution with parameters *note p: 7bc. and
     *note n: 7bc.


File: gsl-ref.info,  Node: The Geometric Distribution,  Next: The Hypergeometric Distribution,  Prev: The Pascal Distribution,  Up: Random Number Distributions

20.36 The Geometric Distribution
================================

 -- Function: unsigned int gsl_ran_geometric (const gsl_rng *r, double
          p)

     This function returns a random integer from the geometric
     distribution, the number of independent trials with probability
     *note p: 7be. until the first success.  The probability
     distribution for geometric variates is,

          p(k) = p (1-p)^{k-1}

     for k \ge 1.  Note that the distribution begins with k = 1 with
     this definition.  There is another convention in which the exponent
     k - 1 is replaced by k.

 -- Function: double gsl_ran_geometric_pdf (unsigned int k, double p)

     This function computes the probability p(k) of obtaining *note k:
     7bf. from a geometric distribution with probability parameter *note
     p: 7bf, using the formula given above.

[gsl-ref-figures/rand-geometric]
 -- Function: double gsl_cdf_geometric_P (unsigned int k, double p)
 -- Function: double gsl_cdf_geometric_Q (unsigned int k, double p)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the geometric distribution with parameter *note p: 7c1.


File: gsl-ref.info,  Node: The Hypergeometric Distribution,  Next: The Logarithmic Distribution,  Prev: The Geometric Distribution,  Up: Random Number Distributions

20.37 The Hypergeometric Distribution
=====================================

 -- Function: unsigned int gsl_ran_hypergeometric (const gsl_rng *r,
          unsigned int n1, unsigned int n2, unsigned int t)

     This function returns a random integer from the hypergeometric
     distribution.  The probability distribution for hypergeometric
     random variates is,

          p(k) = C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)

     where C(a,b) = a!/(b!(a-b)!) and t \leq n_1 + n_2.  The domain of k
     is \max(0, t - n_2), \ldots, \min(t, n_1)

     If a population contains n_1 elements of “type 1” and n_2 elements
     of “type 2” then the hypergeometric distribution gives the
     probability of obtaining k elements of “type 1” in t samples from
     the population without replacement.

 -- Function: double gsl_ran_hypergeometric_pdf (unsigned int k,
          unsigned int n1, unsigned int n2, unsigned int t)

     This function computes the probability p(k) of obtaining *note k:
     7c4. from a hypergeometric distribution with parameters *note n1:
     7c4, *note n2: 7c4, *note t: 7c4, using the formula given above.

[gsl-ref-figures/rand-hypergeometric]
 -- Function: double gsl_cdf_hypergeometric_P (unsigned int k, unsigned
          int n1, unsigned int n2, unsigned int t)
 -- Function: double gsl_cdf_hypergeometric_Q (unsigned int k, unsigned
          int n1, unsigned int n2, unsigned int t)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the hypergeometric distribution with parameters *note n1:
     7c6, *note n2: 7c6. and *note t: 7c6.


File: gsl-ref.info,  Node: The Logarithmic Distribution,  Next: The Wishart Distribution,  Prev: The Hypergeometric Distribution,  Up: Random Number Distributions

20.38 The Logarithmic Distribution
==================================

 -- Function: unsigned int gsl_ran_logarithmic (const gsl_rng *r, double
          p)

     This function returns a random integer from the logarithmic
     distribution.  The probability distribution for logarithmic random
     variates is,

          p(k) = {-1 \over \log(1-p)} {(p^k \over k)}

     for k \ge 1.

 -- Function: double gsl_ran_logarithmic_pdf (unsigned int k, double p)

     This function computes the probability p(k) of obtaining *note k:
     7c9. from a logarithmic distribution with probability parameter
     *note p: 7c9, using the formula given above.

[gsl-ref-figures/rand-logarithmic]

File: gsl-ref.info,  Node: The Wishart Distribution,  Next: Shuffling and Sampling,  Prev: The Logarithmic Distribution,  Up: Random Number Distributions

20.39 The Wishart Distribution
==============================

 -- Function: int gsl_ran_wishart (const gsl_rng *r, const double n,
          const gsl_matrix *L, gsl_matrix *result, gsl_matrix *work)

     This function computes a random symmetric p-by-p matrix from the
     Wishart distribution.  The probability distribution for Wishart
     random variates is,

          p(X) = \frac{|X|^{(n-p-1)/2} e^{-tr( V^{-1} X)/2}}{2^{(np)/2} |V|^{n/2} \Gamma_p(n/2)}

     Here, n > p - 1 is the number of degrees of freedom, V is a
     symmetric positive definite p-by-p scale matrix, whose Cholesky
     factor is specified by *note L: 7cb, and *note work: 7cb. is p-by-p
     workspace.  The p-by-p Wishart distributed matrix X is stored in
     *note result: 7cb. on output.

 -- Function: int gsl_ran_wishart_pdf (const gsl_matrix *X, const
          gsl_matrix *L_X, const double n, const gsl_matrix *L, double
          *result, gsl_matrix *work)
 -- Function: int gsl_ran_wishart_log_pdf (const gsl_matrix *X, const
          gsl_matrix *L_X, const double n, const gsl_matrix *L, double
          *result, gsl_matrix *work)

     These functions compute p(X) or \log{p(X)} for the p-by-p matrix
     *note X: 7cd, whose Cholesky factor is specified in *note L_X: 7cd.
     The degrees of freedom is given by *note n: 7cd, the Cholesky
     factor of the scale matrix V is specified in *note L: 7cd, and
     *note work: 7cd. is p-by-p workspace.  The probably density value
     is returned in *note result: 7cd.


File: gsl-ref.info,  Node: Shuffling and Sampling,  Next: Examples<14>,  Prev: The Wishart Distribution,  Up: Random Number Distributions

20.40 Shuffling and Sampling
============================

The following functions allow the shuffling and sampling of a set of
objects.  The algorithms rely on a random number generator as a source
of randomness and a poor quality generator can lead to correlations in
the output.  In particular it is important to avoid generators with a
short period.  For more information see Knuth, v2, 3rd ed, Section
3.4.2, “Random Sampling and Shuffling”.

 -- Function: void gsl_ran_shuffle (const gsl_rng *r, void *base, size_t
          n, size_t size)

     This function randomly shuffles the order of *note n: 7cf. objects,
     each of size *note size: 7cf, stored in the array ‘base[0..n-1]’.
     The output of the random number generator *note r: 7cf. is used to
     produce the permutation.  The algorithm generates all possible n!
     permutations with equal probability, assuming a perfect source of
     random numbers.

     The following code shows how to shuffle the numbers from 0 to 51:

          int a[52];

          for (i = 0; i < 52; i++)
            {
              a[i] = i;
            }

          gsl_ran_shuffle (r, a, 52, sizeof (int));

 -- Function: int gsl_ran_choose (const gsl_rng *r, void *dest, size_t
          k, void *src, size_t n, size_t size)

     This function fills the array ‘dest[k]’ with *note k: 7d0. objects
     taken randomly from the *note n: 7d0. elements of the array
     ‘src[0..n-1]’.  The objects are each of size *note size: 7d0.  The
     output of the random number generator *note r: 7d0. is used to make
     the selection.  The algorithm ensures all possible samples are
     equally likely, assuming a perfect source of randomness.

     The objects are sampled `without' replacement, thus each object can
     only appear once in *note dest: 7d0.  It is required that *note k:
     7d0. be less than or equal to *note n: 7d0.  The objects in *note
     dest: 7d0. will be in the same relative order as those in *note
     src: 7d0.  You will need to call ‘gsl_ran_shuffle(r, dest, n,
     size)’ if you want to randomize the order.

     The following code shows how to select a random sample of three
     unique numbers from the set 0 to 99:

          double a[3], b[100];

          for (i = 0; i < 100; i++)
            {
              b[i] = (double) i;
            }

          gsl_ran_choose (r, a, 3, b, 100, sizeof (double));

 -- Function: void gsl_ran_sample (const gsl_rng *r, void *dest, size_t
          k, void *src, size_t n, size_t size)

     This function is like *note gsl_ran_choose(): 7d0. but samples
     *note k: 7d1. items from the original array of *note n: 7d1. items
     *note src: 7d1. with replacement, so the same object can appear
     more than once in the output sequence *note dest: 7d1.  There is no
     requirement that *note k: 7d1. be less than *note n: 7d1. in this
     case.


File: gsl-ref.info,  Node: Examples<14>,  Next: References and Further Reading<14>,  Prev: Shuffling and Sampling,  Up: Random Number Distributions

20.41 Examples
==============

The following program demonstrates the use of a random number generator
to produce variates from a distribution.  It prints 10 samples from the
Poisson distribution with a mean of 3.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       int i, n = 10;
       double mu = 3.0;

       /* create a generator chosen by the
          environment variable GSL_RNG_TYPE */

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       /* print n random variates chosen from
          the poisson distribution with mean
          parameter mu */

       for (i = 0; i < n; i++)
         {
           unsigned int k = gsl_ran_poisson (r, mu);
           printf (" %u", k);
         }

       printf ("\n");
       gsl_rng_free (r);
       return 0;
     }

If the library and header files are installed under ‘/usr/local’ (the
default location) then the program can be compiled with these options:

     $ gcc -Wall demo.c -lgsl -lgslcblas -lm

Here is the output of the program,

      2 5 5 2 1 0 3 4 1 1

The variates depend on the seed used by the generator.  The seed for the
default generator type *note gsl_rng_default: 6a7. can be changed with
the *note GSL_RNG_SEED: 690. environment variable to produce a different
stream of variates:

     $ GSL_RNG_SEED=123 ./a.out

giving output

      4 5 6 3 3 1 4 2 5 5

The following program generates a random walk in two dimensions.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     int
     main (void)
     {
       int i;
       double x = 0, y = 0, dx, dy;

       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_rng_env_setup();
       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       printf ("%g %g\n", x, y);

       for (i = 0; i < 10; i++)
         {
           gsl_ran_dir_2d (r, &dx, &dy);
           x += dx; y += dy;
           printf ("%g %g\n", x, y);
         }

       gsl_rng_free (r);
       return 0;
     }

Fig.  %s shows the output from the program.

[gsl-ref-figures/random-walk]
Figure: Four 10-step random walks from the origin.

The following program computes the upper and lower cumulative
distribution functions for the standard normal distribution at x = 2.

     #include <stdio.h>
     #include <gsl/gsl_cdf.h>

     int
     main (void)
     {
       double P, Q;
       double x = 2.0;

       P = gsl_cdf_ugaussian_P (x);
       printf ("prob(x < %f) = %f\n", x, P);

       Q = gsl_cdf_ugaussian_Q (x);
       printf ("prob(x > %f) = %f\n", x, Q);

       x = gsl_cdf_ugaussian_Pinv (P);
       printf ("Pinv(%f) = %f\n", P, x);

       x = gsl_cdf_ugaussian_Qinv (Q);
       printf ("Qinv(%f) = %f\n", Q, x);

       return 0;
     }

Here is the output of the program,

     prob(x < 2.000000) = 0.977250
     prob(x > 2.000000) = 0.022750
     Pinv(0.977250) = 2.000000
     Qinv(0.022750) = 2.000000


File: gsl-ref.info,  Node: References and Further Reading<14>,  Prev: Examples<14>,  Up: Random Number Distributions

20.42 References and Further Reading
====================================

For an encyclopaedic coverage of the subject readers are advised to
consult the book “Non-Uniform Random Variate Generation” by Luc Devroye.
It covers every imaginable distribution and provides hundreds of
algorithms.

   * Luc Devroye, “Non-Uniform Random Variate Generation”,
     Springer-Verlag, ISBN 0-387-96305-7.  Available online at
     ‘http://cg.scs.carleton.ca/~luc/rnbookindex.html’.

The subject of random variate generation is also reviewed by Knuth, who
describes algorithms for all the major distributions.

   * Donald E. Knuth, “The Art of Computer Programming: Seminumerical
     Algorithms” (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

The Particle Data Group provides a short review of techniques for
generating distributions of random numbers in the “Monte Carlo” section
of its Annual Review of Particle Physics.

   * Review of Particle Properties, R.M. Barnett et al., Physical Review
     D54, 1 (1996) ‘http://pdg.lbl.gov/’.

The Review of Particle Physics is available online in postscript and pdf
format.

An overview of methods used to compute cumulative distribution functions
can be found in `Statistical Computing' by W.J. Kennedy and J.E. Gentle.
Another general reference is `Elements of Statistical Computing' by R.A.
Thisted.

   * William E. Kennedy and James E. Gentle, Statistical Computing
     (1980), Marcel Dekker, ISBN 0-8247-6898-1.

   * Ronald A. Thisted, Elements of Statistical Computing (1988),
     Chapman & Hall, ISBN 0-412-01371-1.

The cumulative distribution functions for the Gaussian distribution are
based on the following papers,

   * Rational Chebyshev Approximations Using Linear Equations, W.J.
     Cody, W. Fraser, J.F. Hart.  Numerische Mathematik 12, 242–251
     (1968).

   * Rational Chebyshev Approximations for the Error Function, W.J.
     Cody.  Mathematics of Computation 23, n107, 631–637 (July 1969).


File: gsl-ref.info,  Node: Statistics,  Next: Running Statistics,  Prev: Random Number Distributions,  Up: Top

21 Statistics
*************

This chapter describes the statistical functions in the library.  The
basic statistical functions include routines to compute the mean,
variance and standard deviation.  More advanced functions allow you to
calculate absolute deviations, skewness, and kurtosis as well as the
median and arbitrary percentiles.  The algorithms use recurrence
relations to compute average quantities in a stable way, without large
intermediate values that might overflow.

The functions are available in versions for datasets in the standard
floating-point and integer types.  The versions for double precision
floating-point data have the prefix ‘gsl_stats’ and are declared in the
header file ‘gsl_statistics_double.h’.  The versions for integer data
have the prefix ‘gsl_stats_int’ and are declared in the header file
‘gsl_statistics_int.h’.  All the functions operate on C arrays with a
‘stride’ parameter specifying the spacing between elements.

* Menu:

* Mean, Standard Deviation and Variance: Mean Standard Deviation and Variance.
* Absolute deviation::
* Higher moments (skewness and kurtosis): Higher moments skewness and kurtosis.
* Autocorrelation::
* Covariance::
* Correlation::
* Weighted Samples::
* Maximum and Minimum values::
* Median and Percentiles::
* Order Statistics::
* Robust Location Estimates::
* Robust Scale Estimates::
* Examples: Examples<15>.
* References and Further Reading: References and Further Reading<15>.


File: gsl-ref.info,  Node: Mean Standard Deviation and Variance,  Next: Absolute deviation,  Up: Statistics

21.1 Mean, Standard Deviation and Variance
==========================================

 -- Function: double gsl_stats_mean (const double data[], size_t stride,
          size_t n)

     This function returns the arithmetic mean of *note data: 1d, a
     dataset of length *note n: 1d. with stride *note stride: 1d.  The
     arithmetic mean, or `sample mean', is denoted by \Hat\mu and
     defined as,

          \Hat\mu = {1 \over N} \sum x_i

     where x_i are the elements of the dataset *note data: 1d.  For
     samples drawn from a gaussian distribution the variance of \Hat\mu
     is \sigma^2 / N.

 -- Function: double gsl_stats_variance (const double data[], size_t
          stride, size_t n)

     This function returns the estimated, or `sample', variance of *note
     data: 7d8, a dataset of length *note n: 7d8. with stride *note
     stride: 7d8.  The estimated variance is denoted by \Hat\sigma^2 and
     is defined by,

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2

     where x_i are the elements of the dataset *note data: 7d8.  Note
     that the normalization factor of 1/(N-1) results from the
     derivation of \Hat\sigma^2 as an unbiased estimator of the
     population variance \sigma^2.  For samples drawn from a Gaussian
     distribution the variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.

     This function computes the mean via a call to *note
     gsl_stats_mean(): 1d.  If you have already computed the mean then
     you can pass it directly to *note gsl_stats_variance_m(): 7d9.

 -- Function: double gsl_stats_variance_m (const double data[], size_t
          stride, size_t n, double mean)

     This function returns the sample variance of *note data: 7d9.
     relative to the given value of *note mean: 7d9.  The function is
     computed with \Hat\mu replaced by the value of *note mean: 7d9.
     that you supply,

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2

 -- Function: double gsl_stats_sd (const double data[], size_t stride,
          size_t n)
 -- Function: double gsl_stats_sd_m (const double data[], size_t stride,
          size_t n, double mean)

     The standard deviation is defined as the square root of the
     variance.  These functions return the square root of the
     corresponding variance functions above.

 -- Function: double gsl_stats_tss (const double data[], size_t stride,
          size_t n)
 -- Function: double gsl_stats_tss_m (const double data[], size_t
          stride, size_t n, double mean)

     These functions return the total sum of squares (TSS) of *note
     data: 7dd. about the mean.  For *note gsl_stats_tss_m(): 7dd. the
     user-supplied value of *note mean: 7dd. is used, and for *note
     gsl_stats_tss(): 7dc. it is computed using *note gsl_stats_mean():
     1d.

          TSS =  \sum (x_i - mean)^2

 -- Function: double gsl_stats_variance_with_fixed_mean (const double
          data[], size_t stride, size_t n, double mean)

     This function computes an unbiased estimate of the variance of
     *note data: 7de. when the population mean *note mean: 7de. of the
     underlying distribution is known `a priori'.  In this case the
     estimator for the variance uses the factor 1/N and the sample mean
     \Hat\mu is replaced by the known population mean \mu,

          \Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2

 -- Function: double gsl_stats_sd_with_fixed_mean (const double data[],
          size_t stride, size_t n, double mean)

     This function calculates the standard deviation of *note data: 7df.
     for a fixed population mean *note mean: 7df.  The result is the
     square root of the corresponding variance function.


File: gsl-ref.info,  Node: Absolute deviation,  Next: Higher moments skewness and kurtosis,  Prev: Mean Standard Deviation and Variance,  Up: Statistics

21.2 Absolute deviation
=======================

 -- Function: double gsl_stats_absdev (const double data[], size_t
          stride, size_t n)

     This function computes the absolute deviation from the mean of
     *note data: 7e1, a dataset of length *note n: 7e1. with stride
     *note stride: 7e1.  The absolute deviation from the mean is defined
     as,

          absdev  = (1/N) \sum |x_i - \Hat\mu|

     where x_i are the elements of the dataset *note data: 7e1.  The
     absolute deviation from the mean provides a more robust measure of
     the width of a distribution than the variance.  This function
     computes the mean of *note data: 7e1. via a call to *note
     gsl_stats_mean(): 1d.

 -- Function: double gsl_stats_absdev_m (const double data[], size_t
          stride, size_t n, double mean)

     This function computes the absolute deviation of the dataset *note
     data: 7e2. relative to the given value of *note mean: 7e2,

          absdev  = (1/N) \sum |x_i - mean|

     This function is useful if you have already computed the mean of
     *note data: 7e2. (and want to avoid recomputing it), or wish to
     calculate the absolute deviation relative to another value (such as
     zero, or the median).


File: gsl-ref.info,  Node: Higher moments skewness and kurtosis,  Next: Autocorrelation,  Prev: Absolute deviation,  Up: Statistics

21.3 Higher moments (skewness and kurtosis)
===========================================

 -- Function: double gsl_stats_skew (const double data[], size_t stride,
          size_t n)

     This function computes the skewness of *note data: 7e4, a dataset
     of length *note n: 7e4. with stride *note stride: 7e4.  The
     skewness is defined as,

          skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3

     where x_i are the elements of the dataset *note data: 7e4.  The
     skewness measures the asymmetry of the tails of a distribution.

     The function computes the mean and estimated standard deviation of
     *note data: 7e4. via calls to *note gsl_stats_mean(): 1d. and *note
     gsl_stats_sd(): 7da.

 -- Function: double gsl_stats_skew_m_sd (const double data[], size_t
          stride, size_t n, double mean, double sd)

     This function computes the skewness of the dataset *note data: 7e5.
     using the given values of the mean *note mean: 7e5. and standard
     deviation *note sd: 7e5,

          skew = (1/N) \sum ((x_i - mean)/sd)^3

     These functions are useful if you have already computed the mean
     and standard deviation of *note data: 7e5. and want to avoid
     recomputing them.

 -- Function: double gsl_stats_kurtosis (const double data[], size_t
          stride, size_t n)

     This function computes the kurtosis of *note data: 7e6, a dataset
     of length *note n: 7e6. with stride *note stride: 7e6.  The
     kurtosis is defined as,

          kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3

     The kurtosis measures how sharply peaked a distribution is,
     relative to its width.  The kurtosis is normalized to zero for a
     Gaussian distribution.

 -- Function: double gsl_stats_kurtosis_m_sd (const double data[],
          size_t stride, size_t n, double mean, double sd)

     This function computes the kurtosis of the dataset *note data: 7e7.
     using the given values of the mean *note mean: 7e7. and standard
     deviation *note sd: 7e7,

          kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3

     This function is useful if you have already computed the mean and
     standard deviation of *note data: 7e7. and want to avoid
     recomputing them.


File: gsl-ref.info,  Node: Autocorrelation,  Next: Covariance,  Prev: Higher moments skewness and kurtosis,  Up: Statistics

21.4 Autocorrelation
====================

 -- Function: double gsl_stats_lag1_autocorrelation (const double
          data[], const size_t stride, const size_t n)

     This function computes the lag-1 autocorrelation of the dataset
     *note data: 7e9.

          a_1 = {\sum_{i = 2}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
                 \over
                 \sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}

 -- Function: double gsl_stats_lag1_autocorrelation_m (const double
          data[], const size_t stride, const size_t n, const double
          mean)

     This function computes the lag-1 autocorrelation of the dataset
     *note data: 7ea. using the given value of the mean *note mean: 7ea.


File: gsl-ref.info,  Node: Covariance,  Next: Correlation,  Prev: Autocorrelation,  Up: Statistics

21.5 Covariance
===============

 -- Function: double gsl_stats_covariance (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n)

     This function computes the covariance of the datasets *note data1:
     7ec. and *note data2: 7ec. which must both be of the same length
     *note n: 7ec.

          covar = (1/(n - 1)) \sum_{i = 1}^{n} (x_i - \Hat x) (y_i - \Hat y)

 -- Function: double gsl_stats_covariance_m (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n, const double mean1, const double mean2)

     This function computes the covariance of the datasets *note data1:
     7ed. and *note data2: 7ed. using the given values of the means,
     *note mean1: 7ed. and *note mean2: 7ed.  This is useful if you have
     already computed the means of *note data1: 7ed. and *note data2:
     7ed. and want to avoid recomputing them.


File: gsl-ref.info,  Node: Correlation,  Next: Weighted Samples,  Prev: Covariance,  Up: Statistics

21.6 Correlation
================

 -- Function: double gsl_stats_correlation (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n)

     This function efficiently computes the Pearson correlation
     coefficient between the datasets *note data1: 7ef. and *note data2:
     7ef. which must both be of the same length *note n: 7ef.

          r = cov(x, y) / (\Hat\sigma_x \Hat\sigma_y)
            = {1/(n-1) \sum (x_i - \Hat x) (y_i - \Hat y)
               \over
               \sqrt{1/(n-1) \sum (x_i - \Hat x)^2} \sqrt{1/(n-1) \sum (y_i - \Hat y)^2}
              }

 -- Function: double gsl_stats_spearman (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n, double work[])

     This function computes the Spearman rank correlation coefficient
     between the datasets *note data1: 7f0. and *note data2: 7f0. which
     must both be of the same length *note n: 7f0.  Additional workspace
     of size 2 * *note n: 7f0. is required in *note work: 7f0.  The
     Spearman rank correlation between vectors x and y is equivalent to
     the Pearson correlation between the ranked vectors x_R and y_R,
     where ranks are defined to be the average of the positions of an
     element in the ascending order of the values.


File: gsl-ref.info,  Node: Weighted Samples,  Next: Maximum and Minimum values,  Prev: Correlation,  Up: Statistics

21.7 Weighted Samples
=====================

The functions described in this section allow the computation of
statistics for weighted samples.  The functions accept an array of
samples, x_i, with associated weights, w_i.  Each sample x_i is
considered as having been drawn from a Gaussian distribution with
variance \sigma_i^2.  The sample weight w_i is defined as the reciprocal
of this variance, w_i = 1/\sigma_i^2.  Setting a weight to zero
corresponds to removing a sample from a dataset.

 -- Function: double gsl_stats_wmean (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n)

     This function returns the weighted mean of the dataset *note data:
     7f2. with stride *note stride: 7f2. and length *note n: 7f2, using
     the set of weights *note w: 7f2. with stride *note wstride: 7f2.
     and length *note n: 7f2.  The weighted mean is defined as,

          \Hat\mu = (\sum w_i x_i) / (\sum w_i)

 -- Function: double gsl_stats_wvariance (const double w[], size_t
          wstride, const double data[], size_t stride, size_t n)

     This function returns the estimated variance of the dataset *note
     data: 7f3. with stride *note stride: 7f3. and length *note n: 7f3,
     using the set of weights *note w: 7f3. with stride *note wstride:
     7f3. and length *note n: 7f3.  The estimated variance of a weighted
     dataset is calculated as,

          \Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2)))
                          \sum w_i (x_i - \Hat\mu)^2

     Note that this expression reduces to an unweighted variance with
     the familiar 1/(N-1) factor when there are N equal non-zero
     weights.

 -- Function: double gsl_stats_wvariance_m (const double w[], size_t
          wstride, const double data[], size_t stride, size_t n, double
          wmean)

     This function returns the estimated variance of the weighted
     dataset *note data: 7f4. using the given weighted mean *note wmean:
     7f4.

 -- Function: double gsl_stats_wsd (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n)

     The standard deviation is defined as the square root of the
     variance.  This function returns the square root of the
     corresponding variance function *note gsl_stats_wvariance(): 7f3.
     above.

 -- Function: double gsl_stats_wsd_m (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n, double wmean)

     This function returns the square root of the corresponding variance
     function *note gsl_stats_wvariance_m(): 7f4. above.

 -- Function: double gsl_stats_wvariance_with_fixed_mean (const double
          w[], size_t wstride, const double data[], size_t stride,
          size_t n, const double mean)

     This function computes an unbiased estimate of the variance of the
     weighted dataset *note data: 7f7. when the population mean *note
     mean: 7f7. of the underlying distribution is known `a priori'.  In
     this case the estimator for the variance replaces the sample mean
     \Hat\mu by the known population mean \mu,

          \Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)

 -- Function: double gsl_stats_wsd_with_fixed_mean (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n,
          const double mean)

     The standard deviation is defined as the square root of the
     variance.  This function returns the square root of the
     corresponding variance function above.

 -- Function: double gsl_stats_wtss (const double w[], const size_t
          wstride, const double data[], size_t stride, size_t n)
 -- Function: double gsl_stats_wtss_m (const double w[], const size_t
          wstride, const double data[], size_t stride, size_t n, double
          wmean)

     These functions return the weighted total sum of squares (TSS) of
     *note data: 7fa. about the weighted mean.  For *note
     gsl_stats_wtss_m(): 7fa. the user-supplied value of *note wmean:
     7fa. is used, and for *note gsl_stats_wtss(): 7f9. it is computed
     using *note gsl_stats_wmean(): 7f2.

          TSS =  \sum w_i (x_i - wmean)^2

 -- Function: double gsl_stats_wabsdev (const double w[], size_t
          wstride, const double data[], size_t stride, size_t n)

     This function computes the weighted absolute deviation from the
     weighted mean of *note data: 7fb.  The absolute deviation from the
     mean is defined as,

          absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)

 -- Function: double gsl_stats_wabsdev_m (const double w[], size_t
          wstride, const double data[], size_t stride, size_t n, double
          wmean)

     This function computes the absolute deviation of the weighted
     dataset *note data: 7fc. about the given weighted mean *note wmean:
     7fc.

 -- Function: double gsl_stats_wskew (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n)

     This function computes the weighted skewness of the dataset *note
     data: 7fd.

          skew = (\sum w_i ((x_i - \Hat x)/\Hat \sigma)^3) / (\sum w_i)

 -- Function: double gsl_stats_wskew_m_sd (const double w[], size_t
          wstride, const double data[], size_t stride, size_t n, double
          wmean, double wsd)

     This function computes the weighted skewness of the dataset *note
     data: 7fe. using the given values of the weighted mean and weighted
     standard deviation, *note wmean: 7fe. and *note wsd: 7fe.

 -- Function: double gsl_stats_wkurtosis (const double w[], size_t
          wstride, const double data[], size_t stride, size_t n)

     This function computes the weighted kurtosis of the dataset *note
     data: 7ff.

          kurtosis = ((\sum w_i ((x_i - \Hat x)/\Hat \sigma)^4) / (\sum w_i)) - 3

 -- Function: double gsl_stats_wkurtosis_m_sd (const double w[], size_t
          wstride, const double data[], size_t stride, size_t n, double
          wmean, double wsd)

     This function computes the weighted kurtosis of the dataset *note
     data: 800. using the given values of the weighted mean and weighted
     standard deviation, *note wmean: 800. and *note wsd: 800.


File: gsl-ref.info,  Node: Maximum and Minimum values,  Next: Median and Percentiles,  Prev: Weighted Samples,  Up: Statistics

21.8 Maximum and Minimum values
===============================

The following functions find the maximum and minimum values of a dataset
(or their indices).  If the data contains ‘NaN’-s then a ‘NaN’ will be
returned, since the maximum or minimum value is undefined.  For
functions which return an index, the location of the first ‘NaN’ in the
array is returned.

 -- Function: double gsl_stats_max (const double data[], size_t stride,
          size_t n)

     This function returns the maximum value in *note data: 802, a
     dataset of length *note n: 802. with stride *note stride: 802.  The
     maximum value is defined as the value of the element x_i which
     satisfies x_i \ge x_j for all j.

     If you want instead to find the element with the largest absolute
     magnitude you will need to apply ‘fabs()’ or ‘abs()’ to your data
     before calling this function.

 -- Function: double gsl_stats_min (const double data[], size_t stride,
          size_t n)

     This function returns the minimum value in *note data: 803, a
     dataset of length *note n: 803. with stride *note stride: 803.  The
     minimum value is defined as the value of the element x_i which
     satisfies x_i \le x_j for all j.

     If you want instead to find the element with the smallest absolute
     magnitude you will need to apply ‘fabs()’ or ‘abs()’ to your data
     before calling this function.

 -- Function: void gsl_stats_minmax (double *min, double *max, const
          double data[], size_t stride, size_t n)

     This function finds both the minimum and maximum values *note min:
     804, *note max: 804. in *note data: 804. in a single pass.

 -- Function: size_t gsl_stats_max_index (const double data[], size_t
          stride, size_t n)

     This function returns the index of the maximum value in *note data:
     805, a dataset of length *note n: 805. with stride *note stride:
     805.  The maximum value is defined as the value of the element x_i
     which satisfies x_i \ge x_j for all j.  When there are several
     equal maximum elements then the first one is chosen.

 -- Function: size_t gsl_stats_min_index (const double data[], size_t
          stride, size_t n)

     This function returns the index of the minimum value in *note data:
     806, a dataset of length *note n: 806. with stride *note stride:
     806.  The minimum value is defined as the value of the element x_i
     which satisfies x_i \ge x_j for all j.  When there are several
     equal minimum elements then the first one is chosen.

 -- Function: void gsl_stats_minmax_index (size_t *min_index, size_t
          *max_index, const double data[], size_t stride, size_t n)

     This function returns the indexes *note min_index: 807, *note
     max_index: 807. of the minimum and maximum values in *note data:
     807. in a single pass.


File: gsl-ref.info,  Node: Median and Percentiles,  Next: Order Statistics,  Prev: Maximum and Minimum values,  Up: Statistics

21.9 Median and Percentiles
===========================

The median and percentile functions described in this section operate on
sorted data in O(1) time.  There is also a routine for computing the
median of an unsorted input array in average O(n) time using the
quickselect algorithm.  For convenience we use `quantiles', measured on
a scale of 0 to 1, instead of percentiles (which use a scale of 0 to
100).

 -- Function: double gsl_stats_median_from_sorted_data (const double
          sorted_data[], const size_t stride, const size_t n)

     This function returns the median value of *note sorted_data: 809, a
     dataset of length *note n: 809. with stride *note stride: 809.  The
     elements of the array must be in ascending numerical order.  There
     are no checks to see whether the data are sorted, so the function
     *note gsl_sort(): 462. should always be used first.

     When the dataset has an odd number of elements the median is the
     value of element (n-1)/2.  When the dataset has an even number of
     elements the median is the mean of the two nearest middle values,
     elements (n-1)/2 and n/2.  Since the algorithm for computing the
     median involves interpolation this function always returns a
     floating-point number, even for integer data types.

 -- Function: double gsl_stats_median (double data[], const size_t
          stride, const size_t n)

     This function returns the median value of *note data: 80a, a
     dataset of length *note n: 80a. with stride *note stride: 80a.  The
     median is found using the quickselect algorithm.  The input array
     does not need to be sorted, but note that the algorithm rearranges
     the array and so the input is not preserved on output.

 -- Function: double gsl_stats_quantile_from_sorted_data (const double
          sorted_data[], size_t stride, size_t n, double f)

     This function returns a quantile value of *note sorted_data: 80b, a
     double-precision array of length *note n: 80b. with stride *note
     stride: 80b.  The elements of the array must be in ascending
     numerical order.  The quantile is determined by the *note f: 80b, a
     fraction between 0 and 1.  For example, to compute the value of the
     75th percentile *note f: 80b. should have the value 0.75.

     There are no checks to see whether the data are sorted, so the
     function *note gsl_sort(): 462. should always be used first.

     The quantile is found by interpolation, using the formula

          quantile = (1 - \delta) x_i + \delta x_{i+1}

     where i is ‘floor((n - 1)f)’ and \delta is (n-1)f - i.

     Thus the minimum value of the array (‘data[0*stride]’) is given by
     *note f: 80b. equal to zero, the maximum value
     (‘data[(n-1)*stride]’) is given by *note f: 80b. equal to one and
     the median value is given by *note f: 80b. equal to 0.5.  Since the
     algorithm for computing quantiles involves interpolation this
     function always returns a floating-point number, even for integer
     data types.


File: gsl-ref.info,  Node: Order Statistics,  Next: Robust Location Estimates,  Prev: Median and Percentiles,  Up: Statistics

21.10 Order Statistics
======================

The k-th `order statistic' of a sample is equal to its k-th smallest
value.  The k-th order statistic of a set of n values x = \left\{ x_i
\right\}, 1 \le i \le n is denoted x_{(k)}.  The median of the set x is
equal to x_{\left( \frac{n}{2} \right)} if n is odd, or the average of
x_{\left( \frac{n}{2} \right)} and x_{\left( \frac{n}{2} + 1 \right)} if
n is even.  The k-th smallest element of a length n vector can be found
in average O(n) time using the quickselect algorithm.

 -- Function: double gsl_stats_select (double data[], const size_t
          stride, const size_t n, const size_t k)

     This function finds the *note k: 80d.-th smallest element of the
     input array *note data: 80d. of length *note n: 80d. and stride
     *note stride: 80d. using the quickselect method.  The algorithm
     rearranges the elements of *note data: 80d. and so the input array
     is not preserved on output.


File: gsl-ref.info,  Node: Robust Location Estimates,  Next: Robust Scale Estimates,  Prev: Order Statistics,  Up: Statistics

21.11 Robust Location Estimates
===============================

A `location estimate' refers to a typical or central value which best
describes a given dataset.  The mean and median are both examples of
location estimators.  However, the mean has a severe sensitivity to data
outliers and can give erroneous values when even a small number of
outliers are present.  The median on the other hand, has a strong
insensitivity to data outliers, but due to its non-smoothness it can
behave unexpectedly in certain situations.  GSL offers the following
alternative location estimators, which are robust to the presence of
outliers.

* Menu:

* Trimmed Mean::
* Gastwirth Estimator::


File: gsl-ref.info,  Node: Trimmed Mean,  Next: Gastwirth Estimator,  Up: Robust Location Estimates

21.11.1 Trimmed Mean
--------------------

The trimmed mean, or `truncated mean', discards a certain number of
smallest and largest samples from the input vector before computing the
mean of the remaining samples.  The amount of trimming is specified by a
factor \alpha \in [0,0.5].  Then the number of samples discarded from
both ends of the input vector is \left\lfloor \alpha n \right\rfloor,
where n is the length of the input.  So to discard 25% of the samples
from each end, one would set \alpha = 0.25.

 -- Function: double gsl_stats_trmean_from_sorted_data (const double
          alpha, const double sorted_data[], const size_t stride, const
          size_t n)

     This function returns the trimmed mean of *note sorted_data: 810, a
     dataset of length *note n: 810. with stride *note stride: 810.  The
     elements of the array must be in ascending numerical order.  There
     are no checks to see whether the data are sorted, so the function
     *note gsl_sort(): 462. should always be used first.  The trimming
     factor \alpha is given in *note alpha: 810.  If \alpha \ge 0.5,
     then the median of the input is returned.


File: gsl-ref.info,  Node: Gastwirth Estimator,  Prev: Trimmed Mean,  Up: Robust Location Estimates

21.11.2 Gastwirth Estimator
---------------------------

Gastwirth’s location estimator is a weighted sum of three order
statistics,

     gastwirth = 0.3 * Q_{1/3} + 0.4 * Q_{1/2} + 0.3 * Q_{2/3}

where Q_{\frac{1}{3}} is the one-third quantile, Q_{\frac{1}{2}} is the
one-half quantile (i.e.  median), and Q_{\frac{2}{3}} is the two-thirds
quantile.

 -- Function: double gsl_stats_gastwirth_from_sorted_data (const double
          sorted_data[], const size_t stride, const size_t n)

     This function returns the Gastwirth location estimator of *note
     sorted_data: 812, a dataset of length *note n: 812. with stride
     *note stride: 812.  The elements of the array must be in ascending
     numerical order.  There are no checks to see whether the data are
     sorted, so the function *note gsl_sort(): 462. should always be
     used first.


File: gsl-ref.info,  Node: Robust Scale Estimates,  Next: Examples<15>,  Prev: Robust Location Estimates,  Up: Statistics

21.12 Robust Scale Estimates
============================

A `robust scale estimate', also known as a robust measure of scale,
attempts to quantify the statistical dispersion (variability, scatter,
spread) in a set of data which may contain outliers.  In such datasets,
the usual variance or standard deviation scale estimate can be rendered
useless by even a single outlier.

* Menu:

* Median Absolute Deviation (MAD): Median Absolute Deviation MAD.
* S_n Statistic::
* Q_n Statistic::


File: gsl-ref.info,  Node: Median Absolute Deviation MAD,  Next: S_n Statistic,  Up: Robust Scale Estimates

21.12.1 Median Absolute Deviation (MAD)
---------------------------------------

The median absolute deviation (MAD) is defined as

     MAD = 1.4826 median { | x_i - median(x) | }

In words, first the median of all samples is computed.  Then the median
is subtracted from all samples in the input to find the deviation of
each sample from the median.  The median of all absolute deviations is
then the MAD. The factor 1.4826 makes the MAD an unbiased estimator of
the standard deviation for Gaussian data.  The median absolute deviation
has an asymptotic efficiency of 37%.

 -- Function: double gsl_stats_mad0 (const double data[], const size_t
          stride, const size_t n, double work[])

 -- Function: double gsl_stats_mad (const double data[], const size_t
          stride, const size_t n, double work[])

     These functions return the median absolute deviation of *note data:
     817, a dataset of length *note n: 817. and stride *note stride:
     817.  The ‘mad0’ function calculates \textrm{median} \left\{ \left|
     x_i - \textrm{median} \left( x \right) \right| \right\} (i.e.  the
     MAD statistic without the bias correction scale factor).  These
     functions require additional workspace of size ‘n’ provided in
     *note work: 817.


File: gsl-ref.info,  Node: S_n Statistic,  Next: Q_n Statistic,  Prev: Median Absolute Deviation MAD,  Up: Robust Scale Estimates

21.12.2 S_n Statistic
---------------------

The S_n statistic developed by Croux and Rousseeuw is defined as

     S_n = 1.1926 * c_n * median_i { median_j ( | x_i - x_j | ) }

For each sample x_i, 1 \le i \le n, the median of the values \left| x_i
- x_j \right| is computed for all x_j, 1 \le j \le n.  This yields n
values, whose median then gives the final S_n.  The factor 1.1926 makes
S_n an unbiased estimate of the standard deviation for Gaussian data.
The factor c_n is a correction factor to correct bias in small sample
sizes.  S_n has an asymptotic efficiency of 58%.

 -- Function: double gsl_stats_Sn0_from_sorted_data (const double
          sorted_data[], const size_t stride, const size_t n, double
          work[])

 -- Function: double gsl_stats_Sn_from_sorted_data (const double
          sorted_data[], const size_t stride, const size_t n, double
          work[])

     These functions return the S_n statistic of *note sorted_data: 81b,
     a dataset of length *note n: 81b. with stride *note stride: 81b.
     The elements of the array must be in ascending numerical order.
     There are no checks to see whether the data are sorted, so the
     function *note gsl_sort(): 462. should always be used first.  The
     ‘Sn0’ function calculates \textrm{median}_i \left\{
     \textrm{median}_j \left( \left| x_i - x_j \right| \right) \right\}
     (i.e.  the S_n statistic without the bias correction scale
     factors).  These functions require additional workspace of size ‘n’
     provided in *note work: 81b.


File: gsl-ref.info,  Node: Q_n Statistic,  Prev: S_n Statistic,  Up: Robust Scale Estimates

21.12.3 Q_n Statistic
---------------------

The Q_n statistic developed by Croux and Rousseeuw is defined as

     Q_n = 2.21914 * d_n * { | x_i - x_j |, i < j }_{(k)}

The factor 2.21914 makes Q_n an unbiased estimate of the standard
deviation for Gaussian data.  The factor d_n is a correction factor to
correct bias in small sample sizes.  The order statistic is

     k = ( floor(n/2) + 1 )
         (       2        )

Q_n has an asymptotic efficiency of 82%.

 -- Function: double gsl_stats_Qn0_from_sorted_data (const double
          sorted_data[], const size_t stride, const size_t n, double
          work[], int work_int[])
 -- Function: double gsl_stats_Qn_from_sorted_data (const double
          sorted_data[], const size_t stride, const size_t n, double
          work[], int work_int[])

     These functions return the Q_n statistic of *note sorted_data: 81f,
     a dataset of length *note n: 81f. with stride *note stride: 81f.
     The elements of the array must be in ascending numerical order.
     There are no checks to see whether the data are sorted, so the
     function *note gsl_sort(): 462. should always be used first.  The
     ‘Qn0’ function calculates \left\{ \left| x_i - x_j \right|, i < j
     \right\}_{(k)} (i.e.  Q_n without the bias correction scale
     factors).  These functions require additional workspace of size
     ‘3n’ provided in *note work: 81f. and integer workspace of size
     ‘5n’ provided in *note work_int: 81f.


File: gsl-ref.info,  Node: Examples<15>,  Next: References and Further Reading<15>,  Prev: Robust Scale Estimates,  Up: Statistics

21.13 Examples
==============

Here is a basic example of how to use the statistical functions:

     #include <stdio.h>
     #include <gsl/gsl_statistics.h>

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double mean, variance, largest, smallest;

       mean     = gsl_stats_mean(data, 1, 5);
       variance = gsl_stats_variance(data, 1, 5);
       largest  = gsl_stats_max(data, 1, 5);
       smallest = gsl_stats_min(data, 1, 5);

       printf ("The dataset is %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("The sample mean is %g\n", mean);
       printf ("The estimated variance is %g\n", variance);
       printf ("The largest value is %g\n", largest);
       printf ("The smallest value is %g\n", smallest);
       return 0;
     }

The program should produce the following output,

     The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
     The sample mean is 16.54
     The estimated variance is 5.373
     The largest value is 18.3
     The smallest value is 12.6

Here is an example using sorted data,

     #include <stdio.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_statistics.h>

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double median, upperq, lowerq;

       printf ("Original dataset:  %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       gsl_sort (data, 1, 5);

       printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       median
         = gsl_stats_median_from_sorted_data (data,
                                              1, 5);

       upperq
         = gsl_stats_quantile_from_sorted_data (data,
                                                1, 5,
                                                0.75);
       lowerq
         = gsl_stats_quantile_from_sorted_data (data,
                                                1, 5,
                                                0.25);

       printf ("The median is %g\n", median);
       printf ("The upper quartile is %g\n", upperq);
       printf ("The lower quartile is %g\n", lowerq);
       return 0;
     }

This program should produce the following output,

     Original dataset:  17.2, 18.1, 16.5, 18.3, 12.6
     Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
     The median is 17.2
     The upper quartile is 18.1
     The lower quartile is 16.5


File: gsl-ref.info,  Node: References and Further Reading<15>,  Prev: Examples<15>,  Up: Statistics

21.14 References and Further Reading
====================================

The standard reference for almost any topic in statistics is the
multi-volume `Advanced Theory of Statistics' by Kendall and Stuart.

   * Maurice Kendall, Alan Stuart, and J. Keith Ord.  `The Advanced
     Theory of Statistics' (multiple volumes) reprinted as `Kendall’s
     Advanced Theory of Statistics'.  Wiley, ISBN 047023380X.

Many statistical concepts can be more easily understood by a Bayesian
approach.  The following book by Gelman, Carlin, Stern and Rubin gives a
comprehensive coverage of the subject.

   * Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin.
     `Bayesian Data Analysis'.  Chapman & Hall, ISBN 0412039915.

For physicists the Particle Data Group provides useful reviews of
Probability and Statistics in the “Mathematical Tools” section of its
Annual Review of Particle Physics.

   * `Review of Particle Properties', R.M. Barnett et al., Physical
     Review D54, 1 (1996)

The Review of Particle Physics is available online at the website
‘http://pdg.lbl.gov/’.

The following papers describe robust scale estimation,

   * C. Croux and P. J. Rousseeuw, `Time-Efficient algorithms for two
     highly robust estimators of scale', Comp.  Stat., Physica,
     Heidelberg, 1992.

   * P. J. Rousseeuw and C. Croux, `Explicit scale estimators with high
     breakdown point', L1-Statistical Analysis and Related Methods, pp.
     77-92, 1992.


File: gsl-ref.info,  Node: Running Statistics,  Next: Moving Window Statistics,  Prev: Statistics,  Up: Top

22 Running Statistics
*********************

This chapter describes routines for computing running statistics, also
known as online statistics, of data.  These routines are suitable for
handling large datasets for which it may be inconvenient or impractical
to store in memory all at once.  The data can be processed in a single
pass, one point at a time.  Each time a data point is added to the
accumulator, internal parameters are updated in order to compute the
current mean, variance, standard deviation, skewness, and kurtosis.
These statistics are exact, and are updated with numerically stable
single-pass algorithms.  The median and arbitrary quantiles are also
available, however these calculations use algorithms which provide
approximations, and grow more accurate as more data is added to the
accumulator.

The functions described in this chapter are declared in the header file
‘gsl_rstat.h’.

* Menu:

* Initializing the Accumulator::
* Adding Data to the Accumulator::
* Current Statistics::
* Quantiles::
* Examples: Examples<16>.
* References and Further Reading: References and Further Reading<16>.


File: gsl-ref.info,  Node: Initializing the Accumulator,  Next: Adding Data to the Accumulator,  Up: Running Statistics

22.1 Initializing the Accumulator
=================================

 -- Type: gsl_rstat_workspace

     This workspace contains parameters used to calculate various
     statistics and are updated after each data point is added to the
     accumulator.

 -- Function: *note gsl_rstat_workspace: 825. *gsl_rstat_alloc (void)

     This function allocates a workspace for computing running
     statistics.  The size of the workspace is O(1).

 -- Function: void gsl_rstat_free (gsl_rstat_workspace *w)

     This function frees the memory associated with the workspace *note
     w: 827.

 -- Function: int gsl_rstat_reset (gsl_rstat_workspace *w)

     This function resets the workspace *note w: 828. to its initial
     state, so it can begin working on a new set of data.


File: gsl-ref.info,  Node: Adding Data to the Accumulator,  Next: Current Statistics,  Prev: Initializing the Accumulator,  Up: Running Statistics

22.2 Adding Data to the Accumulator
===================================

 -- Function: int gsl_rstat_add (const double x, gsl_rstat_workspace *w)

     This function adds the data point *note x: 82a. to the statistical
     accumulator, updating calculations of the mean, variance, standard
     deviation, skewness, kurtosis, and median.

 -- Function: size_t gsl_rstat_n (const gsl_rstat_workspace *w)

     This function returns the number of data so far added to the
     accumulator.


File: gsl-ref.info,  Node: Current Statistics,  Next: Quantiles,  Prev: Adding Data to the Accumulator,  Up: Running Statistics

22.3 Current Statistics
=======================

 -- Function: double gsl_rstat_min (const gsl_rstat_workspace *w)

     This function returns the minimum value added to the accumulator.

 -- Function: double gsl_rstat_max (const gsl_rstat_workspace *w)

     This function returns the maximum value added to the accumulator.

 -- Function: double gsl_rstat_mean (const gsl_rstat_workspace *w)

     This function returns the mean of all data added to the
     accumulator, defined as

          \Hat\mu = (1/N) \sum x_i

 -- Function: double gsl_rstat_variance (const gsl_rstat_workspace *w)

     This function returns the variance of all data added to the
     accumulator, defined as

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2

 -- Function: double gsl_rstat_sd (const gsl_rstat_workspace *w)

     This function returns the standard deviation of all data added to
     the accumulator, defined as the square root of the variance given
     above.

 -- Function: double gsl_rstat_sd_mean (const gsl_rstat_workspace *w)

     This function returns the standard deviation of the mean, defined
     as

          sd_mean = \Hat\sigma / \sqrt{N}

 -- Function: double gsl_rstat_rms (const gsl_rstat_workspace *w)

     This function returns the root mean square of all data added to the
     accumulator, defined as

          rms = \sqrt{{1 \over N} \sum x_i^2}

 -- Function: double gsl_rstat_skew (const gsl_rstat_workspace *w)

     This function returns the skewness of all data added to the
     accumulator, defined as

          skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3

 -- Function: double gsl_rstat_kurtosis (const gsl_rstat_workspace *w)

     This function returns the kurtosis of all data added to the
     accumulator, defined as

          kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3

 -- Function: double gsl_rstat_median (gsl_rstat_workspace *w)

     This function returns an estimate of the median of the data added
     to the accumulator.


File: gsl-ref.info,  Node: Quantiles,  Next: Examples<16>,  Prev: Current Statistics,  Up: Running Statistics

22.4 Quantiles
==============

The functions in this section estimate quantiles dynamically without
storing the entire dataset, using the algorithm of Jain and Chlamtec,
1985.  Only five points (markers) are stored which represent the minimum
and maximum of the data, as well as current estimates of the p/2-, p-,
and (1+p)/2-quantiles.  Each time a new data point is added, the marker
positions and heights are updated.

 -- Type: gsl_rstat_quantile_workspace

     This workspace contains parameters for estimating quantiles of the
     current dataset

 -- Function: *note gsl_rstat_quantile_workspace: 838.
          *gsl_rstat_quantile_alloc (const double p)

     This function allocates a workspace for the dynamic estimation of
     *note p: 839.-quantiles, where *note p: 839. is between 0 and 1.
     The median corresponds to p = 0.5.  The size of the workspace is
     O(1).

 -- Function: void gsl_rstat_quantile_free (gsl_rstat_quantile_workspace
          *w)

     This function frees the memory associated with the workspace *note
     w: 83a.

 -- Function: int gsl_rstat_quantile_reset (gsl_rstat_quantile_workspace
          *w)

     This function resets the workspace *note w: 83b. to its initial
     state, so it can begin working on a new set of data.

 -- Function: int gsl_rstat_quantile_add (const double x,
          gsl_rstat_quantile_workspace *w)

     This function updates the estimate of the p-quantile with the new
     data point *note x: 83c.

 -- Function: double gsl_rstat_quantile_get
          (gsl_rstat_quantile_workspace *w)

     This function returns the current estimate of the p-quantile.


File: gsl-ref.info,  Node: Examples<16>,  Next: References and Further Reading<16>,  Prev: Quantiles,  Up: Running Statistics

22.5 Examples
=============

Here is a basic example of how to use the statistical functions:

     #include <stdio.h>
     #include <gsl/gsl_rstat.h>

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double mean, variance, largest, smallest, sd,
              rms, sd_mean, median, skew, kurtosis;
       gsl_rstat_workspace *rstat_p = gsl_rstat_alloc();
       size_t i, n;

       /* add data to rstat accumulator */
       for (i = 0; i < 5; ++i)
         gsl_rstat_add(data[i], rstat_p);

       mean     = gsl_rstat_mean(rstat_p);
       variance = gsl_rstat_variance(rstat_p);
       largest  = gsl_rstat_max(rstat_p);
       smallest = gsl_rstat_min(rstat_p);
       median   = gsl_rstat_median(rstat_p);
       sd       = gsl_rstat_sd(rstat_p);
       sd_mean  = gsl_rstat_sd_mean(rstat_p);
       skew     = gsl_rstat_skew(rstat_p);
       rms      = gsl_rstat_rms(rstat_p);
       kurtosis = gsl_rstat_kurtosis(rstat_p);
       n        = gsl_rstat_n(rstat_p);

       printf ("The dataset is %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("The sample mean is %g\n", mean);
       printf ("The estimated variance is %g\n", variance);
       printf ("The largest value is %g\n", largest);
       printf ("The smallest value is %g\n", smallest);
       printf( "The median is %g\n", median);
       printf( "The standard deviation is %g\n", sd);
       printf( "The root mean square is %g\n", rms);
       printf( "The standard devation of the mean is %g\n", sd_mean);
       printf( "The skew is %g\n", skew);
       printf( "The kurtosis %g\n", kurtosis);
       printf( "There are %zu items in the accumulator\n", n);

       gsl_rstat_reset(rstat_p);
       n = gsl_rstat_n(rstat_p);
       printf( "There are %zu items in the accumulator\n", n);

       gsl_rstat_free(rstat_p);

       return 0;
     }

The program should produce the following output,

     The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
     The sample mean is 16.54
     The estimated variance is 5.373
     The largest value is 18.3
     The smallest value is 12.6
     The median is 17.2
     The standard deviation is 2.31797
     The root mean square is 16.6694
     The standard devation of the mean is 1.03663
     The skew is -0.829058
     The kurtosis -1.2217
     There are 5 items in the accumulator
     There are 0 items in the accumulator

This next program estimates the lower quartile, median and upper
quartile from 10,000 samples of a random Rayleigh distribution, using
the P^2 algorithm of Jain and Chlamtec.  For comparison, the exact
values are also computed from the sorted dataset.

     #include <stdio.h>
     #include <stdlib.h>
     #include <gsl/gsl_rstat.h>
     #include <gsl/gsl_statistics.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_sort.h>

     int
     main(void)
     {
       const size_t N = 10000;
       double *data = malloc(N * sizeof(double));
       gsl_rstat_quantile_workspace *work_25 = gsl_rstat_quantile_alloc(0.25);
       gsl_rstat_quantile_workspace *work_50 = gsl_rstat_quantile_alloc(0.5);
       gsl_rstat_quantile_workspace *work_75 = gsl_rstat_quantile_alloc(0.75);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       double exact_p25, exact_p50, exact_p75;
       double val_p25, val_p50, val_p75;
       size_t i;

       /* add data to quantile accumulators; also store data for exact
        * comparisons */
       for (i = 0; i < N; ++i)
         {
           data[i] = gsl_ran_rayleigh(r, 1.0);
           gsl_rstat_quantile_add(data[i], work_25);
           gsl_rstat_quantile_add(data[i], work_50);
           gsl_rstat_quantile_add(data[i], work_75);
         }

       /* exact values */
       gsl_sort(data, 1, N);
       exact_p25 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.25);
       exact_p50 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.5);
       exact_p75 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.75);

       /* estimated values */
       val_p25 = gsl_rstat_quantile_get(work_25);
       val_p50 = gsl_rstat_quantile_get(work_50);
       val_p75 = gsl_rstat_quantile_get(work_75);

       printf ("The dataset is %g, %g, %g, %g, %g, ...\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("0.25 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p25, val_p25, (val_p25 - exact_p25) / exact_p25);
       printf ("0.50 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p50, val_p50, (val_p50 - exact_p50) / exact_p50);
       printf ("0.75 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p75, val_p75, (val_p75 - exact_p75) / exact_p75);

       gsl_rstat_quantile_free(work_25);
       gsl_rstat_quantile_free(work_50);
       gsl_rstat_quantile_free(work_75);
       gsl_rng_free(r);
       free(data);

       return 0;
     }

The program should produce the following output,

     The dataset is 0.00645272, 0.0074002, 0.0120706, 0.0207256, 0.0227282, ...
     0.25 quartile: exact = 0.75766, estimated = 0.75580, error = -2.450209e-03
     0.50 quartile: exact = 1.17508, estimated = 1.17438, error = -5.995912e-04
     0.75 quartile: exact = 1.65347, estimated = 1.65696, error = 2.110571e-03


File: gsl-ref.info,  Node: References and Further Reading<16>,  Prev: Examples<16>,  Up: Running Statistics

22.6 References and Further Reading
===================================

The algorithm used to dynamically estimate p-quantiles is described in
the paper,

   * R. Jain and I. Chlamtac.  `The P^2 algorithm for dynamic
     calculation of quantiles and histograms without storing
     observations', Communications of the ACM, Volume 28 (October),
     Number 10, 1985, p.  1076-1085.


File: gsl-ref.info,  Node: Moving Window Statistics,  Next: Digital Filtering,  Prev: Running Statistics,  Up: Top

23 Moving Window Statistics
***************************

This chapter describes routines for computing `moving window statistics'
(also called rolling statistics and running statistics), using a window
around a sample which is used to calculate various local statistical
properties of an input data stream.  The window is then slid forward by
one sample to process the next data point and so on.

The functions described in this chapter are declared in the header file
‘gsl_movstat.h’.

* Menu:

* Introduction: Introduction<4>.
* Handling Endpoints::
* Allocation for Moving Window Statistics::
* Moving Mean::
* Moving Variance and Standard Deviation::
* Moving Minimum and Maximum::
* Moving Sum::
* Moving Median::
* Robust Scale Estimation::
* User-defined Moving Statistics::
* Accumulators::
* Examples: Examples<17>.
* References and Further Reading: References and Further Reading<17>.


File: gsl-ref.info,  Node: Introduction<4>,  Next: Handling Endpoints,  Up: Moving Window Statistics

23.1 Introduction
=================

This chapter is concerned with calculating various statistics from
subsets of a given dataset.  The main idea is to compute statistics in
the vicinity of a given data sample by defining a `window' which
includes the sample itself as well as some specified number of samples
before and after the sample in question.  For a sample x_i, we define a
window W_i^{H,J} as

     W_i^{H,J} = {x_{i-H},...,x_i,...,x_{i+J}}

The parameters H and J are non-negative integers specifying the number
of samples to include before and after the sample x_i.  Statistics such
as the mean and standard deviation of the window W_i^{H,J} may be
computed, and then the window is shifted forward by one sample to focus
on x_{i+1}.  The total number of samples in the window is K = H + J + 1.
To define a symmetric window centered on x_i, one would set H = J =
\left\lfloor K / 2 \right\rfloor.


File: gsl-ref.info,  Node: Handling Endpoints,  Next: Allocation for Moving Window Statistics,  Prev: Introduction<4>,  Up: Moving Window Statistics

23.2 Handling Endpoints
=======================

When processing samples near the ends of the input signal, there will
not be enough samples to fill the window W_i^{H,J} defined above.
Therefore the user must specify how to construct the windows near the
end points.  This is done by passing an input argument of type *note
gsl_movstat_end_t: 844.:

 -- Type: gsl_movstat_end_t

     This data type specifies how to construct windows near end points
     and can be selected from the following choices:

      -- Macro: GSL_MOVSTAT_END_PADZERO

          With this option, a full window of length K will be
          constructed by inserting zeros into the window near the signal
          end points.  Effectively, the input signal is modified to

               x~ = {0, ..., 0, x_1, x_2, ..., x_{n-1}, x_n, 0, ..., 0}

          to ensure a well-defined window for all x_i.

      -- Macro: GSL_MOVSTAT_END_PADVALUE

          With this option, a full window of length K will be
          constructed by padding the window with the first and last
          sample in the input signal.  Effectively, the input signal is
          modified to

               x~ = {x_1, ..., x_1, x_1, x_2, ..., x_{n-1}, x_n, x_n, ..., x_n}

      -- Macro: GSL_MOVSTAT_END_TRUNCATE

          With this option, no padding is performed, and the windows are
          simply truncated as the end points are approached.


File: gsl-ref.info,  Node: Allocation for Moving Window Statistics,  Next: Moving Mean,  Prev: Handling Endpoints,  Up: Moving Window Statistics

23.3 Allocation for Moving Window Statistics
============================================

 -- Type: gsl_movstat_workspace

     The moving window statistical routines use a common workspace.

 -- Function: *note gsl_movstat_workspace: 849. *gsl_movstat_alloc
          (const size_t K)

     This function allocates a workspace for computing symmetric,
     centered moving statistics with a window length of K samples.  In
     this case, H = J = \left\lfloor K/2 \right\rfloor.  The size of the
     workspace is O(7K).

 -- Function: *note gsl_movstat_workspace: 849. *gsl_movstat_alloc2
          (const size_t H, const size_t J)

     This function allocates a workspace for computing moving statistics
     using a window with H samples prior to the current sample, and J
     samples after the current sample.  The total window size is K = H +
     J + 1.  The size of the workspace is O(7K).

 -- Function: void *gsl_movstat_free (gsl_movstat_workspace *w)

     This function frees the memory associated with *note w: 84c.


File: gsl-ref.info,  Node: Moving Mean,  Next: Moving Variance and Standard Deviation,  Prev: Allocation for Moving Window Statistics,  Up: Moving Window Statistics

23.4 Moving Mean
================

The moving window mean calculates the mean of the values of each window
W_i^{H,J}.

     \hat{\mu}_i = 1/| W_i^{H,J} | \sum_{x_m \in W_i^{H,J}} x_m

Here, \left| W_i^{H,J} \right| represents the number of elements in the
window W_i^{H,J}.  This will normally be K, unless the
‘GSL_MOVSTAT_END_TRUNCATE’ option is selected, in which case it could be
less than K near the signal end points.

 -- Function: int gsl_movstat_mean (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace *w)

     This function computes the moving window mean of the input vector
     *note x: 84e, storing the output in *note y: 84e.  The parameter
     *note endtype: 84e. specifies how windows near the ends of the
     input should be handled.  It is allowed to have *note x: 84e. =
     *note y: 84e. for an in-place moving mean.


File: gsl-ref.info,  Node: Moving Variance and Standard Deviation,  Next: Moving Minimum and Maximum,  Prev: Moving Mean,  Up: Moving Window Statistics

23.5 Moving Variance and Standard Deviation
===========================================

The moving window variance calculates the `sample variance' of the
values of each window W_i^{H,J}, defined by

     \hat{\sigma}_i^2 = 1/(|W_i^{H,J}| - 1) \sum_{x_m \in W_i^{H,J}} ( x_m - \hat{\mu}_i )^2

where \hat{\mu}_i is the mean of W_i^{H,J} defined above.  The standard
deviation \hat{\sigma}_i is the square root of the variance.

 -- Function: int gsl_movstat_variance (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace *w)

     This function computes the moving window variance of the input
     vector *note x: 850, storing the output in *note y: 850.  The
     parameter *note endtype: 850. specifies how windows near the ends
     of the input should be handled.  It is allowed to have *note x:
     850. = *note y: 850. for an in-place moving variance.

 -- Function: int gsl_movstat_sd (const gsl_movstat_end_t endtype, const
          gsl_vector *x, gsl_vector *y, gsl_movstat_workspace *w)

     This function computes the moving window standard deviation of the
     input vector *note x: 851, storing the output in *note y: 851.  The
     parameter *note endtype: 851. specifies how windows near the ends
     of the input should be handled.  It is allowed to have *note x:
     851. = *note y: 851. for an in-place moving standard deviation.


File: gsl-ref.info,  Node: Moving Minimum and Maximum,  Next: Moving Sum,  Prev: Moving Variance and Standard Deviation,  Up: Moving Window Statistics

23.6 Moving Minimum and Maximum
===============================

The moving minimum/maximum calculates the minimum and maximum values of
each window W_i^{H,J}.

     y_i^{min} = \min W_i^{H,J}
     y_i^{max} = \max W_i^{H,J}

 -- Function: int gsl_movstat_min (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace *w)

     This function computes the moving minimum of the input vector *note
     x: 853, storing the result in *note y: 853.  The parameter *note
     endtype: 853. specifies how windows near the ends of the input
     should be handled.  It is allowed to have *note x: 853. = *note y:
     853. for an in-place moving minimum.

 -- Function: int gsl_movstat_max (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace *w)

     This function computes the moving maximum of the input vector *note
     x: 854, storing the result in *note y: 854.  The parameter *note
     endtype: 854. specifies how windows near the ends of the input
     should be handled.  It is allowed to have *note x: 854. = *note y:
     854. for an in-place moving maximum.

 -- Function: int gsl_movstat_minmax (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *y_min, gsl_vector *y_max,
          gsl_movstat_workspace *w)

     This function computes the moving minimum and maximum of the input
     vector *note x: 855, storing the window minimums in *note y_min:
     855. and the window maximums in *note y_max: 855.  The parameter
     *note endtype: 855. specifies how windows near the ends of the
     input should be handled.


File: gsl-ref.info,  Node: Moving Sum,  Next: Moving Median,  Prev: Moving Minimum and Maximum,  Up: Moving Window Statistics

23.7 Moving Sum
===============

The moving window sum calculates the sum of the values of each window
W_i^{H,J}.

     y_i = \sum_{x_m \in W_i^{H,J}} x_m

 -- Function: int gsl_movstat_sum (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace *w)

     This function computes the moving window sum of the input vector
     *note x: 857, storing the output in *note y: 857.  The parameter
     *note endtype: 857. specifies how windows near the ends of the
     input should be handled.  It is allowed to have *note x: 857. =
     *note y: 857. for an in-place moving sum.


File: gsl-ref.info,  Node: Moving Median,  Next: Robust Scale Estimation,  Prev: Moving Sum,  Up: Moving Window Statistics

23.8 Moving Median
==================

The moving median calculates the median of the window W_i^{H,J} for each
sample x_i:

     y_i = median(W_i^{H,J})

 -- Function: int gsl_movstat_median (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *y, gsl_movstat_workspace *w)

     This function computes the moving median of the input vector *note
     x: 859, storing the output in *note y: 859.  The parameter *note
     endtype: 859. specifies how windows near the ends of the input
     should be handled.  It is allowed for *note x: 859. = *note y: 859.
     for an in-place moving window median.


File: gsl-ref.info,  Node: Robust Scale Estimation,  Next: User-defined Moving Statistics,  Prev: Moving Median,  Up: Moving Window Statistics

23.9 Robust Scale Estimation
============================

A common problem in statistics is to quantify the dispersion (also known
as the variability, scatter, and spread) of a set of data.  Often this
is done by calculating the variance or standard deviation.  However
these statistics are strongly influenced by outliers, and can often
provide erroneous results when even a small number of outliers are
present.

Several useful statistics have emerged to provide robust estimates of
scale which are not as susceptible to data outliers.  A few of these
statistical scale estimators are described below.

* Menu:

* Moving MAD::
* Moving QQR::
* Moving S_n::
* Moving Q_n::


File: gsl-ref.info,  Node: Moving MAD,  Next: Moving QQR,  Up: Robust Scale Estimation

23.9.1 Moving MAD
-----------------

The median absolute deviation (MAD) for the window W_i^{H,J} is defined
to be the median of the absolute deviations from the window’s median:

     MAD_i = 1.4826 * median[ |W_i^{H,J} - median(W_i^{H,J})| ]

The factor of 1.4826 makes the MAD an unbiased estimator of the standard
deviation for Gaussian data.  The MAD has an efficiency of 37%.  See
*note here: 815. for more information.

 -- Function: int gsl_movstat_mad0 (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *xmedian, gsl_vector *xmad,
          gsl_movstat_workspace *w)
 -- Function: int gsl_movstat_mad (const gsl_movstat_end_t endtype,
          const gsl_vector *x, gsl_vector *xmedian, gsl_vector *xmad,
          gsl_movstat_workspace *w)

     These functions compute the moving MAD of the input vector *note x:
     85d. and store the result in *note xmad: 85d.  The medians of each
     window W_i^{H,J} are stored in *note xmedian: 85d. on output.  The
     inputs *note x: 85d, *note xmedian: 85d, and *note xmad: 85d. must
     all be the same length.  The parameter *note endtype: 85d.
     specifies how windows near the ends of the input should be handled.
     The function ‘mad0’ does not include the scale factor of 1.4826,
     while the function ‘mad’ does include this factor.


File: gsl-ref.info,  Node: Moving QQR,  Next: Moving S_n,  Prev: Moving MAD,  Up: Robust Scale Estimation

23.9.2 Moving QQR
-----------------

The q-quantile range (QQR) is the difference between the (1-q) and q
quantiles of a set of data,

     QQR = Q_{1-q} - Q_q

The case q = 0.25 corresponds to the well-known `interquartile range
(IQR)', which is the difference between the 75th and 25th percentiles of
a set of data.  The QQR is a `trimmed estimator', the main idea being to
discard the largest and smallest values in a data window and compute a
scale estimate from the remaining middle values.  In the case of the
IQR, the largest and smallest 25% of the data are discarded and the
scale is estimated from the remaining (middle) 50%.

 -- Function: int gsl_movstat_qqr (const gsl_movstat_end_t endtype,
          const gsl_vector *x, const double q, gsl_vector *xqqr,
          gsl_movstat_workspace *w)

     This function computes the moving QQR of the input vector *note x:
     85f. and stores the q-quantile ranges of each window W_i^{H,J} in
     *note xqqr: 85f.  The quantile parameter *note q: 85f. must be
     between 0 and 0.5.  The input q = 0.25 corresponds to the IQR. The
     inputs *note x: 85f. and *note xqqr: 85f. must be the same length.
     The parameter *note endtype: 85f. specifies how windows near the
     ends of the input should be handled.


File: gsl-ref.info,  Node: Moving S_n,  Next: Moving Q_n,  Prev: Moving QQR,  Up: Robust Scale Estimation

23.9.3 Moving S_n
-----------------

The S_n statistic proposed by Croux and Rousseeuw is based on pairwise
differences between all samples in the window.  It has an efficiency of
58%, significantly higher than the MAD. See *note here: 819. for more
information.

 -- Function: int gsl_movstat_Sn (const gsl_movstat_end_t endtype, const
          gsl_vector *x, gsl_vector *xscale, gsl_movstat_workspace *w)

     This function computes the moving S_n of the input vector *note x:
     861. and stores the output in *note xscale: 861.  The inputs *note
     x: 861. and *note xscale: 861. must be the same length.  The
     parameter *note endtype: 861. specifies how windows near the ends
     of the input should be handled.  It is allowed for *note x: 861. =
     *note xscale: 861. for an in-place moving window S_n.


File: gsl-ref.info,  Node: Moving Q_n,  Prev: Moving S_n,  Up: Robust Scale Estimation

23.9.4 Moving Q_n
-----------------

The Q_n statistic proposed by Croux and Rousseeuw is loosely based on
the Hodges-Lehmann location estimator.  It has a relatively high
efficiency of 82%.  See *note here: 81d. for more information.

 -- Function: int gsl_movstat_Qn (const gsl_movstat_end_t endtype, const
          gsl_vector *x, gsl_vector *xscale, gsl_movstat_workspace *w)

     This function computes the moving Q_n of the input vector *note x:
     863. and stores the output in *note xscale: 863.  The inputs *note
     x: 863. and *note xscale: 863. must be the same length.  The
     parameter *note endtype: 863. specifies how windows near the ends
     of the input should be handled.  It is allowed for *note x: 863. =
     *note xscale: 863. for an in-place moving window Q_n.


File: gsl-ref.info,  Node: User-defined Moving Statistics,  Next: Accumulators,  Prev: Robust Scale Estimation,  Up: Moving Window Statistics

23.10 User-defined Moving Statistics
====================================

GSL offers an interface for users to define their own moving window
statistics functions, without needing to implement the edge-handling and
accumulator machinery.  This can be done by explicitly constructing the
windows W_i^{H,J} for a given input signal (*note gsl_movstat_fill():
865.), or by calculating a user-defined function for each window
automatically.  In order to apply a user-defined function to each
window, users must define a variable of type *note gsl_movstat_function:
866. to pass into *note gsl_movstat_apply(): 867.  This structure is
defined as follows.

 -- Type: gsl_movstat_function

     Structure specifying user-defined moving window statistical
     function:

          typedef struct
          {
            double (* function) (const size_t n, double x[], void * params);
            void * params;
          } gsl_movstat_function;

     This structure contains a pointer to the user-defined function as
     well as possible parameters to pass to the function.

      -- Member: double (*function) (const size_t n, double x[], void
               *params)

          This function returns the user-defined statistic of the array
          ‘x’ of length ‘n’.  User-specified parameters are passed in
          via *note params: 869.  It is allowed to modify the array ‘x’.

      -- Member: void *params

          User-specified parameters to be passed into the function.

 -- Function: int gsl_movstat_apply (const gsl_movstat_end_t endtype,
          const gsl_movstat_function *F, const gsl_vector *x, gsl_vector
          *y, gsl_movstat_workspace *w)

     This function applies the user-defined moving window statistic
     specified in *note F: 867. to the input vector *note x: 867,
     storing the output in *note y: 867.  The parameter *note endtype:
     867. specifies how windows near the ends of the input should be
     handled.  It is allowed for *note x: 867. = *note y: 867. for an
     in-place moving window calculation.

 -- Function: size_t gsl_movstat_fill (const gsl_movstat_end_t endtype,
          const gsl_vector *x, const size_t idx, const size_t H, const
          size_t J, double *window)

     This function explicitly constructs the sliding window for the
     input vector *note x: 865. which is centered on the sample *note
     idx: 865.  On output, the array *note window: 865. will contain
     W_{idx}^{H,J}.  The number of samples to the left and right of the
     sample *note idx: 865. are specified by *note H: 865. and *note J:
     865. respectively.  The parameter *note endtype: 865. specifies how
     windows near the ends of the input should be handled.  The function
     returns the size of the window.


File: gsl-ref.info,  Node: Accumulators,  Next: Examples<17>,  Prev: User-defined Moving Statistics,  Up: Moving Window Statistics

23.11 Accumulators
==================

Many of the algorithms of this chapter are based on an accumulator
design, which process the input vector one sample at a time, updating
calculations of the desired statistic for the current window.  Each
accumulator is stored in the following structure:

 -- Type: gsl_movstat_accum

     Structure specifying accumulator for moving window statistics:

          typedef struct
          {
            size_t (* size) (const size_t n);
            int (* init) (const size_t n, void * vstate);
            int (* insert) (const double x, void * vstate);
            int (* delete) (void * vstate);
            int (* get) (void * params, double * result, const void * vstate);
          } gsl_movstat_accum;

     The structure contains function pointers responsible for performing
     different tasks for the accumulator.

      -- Member: size_t (*size) (const size_t n)

          This function returns the size of the workspace (in bytes)
          needed by the accumulator for a moving window of length ‘n’.

      -- Member: int (*init) (const size_t n, void *vstate)

          This function initializes the workspace ‘vstate’ for a moving
          window of length ‘n’.

      -- Member: int (*insert) (const double x, void *vstate)

          This function inserts a single sample ‘x’ into the
          accumulator, updating internal calculations of the desired
          statistic.  If the accumulator is full (i.e.  n samples have
          already been inserted), then the oldest sample is deleted from
          the accumulator.

      -- Member: int (*delete) (void *vstate)

          This function deletes the oldest sample from the accumulator,
          updating internal calculations of the desired statistic.

      -- Member: int (*get) (void *params, double *result, const void
               *vstate)

          This function stores the desired statistic for the current
          window in ‘result’.  The input ‘params’ specifies optional
          parameters for calculating the statistic.

The following accumulators of type *note gsl_movstat_accum: 86b. are
defined by GSL to perform moving window statistics calculations.

 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_min
 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_max
 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_minmax

     These accumulators calculate moving window minimum/maximums
     efficiently, using the algorithm of D. Lemire.

 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_mean
 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_sd
 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_variance

     These accumulators calculate the moving window mean, standard
     deviation, and variance, using the algorithm of B. P. Welford.

 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_median

     This accumulator calculates the moving window median using the
     min/max heap algorithm of Härdle and Steiger.

 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_Sn
 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_Qn

     These accumulators calculate the moving window S_n and Q_n
     statistics developed by Croux and Rousseeuw.

 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_sum

     This accumulator calculates the moving window sum.

 -- Variable: *note gsl_movstat_accum: 86b. *gsl_movstat_accum_qqr

     This accumulator calculates the moving window q-quantile range.


File: gsl-ref.info,  Node: Examples<17>,  Next: References and Further Reading<17>,  Prev: Accumulators,  Up: Moving Window Statistics

23.12 Examples
==============

* Menu:

* Example 1::
* Example 2; Robust Scale: Example 2 Robust Scale.
* Example 3; User-defined Moving Window: Example 3 User-defined Moving Window.


File: gsl-ref.info,  Node: Example 1,  Next: Example 2 Robust Scale,  Up: Examples<17>

23.12.1 Example 1
-----------------

The following example program computes the moving mean, minimum and
maximum of a noisy sinusoid signal of length N = 500 with a symmetric
moving window of size K = 11.

[gsl-ref-figures/movstat1]
Figure: Original signal time series (gray) with moving mean (green),
moving minimum (blue), and moving maximum (orange).

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 500;                      /* length of time series */
       const size_t K = 11;                       /* window size */
       gsl_movstat_workspace * w = gsl_movstat_alloc(K);
       gsl_vector *x = gsl_vector_alloc(N);
       gsl_vector *xmean = gsl_vector_alloc(N);
       gsl_vector *xmin = gsl_vector_alloc(N);
       gsl_vector *xmax = gsl_vector_alloc(N);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t i;

       for (i = 0; i < N; ++i)
         {
           double xi = cos(4.0 * M_PI * i / (double) N);
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(x, i, xi + ei);
         }

       /* compute moving statistics */
       gsl_movstat_mean(GSL_MOVSTAT_END_PADVALUE, x, xmean, w);
       gsl_movstat_minmax(GSL_MOVSTAT_END_PADVALUE, x, xmin, xmax, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           printf("%zu %f %f %f %f\n",
                  i,
                  gsl_vector_get(x, i),
                  gsl_vector_get(xmean, i),
                  gsl_vector_get(xmin, i),
                  gsl_vector_get(xmax, i));
         }

       gsl_vector_free(x);
       gsl_vector_free(xmean);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }


File: gsl-ref.info,  Node: Example 2 Robust Scale,  Next: Example 3 User-defined Moving Window,  Prev: Example 1,  Up: Examples<17>

23.12.2 Example 2: Robust Scale
-------------------------------

The following example program analyzes a time series of length N = 1000
composed of Gaussian random variates with zero mean whose standard
deviation changes in a piecewise constant fashion as shown in the table
below.

Sample Range     \sigma
                 
------------------------------------
                 
1-200            1.0
                 
                 
201-450          5.0
                 
                 
451-600          1.0
                 
                 
601-850          3.0
                 
                 
851-1000         5.0
                 

Additionally, about 1% of the samples are perturbed to represent
outliers by adding \pm 15 to the random Gaussian variate.  The program
calculates the moving statistics MAD, IQR, S_n, Q_n, and the standard
deviation using a symmetric moving window of length K = 41.  The results
are shown in Fig.  %s.

[gsl-ref-figures/movstat2]
Figure: Top: time series of piecewise constant variance.  Bottom: scale
estimates using a moving window; the true sigma value is in light blue,
MAD in green, IQR in red, S_n in yellow, and Q_n in dark blue.  The
moving standard deviation is shown in gray.

The robust statistics follow the true standard deviation piecewise
changes well, without being influenced by the outliers.  The moving
standard deviation (gray curve) is heavily influenced by the presence of
the outliers.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                                 /* length of time series */
       const double sigma[] = { 1.0, 5.0, 1.0, 3.0, 5.0 };    /* variances */
       const size_t N_sigma[] = { 200, 450, 600, 850, 1000 }; /* samples where variance changes */
       const size_t K = 41;                                   /* window size */
       gsl_vector *x = gsl_vector_alloc(N);
       gsl_vector *xmedian = gsl_vector_alloc(N);
       gsl_vector *xmad = gsl_vector_alloc(N);
       gsl_vector *xiqr = gsl_vector_alloc(N);
       gsl_vector *xSn = gsl_vector_alloc(N);
       gsl_vector *xQn = gsl_vector_alloc(N);
       gsl_vector *xsd = gsl_vector_alloc(N);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_movstat_workspace * w = gsl_movstat_alloc(K);
       size_t idx = 0;
       size_t i;

       for (i = 0; i < N; ++i)
         {
           double gi = gsl_ran_gaussian(r, sigma[idx]);
           double u = gsl_rng_uniform(r);
           double outlier = (u < 0.01) ? 15.0*GSL_SIGN(gi) : 0.0;
           double xi = gi + outlier;

           gsl_vector_set(x, i, xi);

           if (i == N_sigma[idx] - 1)
             ++idx;
         }

       /* compute moving statistics */
       gsl_movstat_mad(GSL_MOVSTAT_END_TRUNCATE, x, xmedian, xmad, w);
       gsl_movstat_qqr(GSL_MOVSTAT_END_TRUNCATE, x, 0.25, xiqr, w);
       gsl_movstat_Sn(GSL_MOVSTAT_END_TRUNCATE, x, xSn, w);
       gsl_movstat_Qn(GSL_MOVSTAT_END_TRUNCATE, x, xQn, w);
       gsl_movstat_sd(GSL_MOVSTAT_END_TRUNCATE, x, xsd, w);

       /* scale IQR by factor to approximate standard deviation */
       gsl_vector_scale(xiqr, 0.7413);

       /* print results */
       idx = 0;
       for (i = 0; i < N; ++i)
         {
           printf("%zu %f %f %f %f %f %f %f\n",
                  i,
                  gsl_vector_get(x, i),
                  sigma[idx],
                  gsl_vector_get(xmad, i),
                  gsl_vector_get(xiqr, i),
                  gsl_vector_get(xSn, i),
                  gsl_vector_get(xQn, i),
                  gsl_vector_get(xsd, i));

           if (i == N_sigma[idx] - 1)
             ++idx;
         }

       gsl_vector_free(x);
       gsl_vector_free(xmedian);
       gsl_vector_free(xmad);
       gsl_vector_free(xiqr);
       gsl_vector_free(xSn);
       gsl_vector_free(xQn);
       gsl_vector_free(xsd);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }


File: gsl-ref.info,  Node: Example 3 User-defined Moving Window,  Prev: Example 2 Robust Scale,  Up: Examples<17>

23.12.3 Example 3: User-defined Moving Window
---------------------------------------------

This example program illustrates how a user can define their own moving
window function to apply to an input vector.  It constructs a random
noisy time series of length N = 1000 with some outliers added.  Then it
applies a moving window trimmed mean to the time series with trim
parameter \alpha = 0.1.  The length of the moving window is K = 11, so
the smallest and largest sample of each window is discarded prior to
computing the mean.  The results are shown in Fig.  %s.

[gsl-ref-figures/movstat3]
Figure: Noisy time series data (black) with moving window trimmed mean
(red)

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_statistics.h>

     double
     func(const size_t n, double x[], void * params)
     {
       const double alpha = *(double *) params;

       gsl_sort(x, 1, n);

       return gsl_stats_trmean_from_sorted_data(alpha, x, 1, n);
     }

     int
     main(void)
     {
       const size_t N = 1000;                /* length of time series */
       const size_t K = 11;                  /* window size */
       double alpha = 0.1;                   /* trimmed mean parameter */
       gsl_vector *x = gsl_vector_alloc(N);  /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);  /* filtered output vector for alpha1 */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_movstat_workspace *w = gsl_movstat_alloc(K);
       gsl_movstat_function F;
       size_t i;
       double sum = 0.0;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ui = gsl_ran_gaussian(r, 1.0);
           double outlier = (gsl_rng_uniform(r) < 0.01) ? 10.0*GSL_SIGN(ui) : 0.0;
           sum += ui;
           gsl_vector_set(x, i, sum + outlier);
         }

       /* apply moving window function */
       F.function = func;
       F.params = &alpha;
       gsl_movstat_apply(GSL_MOVSTAT_END_PADVALUE, &F, x, y, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);

           printf("%f %f\n", xi, yi);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<17>,  Prev: Examples<17>,  Up: Moving Window Statistics

23.13 References and Further Reading
====================================

The following publications are relevant to the algorithms described in
this chapter,

   * W.Hardle and W. Steiger, `Optimal Median Smoothing', Appl.
     Statist., 44 (2), 1995.

   * D. Lemire, `Streaming Maximum-Minimum Filter Using No More than
     Three Comparisons per Element', Nordic Journal of Computing, 13
     (4), 2006 (‘https://arxiv.org/abs/cs/0610046’).

   * B. P. Welford, `Note on a method for calculating corrected sums of
     squares and products', Technometrics, 4 (3), 1962.


File: gsl-ref.info,  Node: Digital Filtering,  Next: Histograms,  Prev: Moving Window Statistics,  Up: Top

24 Digital Filtering
********************

* Menu:

* Introduction: Introduction<5>.
* Handling Endpoints: Handling Endpoints<2>.
* Linear Digital Filters::
* Nonlinear Digital Filters::
* Examples: Examples<18>.
* References and Further Reading: References and Further Reading<18>.


File: gsl-ref.info,  Node: Introduction<5>,  Next: Handling Endpoints<2>,  Up: Digital Filtering

24.1 Introduction
=================

The filters discussed in this chapter are based on the following moving
data window which is centered on i-th sample:

     W_i^H = { x_{i-H}, ..., x_i, ..., x_{i+H} }

Here, H is a non-negative integer called the `window half-length', which
represents the number of samples before and after sample i.  The total
window length is K = 2 H + 1.


File: gsl-ref.info,  Node: Handling Endpoints<2>,  Next: Linear Digital Filters,  Prev: Introduction<5>,  Up: Digital Filtering

24.2 Handling Endpoints
=======================

When processing samples near the ends of the input signal, there will
not be enough samples to fill the window W_i^H defined above.  Therefore
the user must specify how to construct the windows near the end points.
This is done by passing an input argument of type *note
gsl_filter_end_t: 888.:

 -- Type: gsl_filter_end_t

     This data type specifies how to construct windows near end points
     and can be selected from the following choices:

      -- Macro: GSL_FILTER_END_PADZERO

          With this option, a full window of length K will be
          constructed by inserting zeros into the window near the signal
          end points.  Effectively, the input signal is modified to

               x~ = { 0, ..., 0, x_1, x_2, ..., x_{n-1}, x_n, 0, ..., 0 }

          to ensure a well-defined window for all x_i.

      -- Macro: GSL_FILTER_END_PADVALUE

          With this option, a full window of length K will be
          constructed by padding the window with the first and last
          sample in the input signal.  Effectively, the input signal is
          modified to

               x~ = { x_1, ..., x_1, x_1, x_2, ..., x_{n-1}, x_n, x_n, ..., x_n }

      -- Macro: GSL_FILTER_END_TRUNCATE

          With this option, no padding is performed, and the windows are
          simply truncated as the end points are approached.


File: gsl-ref.info,  Node: Linear Digital Filters,  Next: Nonlinear Digital Filters,  Prev: Handling Endpoints<2>,  Up: Digital Filtering

24.3 Linear Digital Filters
===========================

* Menu:

* Gaussian Filter::


File: gsl-ref.info,  Node: Gaussian Filter,  Up: Linear Digital Filters

24.3.1 Gaussian Filter
----------------------

The Gaussian filter convolves the input signal with a Gaussian kernel or
window.  This filter is often used as a smoothing or noise reduction
filter.  The Gaussian kernel is defined by

     G(k) = e^{-1/2 ( \alpha k/((K-1)/2) )^2} = e^{-k^2/2\sigma^2}

for -(K-1)/2 \le k \le (K-1)/2, and K is the size of the kernel.  The
parameter \alpha specifies the number of standard deviations \sigma
desired in the kernel.  So for example setting \alpha = 3 would define a
Gaussian window of length K which spans \pm 3 \sigma.  It is often more
convenient to specify the parameter \alpha rather than the standard
deviation \sigma when constructing the kernel, since a fixed value of
\alpha would correspond to the same shape of Gaussian regardless of the
size K. The appropriate value of the standard deviation depends on K and
is related to \alpha as

     \sigma = (K - 1)/(2 \alpha)

The routines below accept \alpha as an input argument instead of \sigma.

The Gaussian filter offers a convenient way of differentiating and
smoothing an input signal in a single pass.  Using the derivative
property of a convolution,

     d/dt ( G * x ) = dG/dt * x

the input signal x(t) can be smoothed and differentiated at the same
time by convolution with a derivative Gaussian kernel, which can be
readily computed from the analytic expression above.  The same principle
applies to higher order derivatives.

 -- Function: gsl_filter_gaussian_workspace *gsl_filter_gaussian_alloc
          (const size_t K)

     This function initializes a workspace for Gaussian filtering using
     a kernel of size *note K: 88e.  Here, H = K / 2.  If K is even, it
     is rounded up to the next odd integer to ensure a symmetric window.
     The size of the workspace is O(K).

 -- Function: void gsl_filter_gaussian_free
          (gsl_filter_gaussian_workspace *w)

     This function frees the memory associated with *note w: 88f.

 -- Function: int gsl_filter_gaussian (const gsl_filter_end_t endtype,
          const double alpha, const size_t order, const gsl_vector *x,
          gsl_vector *y, gsl_filter_gaussian_workspace *w)

     This function applies a Gaussian filter parameterized by *note
     alpha: 890. to the input vector *note x: 890, storing the output in
     *note y: 890.  The derivative order is specified by *note order:
     890, with ‘0’ corresponding to a Gaussian, ‘1’ corresponding to a
     first derivative Gaussian, and so on.  The parameter *note endtype:
     890. specifies how the signal end points are handled.  It is
     allowed for *note x: 890. = *note y: 890. for an in-place filter.

 -- Function: int gsl_filter_gaussian_kernel (const double alpha, const
          size_t order, const int normalize, gsl_vector *kernel)

     This function constructs a Gaussian kernel parameterized by *note
     alpha: 891. and stores the output in *note kernel: 891.  The
     parameter *note order: 891. specifies the derivative order, with
     ‘0’ corresponding to a Gaussian, ‘1’ corresponding to a first
     derivative Gaussian, and so on.  If *note normalize: 891. is set to
     ‘1’, then the kernel will be normalized to sum to one on output.
     If *note normalize: 891. is set to ‘0’, no normalization is
     performed.


File: gsl-ref.info,  Node: Nonlinear Digital Filters,  Next: Examples<18>,  Prev: Linear Digital Filters,  Up: Digital Filtering

24.4 Nonlinear Digital Filters
==============================

The nonlinear digital filters described below are based on the window
median, which is given by

     m_i = median { W_i^H } = median { x_{i-H}, ..., x_i, ..., x_{i+H} }

The median is considered robust to local outliers, unlike the mean.
Median filters can preserve sharp edges while at the same removing
signal noise, and are used in a wide range of applications.

* Menu:

* Standard Median Filter::
* Recursive Median Filter::
* Impulse Detection Filter::


File: gsl-ref.info,  Node: Standard Median Filter,  Next: Recursive Median Filter,  Up: Nonlinear Digital Filters

24.4.1 Standard Median Filter
-----------------------------

The `standard median filter' (SMF) simply replaces the sample x_i by the
median m_i of the window W_i^H: This filter has one tuning parameter
given by H. The standard median filter is considered highly resistant to
local outliers and local noise in the data sequence \{x_i\}.

 -- Function: gsl_filter_median_workspace *gsl_filter_median_alloc
          (const size_t K)

     This function initializes a workspace for standard median filtering
     using a symmetric centered moving window of size *note K: 894.
     Here, H = K / 2.  If K is even, it is rounded up to the next odd
     integer to ensure a symmetric window.  The size of the workspace is
     O(7K).

 -- Function: void gsl_filter_median_free (gsl_filter_median_workspace
          *w)

     This function frees the memory associated with *note w: 895.

 -- Function: int gsl_filter_median (const gsl_filter_end_t endtype,
          const gsl_vector *x, gsl_vector *y,
          gsl_filter_median_workspace *w)

     This function applies a standard median filter to the input *note
     x: 896, storing the output in *note y: 896.  The parameter *note
     endtype: 896. specifies how the signal end points are handled.  It
     is allowed to have *note x: 896. = *note y: 896. for an in-place
     filter.


File: gsl-ref.info,  Node: Recursive Median Filter,  Next: Impulse Detection Filter,  Prev: Standard Median Filter,  Up: Nonlinear Digital Filters

24.4.2 Recursive Median Filter
------------------------------

The `recursive median filter' (RMF) is a modification of the SMF to
include previous filter outputs in the window before computing the
median.  The filter’s response is

     y_i = median ( y_{i-H}, ..., y_{i-1}, x_i, x_{i+1}, ..., x_{i+H} )

Sometimes, the SMF must be applied several times in a row to achieve
adequate smoothing (i.e.  a cascade filter).  The RMF, on the other
hand, converges to a `root sequence' in one pass, and can sometimes
provide a smoother result than several passes of the SMF. A root
sequence is an input which is left unchanged by the filter.  So there is
no need to apply a recursive median filter twice to an input vector.

 -- Function: gsl_filter_rmedian_workspace *gsl_filter_rmedian_alloc
          (const size_t K)

     This function initializes a workspace for recursive median
     filtering using a symmetric centered moving window of size *note K:
     898.  Here, H = K / 2.  If K is even, it is rounded up to the next
     odd integer to ensure a symmetric window.  The size of the
     workspace is O(K).

 -- Function: void gsl_filter_rmedian_free (gsl_filter_rmedian_workspace
          *w)

     This function frees the memory associated with *note w: 899.

 -- Function: int gsl_filter_rmedian (const gsl_filter_end_t endtype,
          const gsl_vector *x, gsl_vector *y,
          gsl_filter_rmedian_workspace *w)

     This function applies a recursive median filter to the input *note
     x: 89a, storing the output in *note y: 89a.  The parameter *note
     endtype: 89a. specifies how the signal end points are handled.  It
     is allowed to have *note x: 89a. = *note y: 89a. for an in-place
     filter.


File: gsl-ref.info,  Node: Impulse Detection Filter,  Prev: Recursive Median Filter,  Up: Nonlinear Digital Filters

24.4.3 Impulse Detection Filter
-------------------------------

Impulsive noise is characterized by short sequences of data points
distinct from those in the surrounding neighborhood.  This section
describes a powerful class of filters, also known as `impulse rejection
filters' and `decision-based filters', designed to detect and remove
such outliers from data.  The filter’s response is given by

     y_i = { x_i, |x_i - m_i| <= t * S_i
           { m_i, |x_i - m_i| > t * S_i

where m_i is the median value of the window W_i^H, S_i is a robust
estimate of the scatter or dispersion for the window W_i^H, and t is a
tuning parameter specifying the number of scale factors needed to
determine that a point is an outlier.  The main idea is that the median
m_i will be unaffected by a small number of outliers in the window, and
so a given sample x_i is tested to determine how far away it is from the
median in terms of the local scale estimate S_i.  Samples which are more
than t scale estimates away from the median are labeled as outliers and
replaced by the window median m_i.  Samples which are less than t scale
estimates from the median are left unchanged by the filter.

Note that when t = 0, the impulse detection filter is equivalent to the
standard median filter.  When t \rightarrow \infty, it becomes the
identity filter.  This means the impulse detection filter can be viewed
as a “less aggressive” version of the standard median filter, becoming
less aggressive as t is increased.  Note that this filter modifies only
samples identified as outliers, while the standard median filter changes
all samples to the local median, regardless of whether they are
outliers.  This fact, plus the additional flexibility offered by the
additional tuning parameter t can make the impulse detection filter a
better choice for some applications.

It is important to have a robust and accurate scale estimate S_i in
order to detect impulse outliers even in the presence of noise.  The
window standard deviation is not typically a good choice, as it can be
significantly perturbed by the presence of even one outlier.  GSL offers
the following choices (specified by a parameter of type *note
gsl_filter_scale_t: 89c.) for computing the scale estimate S_i, all of
which are robust to the presence of impulse outliers.

 -- Type: gsl_filter_scale_t

     This type specifies how the scale estimate S_i of the window W_i^H
     is calculated.

      -- Macro: GSL_FILTER_SCALE_MAD

          This option specifies the median absolute deviation (MAD)
          scale estimate, defined by

               S_i = 1.4826 median { | W_i^H - m_i | }

          This choice of scale estimate is also known as the `Hampel
          filter' in the statistical literature.  See *note here: 815.
          for more information.

      -- Macro: GSL_FILTER_SCALE_IQR

          This option specifies the interquartile range (IQR) scale
          estimate, defined as the difference between the 75th and 25th
          percentiles of the window W_i^H,

               S_i = 0.7413 ( Q_{0.75} - Q_{0.25} )

          where Q_p is the p-quantile of the window W_i^H. The idea is
          to throw away the largest and smallest 25% of the window
          samples (where the outliers would be), and estimate a scale
          from the middle 50%.  The factor 0.7413 provides an unbiased
          estimate of the standard deviation for Gaussian data.

      -- Macro: GSL_FILTER_SCALE_SN

          This option specifies the so-called S_n statistic proposed by
          Croux and Rousseeuw.  See *note here: 819. for more
          information.

      -- Macro: GSL_FILTER_SCALE_QN

          This option specifies the so-called Q_n statistic proposed by
          Croux and Rousseeuw.  See *note here: 81d. for more
          information.

     Warning: While the scale estimates defined above are much less
     sensitive to outliers than the standard deviation, they can suffer
     from an effect called `implosion'.  The standard deviation of a
     window W_i^H will be zero if and only if all samples in the window
     are equal.  However, it is possible for the MAD of a window to be
     zero even if all the samples in the window are not equal.  For
     example, if K/2 + 1 or more of the K samples in the window are
     equal to some value x^{*}, then the window median will be equal to
     x^{*}.  Consequently, at least K/2 + 1 of the absolute deviations
     |x_j - x^{*}| will be zero, and so the MAD will be zero.  In such a
     case, the Hampel filter will act like the standard median filter
     regardless of the value of t.  Caution should also be exercised if
     dividing by S_i.

 -- Function: gsl_filter_impulse_workspace *gsl_filter_impulse_alloc
          (const size_t K)

     This function initializes a workspace for impulse detection
     filtering using a symmetric moving window of size *note K: 8a1.
     Here, H = K / 2.  If K is even, it is rounded up to the next odd
     integer to ensure a symmetric window.  The size of the workspace is
     O(6K).

 -- Function: void gsl_filter_impulse_free (gsl_filter_impulse_workspace
          *w)

     This function frees the memory associated with *note w: 8a2.

 -- Function: int gsl_filter_impulse (const gsl_filter_end_t endtype,
          const gsl_filter_scale_t scale_type, const double t, const
          gsl_vector *x, gsl_vector *y, gsl_vector *xmedian, gsl_vector
          *xsigma, size_t *noutlier, gsl_vector_int *ioutlier,
          gsl_filter_impulse_workspace *w)

     These functions apply an impulse detection filter to the input
     vector *note x: 8a3, storing the filtered output in *note y: 8a3.
     The tuning parameter t is provided in *note t: 8a3.  The window
     medians m_i are stored in *note xmedian: 8a3. and the S_i are
     stored in *note xsigma: 8a3. on output.  The number of outliers
     detected is stored in *note noutlier: 8a3. on output, while the
     locations of flagged outliers are stored in the boolean array *note
     ioutlier: 8a3.  The input *note ioutlier: 8a3. may be ‘NULL’ if not
     desired.  It is allowed to have *note x: 8a3. = *note y: 8a3. for
     an in-place filter.


File: gsl-ref.info,  Node: Examples<18>,  Next: References and Further Reading<18>,  Prev: Nonlinear Digital Filters,  Up: Digital Filtering

24.5 Examples
=============

* Menu:

* Gaussian Example 1::
* Gaussian Example 2::
* Square Wave Signal Example::
* Impulse Detection Example::


File: gsl-ref.info,  Node: Gaussian Example 1,  Next: Gaussian Example 2,  Up: Examples<18>

24.5.1 Gaussian Example 1
-------------------------

This example program illustrates the Gaussian filter applied to
smoothing a time series of length N = 500 with a kernel size of K = 51.
Three filters are applied with parameters \alpha = 0.5, 3, 10.  The
results are shown in Fig.  %s.

[gsl-ref-figures/gaussfilt]
Figure: Top panel: Gaussian kernels (unnormalized) for \alpha = 0.5, 3,
10.  Bottom panel: Time series (gray) with Gaussian filter output for
same \alpha values.

We see that the filter corresponding to \alpha = 0.5 applies the most
smoothing, while \alpha = 10 corresponds to the least amount of
smoothing.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 500;                        /* length of time series */
       const size_t K = 51;                         /* window size */
       const double alpha[3] = { 0.5, 3.0, 10.0 };  /* alpha values */
       gsl_vector *x = gsl_vector_alloc(N);         /* input vector */
       gsl_vector *y1 = gsl_vector_alloc(N);        /* filtered output vector for alpha1 */
       gsl_vector *y2 = gsl_vector_alloc(N);        /* filtered output vector for alpha2 */
       gsl_vector *y3 = gsl_vector_alloc(N);        /* filtered output vector for alpha3 */
       gsl_vector *k1 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha1 */
       gsl_vector *k2 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha2 */
       gsl_vector *k3 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha3 */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);
       size_t i;
       double sum = 0.0;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ui = gsl_ran_gaussian(r, 1.0);
           sum += ui;
           gsl_vector_set(x, i, sum);
         }

       /* compute kernels without normalization */
       gsl_filter_gaussian_kernel(alpha[0], 0, 0, k1);
       gsl_filter_gaussian_kernel(alpha[1], 0, 0, k2);
       gsl_filter_gaussian_kernel(alpha[2], 0, 0, k3);

       /* apply filters */
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[0], 0, x, y1, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[1], 0, x, y2, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[2], 0, x, y3, gauss_p);

       /* print kernels */
       for (i = 0; i < K; ++i)
         {
           double k1i = gsl_vector_get(k1, i);
           double k2i = gsl_vector_get(k2, i);
           double k3i = gsl_vector_get(k3, i);

           printf("%e %e %e\n", k1i, k2i, k3i);
         }

       printf("\n\n");

       /* print filter results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double y1i = gsl_vector_get(y1, i);
           double y2i = gsl_vector_get(y2, i);
           double y3i = gsl_vector_get(y3, i);

           printf("%.12e %.12e %.12e %.12e\n", xi, y1i, y2i, y3i);
         }

       gsl_vector_free(x);
       gsl_vector_free(y1);
       gsl_vector_free(y2);
       gsl_vector_free(y3);
       gsl_vector_free(k1);
       gsl_vector_free(k2);
       gsl_vector_free(k3);
       gsl_rng_free(r);
       gsl_filter_gaussian_free(gauss_p);

       return 0;
     }


File: gsl-ref.info,  Node: Gaussian Example 2,  Next: Square Wave Signal Example,  Prev: Gaussian Example 1,  Up: Examples<18>

24.5.2 Gaussian Example 2
-------------------------

A common application of the Gaussian filter is to detect edges, or
sudden jumps, in a noisy input signal.  It is used both for 1D edge
detection in time series, as well as 2D edge detection in images.  Here
we will examine a noisy time series of length N = 1000 with a single
edge.  The input signal is defined as

     x(n) = e(n) + { 0,   n <= N/2
                   { 0.5, n >  N/2

where e(n) is Gaussian random noise.  The program smooths the input
signal with order 0,1, and 2 Gaussian filters of length K = 61 with
\alpha = 3.  For comparison, the program also computes finite
differences of the input signal without smoothing.  The results are
shown in Fig.  %s.

[gsl-ref-figures/gaussfilt2]
Figure: Top row: original input signal x(n) (black) with Gaussian
smoothed signal in red.  Second row: First finite differences of input
signal.  Third row: Input signal smoothed with a first order Gaussian
filter.  Fourth row: Input signal smoothed with a second order Gaussian
filter.

The finite difference approximation of the first derivative (second row)
shows the common problem with differentiating a noisy signal.  The noise
is amplified and makes it extremely difficult to detect the sharp
gradient at sample 500.  The third row shows the first order Gaussian
smoothed signal with a clear peak at the location of the edge.
Alternatively, one could examine the second order Gaussian smoothed
signal (fourth row) and look for zero crossings to determine the edge
location.

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                       /* length of time series */
       const size_t K = 61;                         /* window size */
       const double alpha = 3.0;                    /* Gaussian kernel has +/- 3 standard deviations */
       gsl_vector *x = gsl_vector_alloc(N);         /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);         /* filtered output vector */
       gsl_vector *dy = gsl_vector_alloc(N);        /* first derivative filtered vector */
       gsl_vector *d2y = gsl_vector_alloc(N);       /* second derivative filtered vector */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double xi = (i > N / 2) ? 0.5 : 0.0;
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(x, i, xi + ei);
         }

       /* apply filters */
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 0, x, y, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 1, x, dy, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 2, x, d2y, gauss_p);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           double dyi = gsl_vector_get(dy, i);
           double d2yi = gsl_vector_get(d2y, i);
           double dxi;

           /* compute finite difference of x vector */
           if (i == 0)
             dxi = gsl_vector_get(x, i + 1) - xi;
           else if (i == N - 1)
             dxi = gsl_vector_get(x, i) - gsl_vector_get(x, i - 1);
           else
             dxi = 0.5 * (gsl_vector_get(x, i + 1) - gsl_vector_get(x, i - 1));

           printf("%.12e %.12e %.12e %.12e %.12e\n",
                  xi,
                  yi,
                  dxi,
                  dyi,
                  d2yi);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_vector_free(dy);
       gsl_vector_free(d2y);
       gsl_rng_free(r);
       gsl_filter_gaussian_free(gauss_p);

       return 0;
     }


File: gsl-ref.info,  Node: Square Wave Signal Example,  Next: Impulse Detection Example,  Prev: Gaussian Example 2,  Up: Examples<18>

24.5.3 Square Wave Signal Example
---------------------------------

The following example program illustrates the median filters on a noisy
square wave signal.  Median filters are well known for preserving sharp
edges in the input signal while reducing noise.  The program constructs
a 5 Hz square wave signal with Gaussian noise added.  Then the signal is
filtered with a standard median filter and recursive median filter using
a symmetric window of length K = 7.  The results are shown in Fig.  %s.

[gsl-ref-figures/filt_edge]
Figure: Original time series is in gray.  The standard median filter
output is in green and the recursive median filter output is in red.

Both filters preserve the sharp signal edges while reducing the noise.
The recursive median filter achieves a smoother result than the standard
median filter.  The “blocky” nature of the output is characteristic of
all median filters.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                               /* length of time series */
       const size_t K = 7;                                  /* window size */
       const double f = 5.0;                                /* frequency of square wave in Hz */
       gsl_filter_median_workspace *median_p = gsl_filter_median_alloc(K);
       gsl_filter_rmedian_workspace *rmedian_p = gsl_filter_rmedian_alloc(K);
       gsl_vector *t = gsl_vector_alloc(N);                 /* time */
       gsl_vector *x = gsl_vector_alloc(N);                 /* input vector */
       gsl_vector *y_median = gsl_vector_alloc(N);          /* median filtered output */
       gsl_vector *y_rmedian = gsl_vector_alloc(N);         /* recursive median filtered output */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ti = (double) i / (N - 1.0);
           double tmp = sin(2.0 * M_PI * f * ti);
           double xi = (tmp >= 0.0) ? 1.0 : -1.0;
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(t, i, ti);
           gsl_vector_set(x, i, xi + ei);
         }

       gsl_filter_median(GSL_FILTER_END_PADVALUE, x, y_median, median_p);
       gsl_filter_rmedian(GSL_FILTER_END_PADVALUE, x, y_rmedian, rmedian_p);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double ti = gsl_vector_get(t, i);
           double xi = gsl_vector_get(x, i);
           double medi = gsl_vector_get(y_median, i);
           double rmedi = gsl_vector_get(y_rmedian, i);

           printf("%f %f %f %f\n",
                  ti,
                  xi,
                  medi,
                  rmedi);
         }

       gsl_vector_free(t);
       gsl_vector_free(x);
       gsl_vector_free(y_median);
       gsl_vector_free(y_rmedian);
       gsl_rng_free(r);
       gsl_filter_median_free(median_p);

       return 0;
     }


File: gsl-ref.info,  Node: Impulse Detection Example,  Prev: Square Wave Signal Example,  Up: Examples<18>

24.5.4 Impulse Detection Example
--------------------------------

The following example program illustrates the impulse detection filter.
First, it constructs a sinusoid signal of length N = 1000 with Gaussian
noise added.  Then, about 1% of the data are perturbed to represent
large outliers.  An impulse detecting filter is applied with a window
size K = 25 and tuning parameter t = 4, using the Q_n statistic as the
robust measure of scale.  The results are plotted in Fig.  %s.

[gsl-ref-figures/impulse]
Figure: Original time series is in blue, filter output is in green,
upper and lower intervals for detecting outliers are in red and yellow
respectively.  Detected outliers are marked with squares.

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                               /* length of time series */
       const size_t K = 25;                                 /* window size */
       const double t = 4.0;                                /* number of scale factors for outlier detection */
       gsl_vector *x = gsl_vector_alloc(N);                 /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);                 /* output (filtered) vector */
       gsl_vector *xmedian = gsl_vector_alloc(N);           /* window medians */
       gsl_vector *xsigma = gsl_vector_alloc(N);            /* window scale estimates */
       gsl_vector_int *ioutlier = gsl_vector_int_alloc(N);  /* outlier detected? */
       gsl_filter_impulse_workspace * w = gsl_filter_impulse_alloc(K);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t noutlier;
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double xi = 10.0 * sin(2.0 * M_PI * i / (double) N);
           double ei = gsl_ran_gaussian(r, 2.0);
           double u = gsl_rng_uniform(r);
           double outlier = (u < 0.01) ? 15.0*GSL_SIGN(ei) : 0.0;

           gsl_vector_set(x, i, xi + ei + outlier);
         }

       /* apply impulse detection filter */
       gsl_filter_impulse(GSL_FILTER_END_TRUNCATE, GSL_FILTER_SCALE_QN, t, x, y,
                          xmedian, xsigma, &noutlier, ioutlier, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           double xmedi = gsl_vector_get(xmedian, i);
           double xsigmai = gsl_vector_get(xsigma, i);
           int outlier = gsl_vector_int_get(ioutlier, i);

           printf("%zu %f %f %f %f %d\n",
                  i,
                  xi,
                  yi,
                  xmedi + t * xsigmai,
                  xmedi - t * xsigmai,
                  outlier);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_vector_free(xmedian);
       gsl_vector_free(xsigma);
       gsl_vector_int_free(ioutlier);
       gsl_filter_impulse_free(w);
       gsl_rng_free(r);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<18>,  Prev: Examples<18>,  Up: Digital Filtering

24.6 References and Further Reading
===================================

The following publications are relevant to the algorithms described in
this chapter,

   * F. J. Harris, `On the use of windows for harmonic analysis with the
     discrete Fourier transform', Proceedings of the IEEE, 66 (1), 1978.

   * S-J. Ko, Y-H. Lee, and A. T. Fam.  `Efficient implementation of
     one-dimensional recursive median filters', IEEE transactions on
     circuits and systems 37.11 (1990): 1447-1450.

   * R. K. Pearson and M. Gabbouj, `Nonlinear Digital Filtering with
     Python: An Introduction'.  CRC Press, 2015.


File: gsl-ref.info,  Node: Histograms,  Next: N-tuples,  Prev: Digital Filtering,  Up: Top

25 Histograms
*************

This chapter describes functions for creating histograms.  Histograms
provide a convenient way of summarizing the distribution of a set of
data.  A histogram consists of a set of `bins' which count the number of
events falling into a given range of a continuous variable x.  In GSL
the bins of a histogram contain floating-point numbers, so they can be
used to record both integer and non-integer distributions.  The bins can
use arbitrary sets of ranges (uniformly spaced bins are the default).
Both one and two-dimensional histograms are supported.

Once a histogram has been created it can also be converted into a
probability distribution function.  The library provides efficient
routines for selecting random samples from probability distributions.
This can be useful for generating simulations based on real data.

The functions are declared in the header files ‘gsl_histogram.h’ and
‘gsl_histogram2d.h’.

* Menu:

* The histogram struct::
* Histogram allocation::
* Copying Histograms::
* Updating and accessing histogram elements::
* Searching histogram ranges::
* Histogram Statistics::
* Histogram Operations::
* Reading and writing histograms::
* Resampling from histograms::
* The histogram probability distribution struct::
* Example programs for histograms::
* Two dimensional histograms::
* The 2D histogram struct::
* 2D Histogram allocation::
* Copying 2D Histograms::
* Updating and accessing 2D histogram elements::
* Searching 2D histogram ranges::
* 2D Histogram Statistics::
* 2D Histogram Operations::
* Reading and writing 2D histograms::
* Resampling from 2D histograms::
* Example programs for 2D histograms::


File: gsl-ref.info,  Node: The histogram struct,  Next: Histogram allocation,  Up: Histograms

25.1 The histogram struct
=========================

A histogram is defined by the following struct,

 -- Type: gsl_histogram

     size_t n                          This is the number of histogram bins
                                       
                                       
     double * range                    The ranges of the bins are stored in an array of ‘n+1’ elements pointed to by
                                       range.
                                       
                                       
     double * bin                      The counts for each bin are stored in an array of ‘n’ elements pointed to by
                                       ‘bin’.  The bins are floating-point numbers, so you can increment them by
                                       non-integer values if necessary.
                                       

     The range for ‘bin[i]’ is given by ‘range[i]’ to ‘range[i+1]’.  For
     n bins there are ‘n+1’ entries in the array ‘range’.  Each bin is
     inclusive at the lower end and exclusive at the upper end.
     Mathematically this means that the bins are defined by the
     following inequality,

          bin[i] corresponds to range[i] <= x < range[i+1]

     Here is a diagram of the correspondence between ranges and bins on
     the number-line for x:

             [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
          ---|---------|---------|---------|---------|---------|---  x
           r[0]      r[1]      r[2]      r[3]      r[4]      r[5]

     In this picture the values of the ‘range’ array are denoted by r.
     On the left-hand side of each bin the square bracket ‘[’ denotes an
     inclusive lower bound (r \le x), and the round parentheses ‘)’ on
     the right-hand side denote an exclusive upper bound (x < r).  Thus
     any samples which fall on the upper end of the histogram are
     excluded.  If you want to include this value for the last bin you
     will need to add an extra bin to your histogram.

     The *note gsl_histogram: 8b1. struct and its associated functions
     are defined in the header file ‘gsl_histogram.h’.


File: gsl-ref.info,  Node: Histogram allocation,  Next: Copying Histograms,  Prev: The histogram struct,  Up: Histograms

25.2 Histogram allocation
=========================

The functions for allocating memory to a histogram follow the style of
‘malloc()’ and ‘free()’.  In addition they also perform their own error
checking.  If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of *note GSL_ENOMEM: 2a.) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your
program then it isn’t necessary to check every histogram ‘alloc’.

 -- Function: *note gsl_histogram: 8b1. *gsl_histogram_alloc (size_t n)

     This function allocates memory for a histogram with *note n: 8b3.
     bins, and returns a pointer to a newly created *note gsl_histogram:
     8b1. struct.  If insufficient memory is available a null pointer is
     returned and the error handler is invoked with an error code of
     *note GSL_ENOMEM: 2a.  The bins and ranges are not initialized, and
     should be prepared using one of the range-setting functions below
     in order to make the histogram ready for use.

 -- Function: int gsl_histogram_set_ranges (gsl_histogram *h, const
          double range[], size_t size)

     This function sets the ranges of the existing histogram *note h:
     8b4. using the array *note range: 8b4. of size *note size: 8b4.
     The values of the histogram bins are reset to zero.  The *note
     range: 8b4. array should contain the desired bin limits.  The
     ranges can be arbitrary, subject to the restriction that they are
     monotonically increasing.

     The following example shows how to create a histogram with
     logarithmic bins with ranges [1,10), [10,100) and [100,1000):

          gsl_histogram * h = gsl_histogram_alloc (3);

          /* bin[0] covers the range 1 <= x < 10 */
          /* bin[1] covers the range 10 <= x < 100 */
          /* bin[2] covers the range 100 <= x < 1000 */

          double range[4] = { 1.0, 10.0, 100.0, 1000.0 };

          gsl_histogram_set_ranges (h, range, 4);

     Note that the size of the *note range: 8b4. array should be defined
     to be one element bigger than the number of bins.  The additional
     element is required for the upper value of the final bin.

 -- Function: int gsl_histogram_set_ranges_uniform (gsl_histogram *h,
          double xmin, double xmax)

     This function sets the ranges of the existing histogram *note h:
     8b5. to cover the range *note xmin: 8b5. to *note xmax: 8b5.
     uniformly.  The values of the histogram bins are reset to zero.
     The bin ranges are shown in the table below,

          bin[0] corresponds to xmin <= x < xmin + d
          bin[1] corresponds to xmin + d <= x < xmin + 2 d
          ......
          bin[n-1] corresponds to xmin + (n-1)d <= x < xmax

     where d is the bin spacing, d = (xmax-xmin)/n.

 -- Function: void gsl_histogram_free (gsl_histogram *h)

     This function frees the histogram *note h: 8b6. and all of the
     memory associated with it.


File: gsl-ref.info,  Node: Copying Histograms,  Next: Updating and accessing histogram elements,  Prev: Histogram allocation,  Up: Histograms

25.3 Copying Histograms
=======================

 -- Function: int gsl_histogram_memcpy (gsl_histogram *dest, const
          gsl_histogram *src)

     This function copies the histogram *note src: 8b8. into the
     pre-existing histogram *note dest: 8b8, making *note dest: 8b8.
     into an exact copy of *note src: 8b8.  The two histograms must be
     of the same size.

 -- Function: *note gsl_histogram: 8b1. *gsl_histogram_clone (const
          gsl_histogram *src)

     This function returns a pointer to a newly created histogram which
     is an exact copy of the histogram *note src: 8b9.


File: gsl-ref.info,  Node: Updating and accessing histogram elements,  Next: Searching histogram ranges,  Prev: Copying Histograms,  Up: Histograms

25.4 Updating and accessing histogram elements
==============================================

There are two ways to access histogram bins, either by specifying an x
coordinate or by using the bin-index directly.  The functions for
accessing the histogram through x coordinates use a binary search to
identify the bin which covers the appropriate range.

 -- Function: int gsl_histogram_increment (gsl_histogram *h, double x)

     This function updates the histogram *note h: 8bb. by adding one
     (1.0) to the bin whose range contains the coordinate *note x: 8bb.

     If *note x: 8bb. lies in the valid range of the histogram then the
     function returns zero to indicate success.  If *note x: 8bb. is
     less than the lower limit of the histogram then the function
     returns *note GSL_EDOM: 28, and none of bins are modified.
     Similarly, if the value of *note x: 8bb. is greater than or equal
     to the upper limit of the histogram then the function returns *note
     GSL_EDOM: 28, and none of the bins are modified.  The error handler
     is not called, however, since it is often necessary to compute
     histograms for a small range of a larger dataset, ignoring the
     values outside the range of interest.

 -- Function: int gsl_histogram_accumulate (gsl_histogram *h, double x,
          double weight)

     This function is similar to *note gsl_histogram_increment(): 8bb.
     but increases the value of the appropriate bin in the histogram
     *note h: 8bc. by the floating-point number *note weight: 8bc.

 -- Function: double gsl_histogram_get (const gsl_histogram *h, size_t
          i)

     This function returns the contents of the *note i: 8bd.-th bin of
     the histogram *note h: 8bd.  If *note i: 8bd. lies outside the
     valid range of indices for the histogram then the error handler is
     called with an error code of *note GSL_EDOM: 28. and the function
     returns 0.

 -- Function: int gsl_histogram_get_range (const gsl_histogram *h,
          size_t i, double *lower, double *upper)

     This function finds the upper and lower range limits of the *note
     i: 8be.-th bin of the histogram *note h: 8be.  If the index *note
     i: 8be. is valid then the corresponding range limits are stored in
     *note lower: 8be. and *note upper: 8be.  The lower limit is
     inclusive (i.e.  events with this coordinate are included in the
     bin) and the upper limit is exclusive (i.e.  events with the
     coordinate of the upper limit are excluded and fall in the
     neighboring higher bin, if it exists).  The function returns 0 to
     indicate success.  If *note i: 8be. lies outside the valid range of
     indices for the histogram then the error handler is called and the
     function returns an error code of *note GSL_EDOM: 28.

 -- Function: double gsl_histogram_max (const gsl_histogram *h)
 -- Function: double gsl_histogram_min (const gsl_histogram *h)
 -- Function: size_t gsl_histogram_bins (const gsl_histogram *h)

     These functions return the maximum upper and minimum lower range
     limits and the number of bins of the histogram *note h: 8c1.  They
     provide a way of determining these values without accessing the
     *note gsl_histogram: 8b1. struct directly.

 -- Function: void gsl_histogram_reset (gsl_histogram *h)

     This function resets all the bins in the histogram *note h: 8c2. to
     zero.


File: gsl-ref.info,  Node: Searching histogram ranges,  Next: Histogram Statistics,  Prev: Updating and accessing histogram elements,  Up: Histograms

25.5 Searching histogram ranges
===============================

The following functions are used by the access and update routines to
locate the bin which corresponds to a given x coordinate.

 -- Function: int gsl_histogram_find (const gsl_histogram *h, double x,
          size_t *i)

     This function finds and sets the index *note i: 8c4. to the bin
     number which covers the coordinate *note x: 8c4. in the histogram
     *note h: 8c4.  The bin is located using a binary search.  The
     search includes an optimization for histograms with uniform range,
     and will return the correct bin immediately in this case.  If *note
     x: 8c4. is found in the range of the histogram then the function
     sets the index *note i: 8c4. and returns ‘GSL_SUCCESS’.  If *note
     x: 8c4. lies outside the valid range of the histogram then the
     function returns *note GSL_EDOM: 28. and the error handler is
     invoked.


File: gsl-ref.info,  Node: Histogram Statistics,  Next: Histogram Operations,  Prev: Searching histogram ranges,  Up: Histograms

25.6 Histogram Statistics
=========================

 -- Function: double gsl_histogram_max_val (const gsl_histogram *h)

     This function returns the maximum value contained in the histogram
     bins.

 -- Function: size_t gsl_histogram_max_bin (const gsl_histogram *h)

     This function returns the index of the bin containing the maximum
     value.  In the case where several bins contain the same maximum
     value the smallest index is returned.

 -- Function: double gsl_histogram_min_val (const gsl_histogram *h)

     This function returns the minimum value contained in the histogram
     bins.

 -- Function: size_t gsl_histogram_min_bin (const gsl_histogram *h)

     This function returns the index of the bin containing the minimum
     value.  In the case where several bins contain the same minimum
     value the smallest index is returned.

 -- Function: double gsl_histogram_mean (const gsl_histogram *h)

     This function returns the mean of the histogrammed variable, where
     the histogram is regarded as a probability distribution.  Negative
     bin values are ignored for the purposes of this calculation.  The
     accuracy of the result is limited by the bin width.

 -- Function: double gsl_histogram_sigma (const gsl_histogram *h)

     This function returns the standard deviation of the histogrammed
     variable, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.  The accuracy of the result is limited by the bin
     width.

 -- Function: double gsl_histogram_sum (const gsl_histogram *h)

     This function returns the sum of all bin values.  Negative bin
     values are included in the sum.


File: gsl-ref.info,  Node: Histogram Operations,  Next: Reading and writing histograms,  Prev: Histogram Statistics,  Up: Histograms

25.7 Histogram Operations
=========================

 -- Function: int gsl_histogram_equal_bins_p (const gsl_histogram *h1,
          const gsl_histogram *h2)

     This function returns 1 if the all of the individual bin ranges of
     the two histograms are identical, and 0 otherwise.

 -- Function: int gsl_histogram_add (gsl_histogram *h1, const
          gsl_histogram *h2)

     This function adds the contents of the bins in histogram *note h2:
     8cf. to the corresponding bins of histogram *note h1: 8cf, i.e.
     h'_1(i) = h_1(i) + h_2(i).  The two histograms must have identical
     bin ranges.

 -- Function: int gsl_histogram_sub (gsl_histogram *h1, const
          gsl_histogram *h2)

     This function subtracts the contents of the bins in histogram *note
     h2: 8d0. from the corresponding bins of histogram *note h1: 8d0,
     i.e.  h'_1(i) = h_1(i) - h_2(i).  The two histograms must have
     identical bin ranges.

 -- Function: int gsl_histogram_mul (gsl_histogram *h1, const
          gsl_histogram *h2)

     This function multiplies the contents of the bins of histogram
     *note h1: 8d1. by the contents of the corresponding bins in
     histogram *note h2: 8d1, i.e.  h'_1(i) = h_1(i) * h_2(i).  The two
     histograms must have identical bin ranges.

 -- Function: int gsl_histogram_div (gsl_histogram *h1, const
          gsl_histogram *h2)

     This function divides the contents of the bins of histogram *note
     h1: 8d2. by the contents of the corresponding bins in histogram
     *note h2: 8d2, i.e.  h'_1(i) = h_1(i) / h_2(i).  The two histograms
     must have identical bin ranges.

 -- Function: int gsl_histogram_scale (gsl_histogram *h, double scale)

     This function multiplies the contents of the bins of histogram
     *note h: 8d3. by the constant *note scale: 8d3, i.e.

          h'_1(i) = h_1(i) * scale

 -- Function: int gsl_histogram_shift (gsl_histogram *h, double offset)

     This function shifts the contents of the bins of histogram *note h:
     8d4. by the constant *note offset: 8d4, i.e.

          h'_1(i) = h_1(i) + offset


File: gsl-ref.info,  Node: Reading and writing histograms,  Next: Resampling from histograms,  Prev: Histogram Operations,  Up: Histograms

25.8 Reading and writing histograms
===================================

The library provides functions for reading and writing histograms to a
file as binary data or formatted text.

 -- Function: int gsl_histogram_fwrite (FILE *stream, const
          gsl_histogram *h)

     This function writes the ranges and bins of the histogram *note h:
     8d6. to the stream *note stream: 8d6. in binary format.  The return
     value is 0 for success and ‘GSL_EFAILED’ if there was a problem
     writing to the file.  Since the data is written in the native
     binary format it may not be portable between different
     architectures.

 -- Function: int gsl_histogram_fread (FILE *stream, gsl_histogram *h)

     This function reads into the histogram *note h: 8d7. from the open
     stream *note stream: 8d7. in binary format.  The histogram *note h:
     8d7. must be preallocated with the correct size since the function
     uses the number of bins in *note h: 8d7. to determine how many
     bytes to read.  The return value is 0 for success and ‘GSL_EFAILED’
     if there was a problem reading from the file.  The data is assumed
     to have been written in the native binary format on the same
     architecture.

 -- Function: int gsl_histogram_fprintf (FILE *stream, const
          gsl_histogram *h, const char *range_format, const char
          *bin_format)

     This function writes the ranges and bins of the histogram *note h:
     8d8. line-by-line to the stream *note stream: 8d8. using the format
     specifiers *note range_format: 8d8. and *note bin_format: 8d8.
     These should be one of the ‘%g’, ‘%e’ or ‘%f’ formats for floating
     point numbers.  The function returns 0 for success and
     ‘GSL_EFAILED’ if there was a problem writing to the file.  The
     histogram output is formatted in three columns, and the columns are
     separated by spaces, like this:

          range[0] range[1] bin[0]
          range[1] range[2] bin[1]
          range[2] range[3] bin[2]
          ....
          range[n-1] range[n] bin[n-1]

     The values of the ranges are formatted using *note range_format:
     8d8. and the value of the bins are formatted using *note
     bin_format: 8d8.  Each line contains the lower and upper limit of
     the range of the bins and the value of the bin itself.  Since the
     upper limit of one bin is the lower limit of the next there is
     duplication of these values between lines but this allows the
     histogram to be manipulated with line-oriented tools.

 -- Function: int gsl_histogram_fscanf (FILE *stream, gsl_histogram *h)

     This function reads formatted data from the stream *note stream:
     8d9. into the histogram *note h: 8d9.  The data is assumed to be in
     the three-column format used by *note gsl_histogram_fprintf(): 8d8.
     The histogram *note h: 8d9. must be preallocated with the correct
     length since the function uses the size of *note h: 8d9. to
     determine how many numbers to read.  The function returns 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.


File: gsl-ref.info,  Node: Resampling from histograms,  Next: The histogram probability distribution struct,  Prev: Reading and writing histograms,  Up: Histograms

25.9 Resampling from histograms
===============================

A histogram made by counting events can be regarded as a measurement of
a probability distribution.  Allowing for statistical error, the height
of each bin represents the probability of an event where the value of x
falls in the range of that bin.  The probability distribution function
has the one-dimensional form p(x)dx where,

     p(x) = n_i / (N w_i)

In this equation n_i is the number of events in the bin which contains
x, w_i is the width of the bin and N is the total number of events.  The
distribution of events within each bin is assumed to be uniform.


File: gsl-ref.info,  Node: The histogram probability distribution struct,  Next: Example programs for histograms,  Prev: Resampling from histograms,  Up: Histograms

25.10 The histogram probability distribution struct
===================================================

The probability distribution function for a histogram consists of a set
of `bins' which measure the probability of an event falling into a given
range of a continuous variable x.  A probability distribution function
is defined by the following struct, which actually stores the cumulative
probability distribution function.  This is the natural quantity for
generating samples via the inverse transform method, because there is a
one-to-one mapping between the cumulative probability distribution and
the range [0,1].  It can be shown that by taking a uniform random number
in this range and finding its corresponding coordinate in the cumulative
probability distribution we obtain samples with the desired probability
distribution.

 -- Type: gsl_histogram_pdf

     ‘size_t n’                           This is the number of bins used to approximate the probability
                                          distribution function.
                                          
                                          
     ‘double * range’                     The ranges of the bins are stored in an array of n + 1 elements pointed
                                          to by ‘range’.
                                          
                                          
     ‘double * sum’                       The cumulative probability for the bins is stored in an array of ‘n’
                                          elements pointed to by ‘sum’.
                                          

The following functions allow you to create a *note gsl_histogram_pdf:
8dc. struct which represents this probability distribution and generate
random samples from it.

 -- Function: *note gsl_histogram_pdf: 8dc. *gsl_histogram_pdf_alloc
          (size_t n)

     This function allocates memory for a probability distribution with
     *note n: 8dd. bins and returns a pointer to a newly initialized
     *note gsl_histogram_pdf: 8dc. struct.  If insufficient memory is
     available a null pointer is returned and the error handler is
     invoked with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_histogram_pdf_init (gsl_histogram_pdf *p, const
          gsl_histogram *h)

     This function initializes the probability distribution *note p:
     8de. with the contents of the histogram *note h: 8de.  If any of
     the bins of *note h: 8de. are negative then the error handler is
     invoked with an error code of *note GSL_EDOM: 28. because a
     probability distribution cannot contain negative values.

 -- Function: void gsl_histogram_pdf_free (gsl_histogram_pdf *p)

     This function frees the probability distribution function *note p:
     8df. and all of the memory associated with it.

 -- Function: double gsl_histogram_pdf_sample (const gsl_histogram_pdf
          *p, double r)

     This function uses *note r: 8e0, a uniform random number between
     zero and one, to compute a single random sample from the
     probability distribution *note p: 8e0.  The algorithm used to
     compute the sample s is given by the following formula,

          s = range[i] + delta * (range[i+1] - range[i])

     where i is the index which satisfies sum[i] \le r < sum[i+1] and
     delta is (r - sum[i])/(sum[i+1] - sum[i]).


File: gsl-ref.info,  Node: Example programs for histograms,  Next: Two dimensional histograms,  Prev: The histogram probability distribution struct,  Up: Histograms

25.11 Example programs for histograms
=====================================

The following program shows how to make a simple histogram of a column
of numerical data supplied on ‘stdin’.  The program takes three
arguments, specifying the upper and lower bounds of the histogram and
the number of bins.  It then reads numbers from ‘stdin’, one line at a
time, and adds them to the histogram.  When there is no more data to
read it prints out the accumulated histogram using *note
gsl_histogram_fprintf(): 8d8.

     #include <stdio.h>
     #include <stdlib.h>
     #include <gsl/gsl_histogram.h>

     int
     main (int argc, char **argv)
     {
       double a, b;
       size_t n;

       if (argc != 4)
         {
           printf ("Usage: gsl-histogram xmin xmax n\n"
                   "Computes a histogram of the data "
                   "on stdin using n bins from xmin "
                   "to xmax\n");
           exit (0);
         }

       a = atof (argv[1]);
       b = atof (argv[2]);
       n = atoi (argv[3]);

       {
         double x;
         gsl_histogram * h = gsl_histogram_alloc (n);
         gsl_histogram_set_ranges_uniform (h, a, b);

         while (fscanf (stdin, "%lg", &x) == 1)
           {
             gsl_histogram_increment (h, x);
           }
         gsl_histogram_fprintf (stdout, h, "%g", "%g");
         gsl_histogram_free (h);
       }
       exit (0);
     }

Here is an example of the program in use.  We generate 10000 random
samples from a Cauchy distribution with a width of 30 and histogram them
over the range -100 to 100, using 200 bins:

     $ gsl-randist 0 10000 cauchy 30
        | gsl-histogram -- -100 100 200 > histogram.dat

Fig.  %s shows the familiar shape of the Cauchy distribution and the
fluctuations caused by the finite sample size.

[gsl-ref-figures/histogram]
Figure: Histogram output from example program


File: gsl-ref.info,  Node: Two dimensional histograms,  Next: The 2D histogram struct,  Prev: Example programs for histograms,  Up: Histograms

25.12 Two dimensional histograms
================================

A two dimensional histogram consists of a set of `bins' which count the
number of events falling in a given area of the (x,y) plane.  The
simplest way to use a two dimensional histogram is to record
two-dimensional position information, n(x,y).  Another possibility is to
form a `joint distribution' by recording related variables.  For example
a detector might record both the position of an event (x) and the amount
of energy it deposited E. These could be histogrammed as the joint
distribution n(x,E).


File: gsl-ref.info,  Node: The 2D histogram struct,  Next: 2D Histogram allocation,  Prev: Two dimensional histograms,  Up: Histograms

25.13 The 2D histogram struct
=============================

Two dimensional histograms are defined by the following struct,

 -- Type: gsl_histogram2d

     ‘size_t nx, ny’                 This is the number of histogram bins in the x and y directions.
                                     
                                     
     ‘double * xrange’               The ranges of the bins in the x-direction are stored in an array of ‘nx + 1’
                                     elements pointed to by ‘xrange’.
                                     
                                     
     ‘double * yrange’               The ranges of the bins in the y-direction are stored in an array of ‘ny + 1’
                                     elements pointed to by ‘yrange’.
                                     
                                     
     ‘double * bin’                  The counts for each bin are stored in an array pointed to by ‘bin’.  The bins
                                     are floating-point numbers, so you can increment them by non-integer values if
                                     necessary.  The array ‘bin’ stores the two dimensional array of bins in a
                                     single block of memory according to the mapping ‘bin(i,j)’ = ‘bin[i * ny +
                                     j]’.
                                     

The range for ‘bin(i,j)’ is given by ‘xrange[i]’ to ‘xrange[i+1]’ in the
x-direction and ‘yrange[j]’ to ‘yrange[j+1]’ in the y-direction.  Each
bin is inclusive at the lower end and exclusive at the upper end.
Mathematically this means that the bins are defined by the following
inequality,

     bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
                         and yrange[j] <= y < yrange[j+1]

Note that any samples which fall on the upper sides of the histogram are
excluded.  If you want to include these values for the side bins you
will need to add an extra row or column to your histogram.

The *note gsl_histogram2d: 8e5. struct and its associated functions are
defined in the header file ‘gsl_histogram2d.h’.


File: gsl-ref.info,  Node: 2D Histogram allocation,  Next: Copying 2D Histograms,  Prev: The 2D histogram struct,  Up: Histograms

25.14 2D Histogram allocation
=============================

The functions for allocating memory to a 2D histogram follow the style
of ‘malloc()’ and ‘free()’.  In addition they also perform their own
error checking.  If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of *note GSL_ENOMEM: 2a.) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your
program then it isn’t necessary to check every 2D histogram ‘alloc’.

 -- Function: *note gsl_histogram2d: 8e5. *gsl_histogram2d_alloc (size_t
          nx, size_t ny)

     This function allocates memory for a two-dimensional histogram with
     *note nx: 8e7. bins in the x direction and *note ny: 8e7. bins in
     the y direction.  The function returns a pointer to a newly created
     *note gsl_histogram2d: 8e5. struct.  If insufficient memory is
     available a null pointer is returned and the error handler is
     invoked with an error code of *note GSL_ENOMEM: 2a.  The bins and
     ranges must be initialized with one of the functions below before
     the histogram is ready for use.

 -- Function: int gsl_histogram2d_set_ranges (gsl_histogram2d *h, const
          double xrange[], size_t xsize, const double yrange[], size_t
          ysize)

     This function sets the ranges of the existing histogram *note h:
     8e8. using the arrays *note xrange: 8e8. and *note yrange: 8e8. of
     size *note xsize: 8e8. and *note ysize: 8e8. respectively.  The
     values of the histogram bins are reset to zero.

 -- Function: int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d
          *h, double xmin, double xmax, double ymin, double ymax)

     This function sets the ranges of the existing histogram *note h:
     8e9. to cover the ranges *note xmin: 8e9. to *note xmax: 8e9. and
     *note ymin: 8e9. to *note ymax: 8e9. uniformly.  The values of the
     histogram bins are reset to zero.

 -- Function: void gsl_histogram2d_free (gsl_histogram2d *h)

     This function frees the 2D histogram *note h: 8ea. and all of the
     memory associated with it.


File: gsl-ref.info,  Node: Copying 2D Histograms,  Next: Updating and accessing 2D histogram elements,  Prev: 2D Histogram allocation,  Up: Histograms

25.15 Copying 2D Histograms
===========================

 -- Function: int gsl_histogram2d_memcpy (gsl_histogram2d *dest, const
          gsl_histogram2d *src)

     This function copies the histogram *note src: 8ec. into the
     pre-existing histogram *note dest: 8ec, making *note dest: 8ec.
     into an exact copy of *note src: 8ec.  The two histograms must be
     of the same size.

 -- Function: *note gsl_histogram2d: 8e5. *gsl_histogram2d_clone (const
          gsl_histogram2d *src)

     This function returns a pointer to a newly created histogram which
     is an exact copy of the histogram *note src: 8ed.


File: gsl-ref.info,  Node: Updating and accessing 2D histogram elements,  Next: Searching 2D histogram ranges,  Prev: Copying 2D Histograms,  Up: Histograms

25.16 Updating and accessing 2D histogram elements
==================================================

You can access the bins of a two-dimensional histogram either by
specifying a pair of (x,y) coordinates or by using the bin indices (i,j)
directly.  The functions for accessing the histogram through (x,y)
coordinates use binary searches in the x and y directions to identify
the bin which covers the appropriate range.

 -- Function: int gsl_histogram2d_increment (gsl_histogram2d *h, double
          x, double y)

     This function updates the histogram *note h: 8ef. by adding one
     (1.0) to the bin whose x and y ranges contain the coordinates
     (*note x: 8ef, *note y: 8ef.).

     If the point (x,y) lies inside the valid ranges of the histogram
     then the function returns zero to indicate success.  If (x,y) lies
     outside the limits of the histogram then the function returns *note
     GSL_EDOM: 28, and none of the bins are modified.  The error handler
     is not called, since it is often necessary to compute histograms
     for a small range of a larger dataset, ignoring any coordinates
     outside the range of interest.

 -- Function: int gsl_histogram2d_accumulate (gsl_histogram2d *h, double
          x, double y, double weight)

     This function is similar to *note gsl_histogram2d_increment(): 8ef.
     but increases the value of the appropriate bin in the histogram
     *note h: 8f0. by the floating-point number *note weight: 8f0.

 -- Function: double gsl_histogram2d_get (const gsl_histogram2d *h,
          size_t i, size_t j)

     This function returns the contents of the (*note i: 8f1, *note j:
     8f1.)-th bin of the histogram *note h: 8f1.  If (*note i: 8f1,
     *note j: 8f1.) lies outside the valid range of indices for the
     histogram then the error handler is called with an error code of
     *note GSL_EDOM: 28. and the function returns 0.

 -- Function: int gsl_histogram2d_get_xrange (const gsl_histogram2d *h,
          size_t i, double *xlower, double *xupper)
 -- Function: int gsl_histogram2d_get_yrange (const gsl_histogram2d *h,
          size_t j, double *ylower, double *yupper)

     These functions find the upper and lower range limits of the ‘i’-th
     and *note j: 8f3.-th bins in the x and y directions of the
     histogram *note h: 8f3.  The range limits are stored in ‘xlower’
     and ‘xupper’ or *note ylower: 8f3. and *note yupper: 8f3.  The
     lower limits are inclusive (i.e.  events with these coordinates are
     included in the bin) and the upper limits are exclusive (i.e.
     events with the value of the upper limit are not included and fall
     in the neighboring higher bin, if it exists).  The functions return
     0 to indicate success.  If ‘i’ or *note j: 8f3. lies outside the
     valid range of indices for the histogram then the error handler is
     called with an error code of *note GSL_EDOM: 28.

 -- Function: double gsl_histogram2d_xmax (const gsl_histogram2d *h)
 -- Function: double gsl_histogram2d_xmin (const gsl_histogram2d *h)
 -- Function: size_t gsl_histogram2d_nx (const gsl_histogram2d *h)
 -- Function: double gsl_histogram2d_ymax (const gsl_histogram2d *h)
 -- Function: double gsl_histogram2d_ymin (const gsl_histogram2d *h)
 -- Function: size_t gsl_histogram2d_ny (const gsl_histogram2d *h)

     These functions return the maximum upper and minimum lower range
     limits and the number of bins for the x and y directions of the
     histogram *note h: 8f9.  They provide a way of determining these
     values without accessing the *note gsl_histogram2d: 8e5. struct
     directly.

 -- Function: void gsl_histogram2d_reset (gsl_histogram2d *h)

     This function resets all the bins of the histogram *note h: 8fa. to
     zero.


File: gsl-ref.info,  Node: Searching 2D histogram ranges,  Next: 2D Histogram Statistics,  Prev: Updating and accessing 2D histogram elements,  Up: Histograms

25.17 Searching 2D histogram ranges
===================================

The following functions are used by the access and update routines to
locate the bin which corresponds to a given (x,y) coordinate.

 -- Function: int gsl_histogram2d_find (const gsl_histogram2d *h, double
          x, double y, size_t *i, size_t *j)

     This function finds and sets the indices *note i: 8fc. and *note j:
     8fc. to the bin which covers the coordinates (*note x: 8fc, *note
     y: 8fc.).  The bin is located using a binary search.  The search
     includes an optimization for histograms with uniform ranges, and
     will return the correct bin immediately in this case.  If (x,y) is
     found then the function sets the indices (*note i: 8fc, *note j:
     8fc.) and returns ‘GSL_SUCCESS’.  If (x,y) lies outside the valid
     range of the histogram then the function returns *note GSL_EDOM:
     28. and the error handler is invoked.


File: gsl-ref.info,  Node: 2D Histogram Statistics,  Next: 2D Histogram Operations,  Prev: Searching 2D histogram ranges,  Up: Histograms

25.18 2D Histogram Statistics
=============================

 -- Function: double gsl_histogram2d_max_val (const gsl_histogram2d *h)

     This function returns the maximum value contained in the histogram
     bins.

 -- Function: void gsl_histogram2d_max_bin (const gsl_histogram2d *h,
          size_t *i, size_t *j)

     This function finds the indices of the bin containing the maximum
     value in the histogram *note h: 8ff. and stores the result in
     (*note i: 8ff, *note j: 8ff.).  In the case where several bins
     contain the same maximum value the first bin found is returned.

 -- Function: double gsl_histogram2d_min_val (const gsl_histogram2d *h)

     This function returns the minimum value contained in the histogram
     bins.

 -- Function: void gsl_histogram2d_min_bin (const gsl_histogram2d *h,
          size_t *i, size_t *j)

     This function finds the indices of the bin containing the minimum
     value in the histogram *note h: 901. and stores the result in
     (*note i: 901, *note j: 901.).  In the case where several bins
     contain the same maximum value the first bin found is returned.

 -- Function: double gsl_histogram2d_xmean (const gsl_histogram2d *h)

     This function returns the mean of the histogrammed x variable,
     where the histogram is regarded as a probability distribution.
     Negative bin values are ignored for the purposes of this
     calculation.

 -- Function: double gsl_histogram2d_ymean (const gsl_histogram2d *h)

     This function returns the mean of the histogrammed y variable,
     where the histogram is regarded as a probability distribution.
     Negative bin values are ignored for the purposes of this
     calculation.

 -- Function: double gsl_histogram2d_xsigma (const gsl_histogram2d *h)

     This function returns the standard deviation of the histogrammed x
     variable, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_ysigma (const gsl_histogram2d *h)

     This function returns the standard deviation of the histogrammed y
     variable, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_cov (const gsl_histogram2d *h)

     This function returns the covariance of the histogrammed x and y
     variables, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_sum (const gsl_histogram2d *h)

     This function returns the sum of all bin values.  Negative bin
     values are included in the sum.


File: gsl-ref.info,  Node: 2D Histogram Operations,  Next: Reading and writing 2D histograms,  Prev: 2D Histogram Statistics,  Up: Histograms

25.19 2D Histogram Operations
=============================

 -- Function: int gsl_histogram2d_equal_bins_p (const gsl_histogram2d
          *h1, const gsl_histogram2d *h2)

     This function returns 1 if all the individual bin ranges of the two
     histograms are identical, and 0 otherwise.

 -- Function: int gsl_histogram2d_add (gsl_histogram2d *h1, const
          gsl_histogram2d *h2)

     This function adds the contents of the bins in histogram *note h2:
     90a. to the corresponding bins of histogram *note h1: 90a, i.e.
     h'_1(i,j) = h_1(i,j) + h_2(i,j).  The two histograms must have
     identical bin ranges.

 -- Function: int gsl_histogram2d_sub (gsl_histogram2d *h1, const
          gsl_histogram2d *h2)

     This function subtracts the contents of the bins in histogram *note
     h2: 90b. from the corresponding bins of histogram *note h1: 90b,
     i.e.  h'_1(i,j) = h_1(i,j) - h_2(i,j).  The two histograms must
     have identical bin ranges.

 -- Function: int gsl_histogram2d_mul (gsl_histogram2d *h1, const
          gsl_histogram2d *h2)

     This function multiplies the contents of the bins of histogram
     *note h1: 90c. by the contents of the corresponding bins in
     histogram *note h2: 90c, i.e.  h'_1(i,j) = h_1(i,j) * h_2(i,j).
     The two histograms must have identical bin ranges.

 -- Function: int gsl_histogram2d_div (gsl_histogram2d *h1, const
          gsl_histogram2d *h2)

     This function divides the contents of the bins of histogram *note
     h1: 90d. by the contents of the corresponding bins in histogram
     *note h2: 90d, i.e.  h'_1(i,j) = h_1(i,j) / h_2(i,j).  The two
     histograms must have identical bin ranges.

 -- Function: int gsl_histogram2d_scale (gsl_histogram2d *h, double
          scale)

     This function multiplies the contents of the bins of histogram
     *note h: 90e. by the constant *note scale: 90e, i.e.

          h'_1(i,j) = h_1(i,j) scale

 -- Function: int gsl_histogram2d_shift (gsl_histogram2d *h, double
          offset)

     This function shifts the contents of the bins of histogram *note h:
     90f. by the constant *note offset: 90f, i.e.

          h'_1(i,j) = h_1(i,j) + offset


File: gsl-ref.info,  Node: Reading and writing 2D histograms,  Next: Resampling from 2D histograms,  Prev: 2D Histogram Operations,  Up: Histograms

25.20 Reading and writing 2D histograms
=======================================

The library provides functions for reading and writing two dimensional
histograms to a file as binary data or formatted text.

 -- Function: int gsl_histogram2d_fwrite (FILE *stream, const
          gsl_histogram2d *h)

     This function writes the ranges and bins of the histogram *note h:
     911. to the stream *note stream: 911. in binary format.  The return
     value is 0 for success and ‘GSL_EFAILED’ if there was a problem
     writing to the file.  Since the data is written in the native
     binary format it may not be portable between different
     architectures.

 -- Function: int gsl_histogram2d_fread (FILE *stream, gsl_histogram2d
          *h)

     This function reads into the histogram *note h: 912. from the
     stream *note stream: 912. in binary format.  The histogram *note h:
     912. must be preallocated with the correct size since the function
     uses the number of x and y bins in *note h: 912. to determine how
     many bytes to read.  The return value is 0 for success and
     ‘GSL_EFAILED’ if there was a problem reading from the file.  The
     data is assumed to have been written in the native binary format on
     the same architecture.

 -- Function: int gsl_histogram2d_fprintf (FILE *stream, const
          gsl_histogram2d *h, const char *range_format, const char
          *bin_format)

     This function writes the ranges and bins of the histogram *note h:
     913. line-by-line to the stream *note stream: 913. using the format
     specifiers *note range_format: 913. and *note bin_format: 913.
     These should be one of the ‘%g’, ‘%e’ or ‘%f’ formats for floating
     point numbers.  The function returns 0 for success and
     ‘GSL_EFAILED’ if there was a problem writing to the file.  The
     histogram output is formatted in five columns, and the columns are
     separated by spaces, like this:

          xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
          xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
          xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
          ....
          xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)

          xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
          ....
          xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)

          ....

          xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
          ....
          xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)

     Each line contains the lower and upper limits of the bin and the
     contents of the bin.  Since the upper limits of the each bin are
     the lower limits of the neighboring bins there is duplication of
     these values but this allows the histogram to be manipulated with
     line-oriented tools.

 -- Function: int gsl_histogram2d_fscanf (FILE *stream, gsl_histogram2d
          *h)

     This function reads formatted data from the stream *note stream:
     914. into the histogram *note h: 914.  The data is assumed to be in
     the five-column format used by *note gsl_histogram2d_fprintf():
     913.  The histogram *note h: 914. must be preallocated with the
     correct lengths since the function uses the sizes of *note h: 914.
     to determine how many numbers to read.  The function returns 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.


File: gsl-ref.info,  Node: Resampling from 2D histograms,  Next: Example programs for 2D histograms,  Prev: Reading and writing 2D histograms,  Up: Histograms

25.21 Resampling from 2D histograms
===================================

As in the one-dimensional case, a two-dimensional histogram made by
counting events can be regarded as a measurement of a probability
distribution.  Allowing for statistical error, the height of each bin
represents the probability of an event where (x, y) falls in the range
of that bin.  For a two-dimensional histogram the probability
distribution takes the form p(x,y) dx dy where,

     p(x,y) = n_{ij} / (N A_{ij})

In this equation n_{ij} is the number of events in the bin which
contains (x,y), A_{ij} is the area of the bin and N is the total number
of events.  The distribution of events within each bin is assumed to be
uniform.

 -- Type: gsl_histogram2d_pdf

     ‘size_t nx, ny’                   This is the number of histogram bins used to approximate the probability
                                       distribution function in the x and y directions.
                                       
                                       
     ‘double * xrange’                 The ranges of the bins in the x-direction are stored in an array of ‘nx + 1’
                                       elements pointed to by ‘xrange’.
                                       
                                       
     ‘double * yrange’                 The ranges of the bins in the y-direction are stored in an array of ‘ny + 1’
                                       pointed to by ‘yrange’.
                                       
                                       
     ‘double * sum’                    The cumulative probability for the bins is stored in an array of ‘nx’ * ‘ny’
                                       elements pointed to by ‘sum’.
                                       

The following functions allow you to create a *note gsl_histogram2d_pdf:
916. struct which represents a two dimensional probability distribution
and generate random samples from it.

 -- Function: *note gsl_histogram2d_pdf: 916. *gsl_histogram2d_pdf_alloc
          (size_t nx, size_t ny)

     This function allocates memory for a two-dimensional probability
     distribution of size *note nx: 917.-by-*note ny: 917. and returns a
     pointer to a newly initialized *note gsl_histogram2d_pdf: 916.
     struct.  If insufficient memory is available a null pointer is
     returned and the error handler is invoked with an error code of
     *note GSL_ENOMEM: 2a.

 -- Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf *p,
          const gsl_histogram2d *h)

     This function initializes the two-dimensional probability
     distribution calculated *note p: 918. from the histogram *note h:
     918.  If any of the bins of *note h: 918. are negative then the
     error handler is invoked with an error code of *note GSL_EDOM: 28.
     because a probability distribution cannot contain negative values.

 -- Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf *p)

     This function frees the two-dimensional probability distribution
     function *note p: 919. and all of the memory associated with it.

 -- Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf
          *p, double r1, double r2, double *x, double *y)

     This function uses two uniform random numbers between zero and one,
     *note r1: 91a. and *note r2: 91a, to compute a single random sample
     from the two-dimensional probability distribution *note p: 91a.


File: gsl-ref.info,  Node: Example programs for 2D histograms,  Prev: Resampling from 2D histograms,  Up: Histograms

25.22 Example programs for 2D histograms
========================================

This program demonstrates two features of two-dimensional histograms.
First a 10-by-10 two-dimensional histogram is created with x and y
running from 0 to 1.  Then a few sample points are added to the
histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height
of 5 and at (0.7,0.9) with a height of 0.5.  This histogram with three
events is used to generate a random sample of 1000 simulated events,
which are printed out.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_histogram2d.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);

       gsl_histogram2d_set_ranges_uniform (h,
                                           0.0, 1.0,
                                           0.0, 1.0);

       gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
       gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
       gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       {
         int i;
         gsl_histogram2d_pdf * p
           = gsl_histogram2d_pdf_alloc (h->nx, h->ny);

         gsl_histogram2d_pdf_init (p, h);

         for (i = 0; i < 1000; i++) {
           double x, y;
           double u = gsl_rng_uniform (r);
           double v = gsl_rng_uniform (r);

           gsl_histogram2d_pdf_sample (p, u, v, &x, &y);

           printf ("%g %g\n", x, y);
         }

         gsl_histogram2d_pdf_free (p);
       }

       gsl_histogram2d_free (h);
       gsl_rng_free (r);

       return 0;
     }

The following plot shows the distribution of the simulated events.
Using a higher resolution grid we can see the original underlying
histogram and also the statistical fluctuations caused by the events
being uniformly distributed over the area of the original bins.

[gsl-ref-figures/histogram2d]
Figure: Distribution of simulated events from example program


File: gsl-ref.info,  Node: N-tuples,  Next: Monte Carlo Integration,  Prev: Histograms,  Up: Top

26 N-tuples
***********

This chapter describes functions for creating and manipulating
`ntuples', sets of values associated with events.  The ntuples are
stored in files.  Their values can be extracted in any combination and
`booked' in a histogram using a selection function.

The values to be stored are held in a user-defined data structure, and
an ntuple is created associating this data structure with a file.  The
values are then written to the file (normally inside a loop) using the
ntuple functions described below.

A histogram can be created from ntuple data by providing a selection
function and a value function.  The selection function specifies whether
an event should be included in the subset to be analyzed or not.  The
value function computes the entry to be added to the histogram for each
event.

All the ntuple functions are defined in the header file ‘gsl_ntuple.h’.

* Menu:

* The ntuple struct::
* Creating ntuples::
* Opening an existing ntuple file::
* Writing ntuples::
* Reading ntuples::
* Closing an ntuple file::
* Histogramming ntuple values::
* Examples: Examples<19>.
* References and Further Reading: References and Further Reading<19>.


File: gsl-ref.info,  Node: The ntuple struct,  Next: Creating ntuples,  Up: N-tuples

26.1 The ntuple struct
======================

 -- Type: gsl_ntuple

     Ntuples are manipulated using the *note gsl_ntuple: 91f. struct.
     This struct contains information on the file where the ntuple data
     is stored, a pointer to the current ntuple data row and the size of
     the user-defined ntuple data struct:

          typedef struct
            {
              FILE * file;
              void * ntuple_data;
              size_t size;
            } gsl_ntuple;


File: gsl-ref.info,  Node: Creating ntuples,  Next: Opening an existing ntuple file,  Prev: The ntuple struct,  Up: N-tuples

26.2 Creating ntuples
=====================

 -- Function: *note gsl_ntuple: 91f. *gsl_ntuple_create (char *filename,
          void *ntuple_data, size_t size)

     This function creates a new write-only ntuple file *note filename:
     921. for ntuples of size *note size: 921. and returns a pointer to
     the newly created ntuple struct.  Any existing file with the same
     name is truncated to zero length and overwritten.  A pointer to
     memory for the current ntuple row *note ntuple_data: 921. must be
     supplied—this is used to copy ntuples in and out of the file.


File: gsl-ref.info,  Node: Opening an existing ntuple file,  Next: Writing ntuples,  Prev: Creating ntuples,  Up: N-tuples

26.3 Opening an existing ntuple file
====================================

 -- Function: *note gsl_ntuple: 91f. *gsl_ntuple_open (char *filename,
          void *ntuple_data, size_t size)

     This function opens an existing ntuple file *note filename: 923.
     for reading and returns a pointer to a corresponding ntuple struct.
     The ntuples in the file must have size *note size: 923.  A pointer
     to memory for the current ntuple row *note ntuple_data: 923. must
     be supplied—this is used to copy ntuples in and out of the file.


File: gsl-ref.info,  Node: Writing ntuples,  Next: Reading ntuples,  Prev: Opening an existing ntuple file,  Up: N-tuples

26.4 Writing ntuples
====================

 -- Function: int gsl_ntuple_write (gsl_ntuple *ntuple)

     This function writes the current ntuple ‘ntuple->ntuple_data’ of
     size ‘ntuple->size’ to the corresponding file.

 -- Function: int gsl_ntuple_bookdata (gsl_ntuple *ntuple)

     This function is a synonym for *note gsl_ntuple_write(): 925.


File: gsl-ref.info,  Node: Reading ntuples,  Next: Closing an ntuple file,  Prev: Writing ntuples,  Up: N-tuples

26.5 Reading ntuples
====================

 -- Function: int gsl_ntuple_read (gsl_ntuple *ntuple)

     This function reads the current row of the ntuple file for *note
     ntuple: 928. and stores the values in ‘ntuple->data’.


File: gsl-ref.info,  Node: Closing an ntuple file,  Next: Histogramming ntuple values,  Prev: Reading ntuples,  Up: N-tuples

26.6 Closing an ntuple file
===========================

 -- Function: int gsl_ntuple_close (gsl_ntuple *ntuple)

     This function closes the ntuple file *note ntuple: 92a. and frees
     its associated allocated memory.


File: gsl-ref.info,  Node: Histogramming ntuple values,  Next: Examples<19>,  Prev: Closing an ntuple file,  Up: N-tuples

26.7 Histogramming ntuple values
================================

Once an ntuple has been created its contents can be histogrammed in
various ways using the function *note gsl_ntuple_project(): 92c.  Two
user-defined functions must be provided, a function to select events and
a function to compute scalar values.  The selection function and the
value function both accept the ntuple row as a first argument and other
parameters as a second argument.

 -- Type: gsl_ntuple_select_fn

     The `selection function' determines which ntuple rows are selected
     for histogramming.  It is defined by the following struct:

          typedef struct
            {
              int (* function) (void * ntuple_data, void * params);
              void * params;
            } gsl_ntuple_select_fn;

     The struct component ‘function’ should return a non-zero value for
     each ntuple row that is to be included in the histogram.

 -- Type: gsl_ntuple_value_fn

     The `value function' computes scalar values for those ntuple rows
     selected by the selection function:

          typedef struct
            {
              double (* function) (void * ntuple_data, void * params);
              void * params;
            } gsl_ntuple_value_fn;

     In this case the struct component ‘function’ should return the
     value to be added to the histogram for the ntuple row.

 -- Function: int gsl_ntuple_project (gsl_histogram *h, gsl_ntuple
          *ntuple, gsl_ntuple_value_fn *value_func, gsl_ntuple_select_fn
          *select_func)

     This function updates the histogram *note h: 92c. from the ntuple
     *note ntuple: 92c. using the functions *note value_func: 92c. and
     *note select_func: 92c.  For each ntuple row where the selection
     function *note select_func: 92c. is non-zero the corresponding
     value of that row is computed using the function *note value_func:
     92c. and added to the histogram.  Those ntuple rows where *note
     select_func: 92c. returns zero are ignored.  New entries are added
     to the histogram, so subsequent calls can be used to accumulate
     further data in the same histogram.


File: gsl-ref.info,  Node: Examples<19>,  Next: References and Further Reading<19>,  Prev: Histogramming ntuple values,  Up: N-tuples

26.8 Examples
=============

The following example programs demonstrate the use of ntuples in
managing a large dataset.  The first program creates a set of 10,000
simulated “events”, each with 3 associated values (x,y,z).  These are
generated from a Gaussian distribution with unit variance, for
demonstration purposes, and written to the ntuple file ‘test.dat’.

     #include <gsl/gsl_ntuple.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     struct data
     {
       double x;
       double y;
       double z;
     };

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       struct data ntuple_row;
       int i;

       gsl_ntuple *ntuple
         = gsl_ntuple_create ("test.dat", &ntuple_row,
                              sizeof (ntuple_row));

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < 10000; i++)
         {
           ntuple_row.x = gsl_ran_ugaussian (r);
           ntuple_row.y = gsl_ran_ugaussian (r);
           ntuple_row.z = gsl_ran_ugaussian (r);

           gsl_ntuple_write (ntuple);
         }

       gsl_ntuple_close (ntuple);
       gsl_rng_free (r);

       return 0;
     }

The next program analyses the ntuple data in the file ‘test.dat’.  The
analysis procedure is to compute the squared-magnitude of each event,
E^2=x^2+y^2+z^2, and select only those which exceed a lower limit of
1.5.  The selected events are then histogrammed using their E^2 values.

     #include <math.h>
     #include <gsl/gsl_ntuple.h>
     #include <gsl/gsl_histogram.h>

     struct data
     {
       double x;
       double y;
       double z;
     };

     int sel_func (void *ntuple_data, void *params);
     double val_func (void *ntuple_data, void *params);

     int
     main (void)
     {
       struct data ntuple_row;

       gsl_ntuple *ntuple
         = gsl_ntuple_open ("test.dat", &ntuple_row,
                            sizeof (ntuple_row));
       double lower = 1.5;

       gsl_ntuple_select_fn S;
       gsl_ntuple_value_fn V;

       gsl_histogram *h = gsl_histogram_alloc (100);
       gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);

       S.function = &sel_func;
       S.params = &lower;

       V.function = &val_func;
       V.params = 0;

       gsl_ntuple_project (h, ntuple, &V, &S);
       gsl_histogram_fprintf (stdout, h, "%f", "%f");
       gsl_histogram_free (h);
       gsl_ntuple_close (ntuple);

       return 0;
     }

     int
     sel_func (void *ntuple_data, void *params)
     {
       struct data * data = (struct data *) ntuple_data;
       double x, y, z, E2, scale;
       scale = *(double *) params;

       x = data->x;
       y = data->y;
       z = data->z;

       E2 = x * x + y * y + z * z;

       return E2 > scale;
     }

     double
     val_func (void *ntuple_data, void *params)
     {
       (void)(params); /* avoid unused parameter warning */
       struct data * data = (struct data *) ntuple_data;
       double x, y, z;

       x = data->x;
       y = data->y;
       z = data->z;

       return x * x + y * y + z * z;
     }

Fig.  %s shows the distribution of the selected events.  Note the
cut-off at the lower bound.

[gsl-ref-figures/ntuple]
Figure: Distribution of selected events


File: gsl-ref.info,  Node: References and Further Reading<19>,  Prev: Examples<19>,  Up: N-tuples

26.9 References and Further Reading
===================================

Further information on the use of ntuples can be found in the
documentation for the CERN packages PAW and HBOOK (available online).


File: gsl-ref.info,  Node: Monte Carlo Integration,  Next: Simulated Annealing,  Prev: N-tuples,  Up: Top

27 Monte Carlo Integration
**************************

This chapter describes routines for multidimensional Monte Carlo
integration.  These include the traditional Monte Carlo method and
adaptive algorithms such as VEGAS and MISER which use importance
sampling and stratified sampling techniques.  Each algorithm computes an
estimate of a multidimensional definite integral of the form,

     I = \int_{x_l}^{x_u} dx \int_{y_l}^{y_u} dy ... f(x, y, ...)

over a hypercubic region ((x_l,x_u), (y_l,y_u), ...) using a fixed
number of function calls.  The routines also provide a statistical
estimate of the error on the result.  This error estimate should be
taken as a guide rather than as a strict error bound—random sampling of
the region may not uncover all the important features of the function,
resulting in an underestimate of the error.

The functions are defined in separate header files for each routine,
‘gsl_monte_plain.h’, ‘gsl_monte_miser.h’ and ‘gsl_monte_vegas.h’.

* Menu:

* Interface::
* PLAIN Monte Carlo::
* MISER::
* VEGAS::
* Examples: Examples<20>.
* References and Further Reading: References and Further Reading<20>.


File: gsl-ref.info,  Node: Interface,  Next: PLAIN Monte Carlo,  Up: Monte Carlo Integration

27.1 Interface
==============

All of the Monte Carlo integration routines use the same general form of
interface.  There is an allocator to allocate memory for control
variables and workspace, a routine to initialize those control
variables, the integrator itself, and a function to free the space when
done.

Each integration function requires a random number generator to be
supplied, and returns an estimate of the integral and its standard
deviation.  The accuracy of the result is determined by the number of
function calls specified by the user.  If a known level of accuracy is
required this can be achieved by calling the integrator several times
and averaging the individual results until the desired accuracy is
obtained.

Random sample points used within the Monte Carlo routines are always
chosen strictly within the integration region, so that endpoint
singularities are automatically avoided.

The function to be integrated has its own datatype, defined in the
header file ‘gsl_monte.h’.

 -- Type: gsl_monte_function

     This data type defines a general function with parameters for Monte
     Carlo integration.

     ‘double (* f) (double * x, size_t dim, void * params)’           this function should return the value
                                                                      f(x,params) for the argument ‘x’ and
                                                                      parameters ‘params’, where ‘x’ is an array
                                                                      of size ‘dim’ giving the coordinates of the
                                                                      point where the function is to be evaluated.
                                                                      
                                                                      
     ‘size_t dim’                                                     the number of dimensions for ‘x’.
                                                                      
                                                                      
     ‘void * params’                                                  a pointer to the parameters of the function.
                                                                      

Here is an example for a quadratic function in two dimensions,

     f(x,y) = a x^2 + b x y + c y^2

with a = 3, b = 2, c = 1.  The following code defines a *note
gsl_monte_function: 935. ‘F’ which you could pass to an integrator:

     struct my_f_params { double a; double b; double c; };

     double
     my_f (double x[], size_t dim, void * p) {
        struct my_f_params * fp = (struct my_f_params *)p;

        if (dim != 2)
           {
             fprintf (stderr, "error: dim != 2");
             abort ();
           }

        return  fp->a * x[0] * x[0]
                  + fp->b * x[0] * x[1]
                    + fp->c * x[1] * x[1];
     }

     gsl_monte_function F;
     struct my_f_params params = { 3.0, 2.0, 1.0 };

     F.f = &my_f;
     F.dim = 2;
     F.params = &params;

The function f(x) can be evaluated using the following macro:

     #define GSL_MONTE_FN_EVAL(F,x)
         (*((F)->f))(x,(F)->dim,(F)->params)


File: gsl-ref.info,  Node: PLAIN Monte Carlo,  Next: MISER,  Prev: Interface,  Up: Monte Carlo Integration

27.2 PLAIN Monte Carlo
======================

The plain Monte Carlo algorithm samples points randomly from the
integration region to estimate the integral and its error.  Using this
algorithm the estimate of the integral E(f; N) for N randomly
distributed points x_i is given by,

     E(f; N) = =  V <f> = (V / N) \sum_i^N f(x_i)

where V is the volume of the integration region.  The error on this
estimate \sigma(E;N) is calculated from the estimated variance of the
mean,

     \sigma^2 (E; N) = (V^2 / N^2) \sum_i^N (f(x_i) -  <f>)^2.

For large N this variance decreases asymptotically as \Var(f)/N, where
\Var(f) is the true variance of the function over the integration
region.  The error estimate itself should decrease as
\sigma(f)/\sqrt{N}.  The familiar law of errors decreasing as 1/\sqrt{N}
applies—to reduce the error by a factor of 10 requires a 100-fold
increase in the number of sample points.

The functions described in this section are declared in the header file
‘gsl_monte_plain.h’.

 -- Type: gsl_monte_plain_state

     This is a workspace for plain Monte Carlo integration

 -- Function: *note gsl_monte_plain_state: 937. *gsl_monte_plain_alloc
          (size_t dim)

     This function allocates and initializes a workspace for Monte Carlo
     integration in *note dim: 938. dimensions.

 -- Function: int gsl_monte_plain_init (gsl_monte_plain_state *s)

     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_plain_integrate (gsl_monte_function *f,
          const double xl[], const double xu[], size_t dim, size_t
          calls, gsl_rng *r, gsl_monte_plain_state *s, double *result,
          double *abserr)

     This routines uses the plain Monte Carlo algorithm to integrate the
     function *note f: 93a. over the *note dim: 93a.-dimensional
     hypercubic region defined by the lower and upper limits in the
     arrays *note xl: 93a. and *note xu: 93a, each of size *note dim:
     93a.  The integration uses a fixed number of function calls *note
     calls: 93a, and obtains random sampling points using the random
     number generator *note r: 93a.  A previously allocated workspace
     *note s: 93a. must be supplied.  The result of the integration is
     returned in *note result: 93a, with an estimated absolute error
     *note abserr: 93a.

 -- Function: void gsl_monte_plain_free (gsl_monte_plain_state *s)

     This function frees the memory associated with the integrator state
     *note s: 93b.


File: gsl-ref.info,  Node: MISER,  Next: VEGAS,  Prev: PLAIN Monte Carlo,  Up: Monte Carlo Integration

27.3 MISER
==========

The MISER algorithm of Press and Farrar is based on recursive stratified
sampling.  This technique aims to reduce the overall integration error
by concentrating integration points in the regions of highest variance.

The idea of stratified sampling begins with the observation that for two
disjoint regions a and b with Monte Carlo estimates of the integral
E_a(f) and E_b(f) and variances \sigma_a^2(f) and \sigma_b^2(f), the
variance \Var(f) of the combined estimate E(f) = {1\over 2} (E_a(f) +
E_b(f)) is given by,

     \Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).

It can be shown that this variance is minimized by distributing the
points such that,

     N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).

Hence the smallest error estimate is obtained by allocating sample
points in proportion to the standard deviation of the function in each
sub-region.

The MISER algorithm proceeds by bisecting the integration region along
one coordinate axis to give two sub-regions at each step.  The direction
is chosen by examining all d possible bisections and selecting the one
which will minimize the combined variance of the two sub-regions.  The
variance in the sub-regions is estimated by sampling with a fraction of
the total number of points available to the current step.  The same
procedure is then repeated recursively for each of the two half-spaces
from the best bisection.  The remaining sample points are allocated to
the sub-regions using the formula for N_a and N_b.  This recursive
allocation of integration points continues down to a user-specified
depth where each sub-region is integrated using a plain Monte Carlo
estimate.  These individual values and their error estimates are then
combined upwards to give an overall result and an estimate of its error.

The functions described in this section are declared in the header file
‘gsl_monte_miser.h’.

 -- Type: gsl_monte_miser_state

     This workspace is used for MISER Monte Carlo integration

 -- Function: *note gsl_monte_miser_state: 93d. *gsl_monte_miser_alloc
          (size_t dim)

     This function allocates and initializes a workspace for Monte Carlo
     integration in *note dim: 93e. dimensions.  The workspace is used
     to maintain the state of the integration.

 -- Function: int gsl_monte_miser_init (gsl_monte_miser_state *s)

     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_miser_integrate (gsl_monte_function *f,
          const double xl[], const double xu[], size_t dim, size_t
          calls, gsl_rng *r, gsl_monte_miser_state *s, double *result,
          double *abserr)

     This routines uses the MISER Monte Carlo algorithm to integrate the
     function *note f: 940. over the *note dim: 940.-dimensional
     hypercubic region defined by the lower and upper limits in the
     arrays *note xl: 940. and *note xu: 940, each of size *note dim:
     940.  The integration uses a fixed number of function calls *note
     calls: 940, and obtains random sampling points using the random
     number generator *note r: 940.  A previously allocated workspace
     *note s: 940. must be supplied.  The result of the integration is
     returned in *note result: 940, with an estimated absolute error
     *note abserr: 940.

 -- Function: void gsl_monte_miser_free (gsl_monte_miser_state *s)

     This function frees the memory associated with the integrator state
     *note s: 941.

The MISER algorithm has several configurable parameters which can be
changed using the following two functions (1).

 -- Function: void gsl_monte_miser_params_get (const
          gsl_monte_miser_state *s, gsl_monte_miser_params *params)

     This function copies the parameters of the integrator state into
     the user-supplied *note params: 942. structure.

 -- Function: void gsl_monte_miser_params_set (gsl_monte_miser_state *s,
          const gsl_monte_miser_params *params)

     This function sets the integrator parameters based on values
     provided in the *note params: 943. structure.

Typically the values of the parameters are first read using *note
gsl_monte_miser_params_get(): 942, the necessary changes are made to the
fields of the ‘params’ structure, and the values are copied back into
the integrator state using *note gsl_monte_miser_params_set(): 943.  The
functions use the *note gsl_monte_miser_params: 944. structure which
contains the following fields:

 -- Type: gsl_monte_miser_params

      -- Variable: double estimate_frac

          This parameter specifies the fraction of the currently
          available number of function calls which are allocated to
          estimating the variance at each recursive step.  The default
          value is 0.1.

      -- Variable: size_t min_calls

          This parameter specifies the minimum number of function calls
          required for each estimate of the variance.  If the number of
          function calls allocated to the estimate using *note
          estimate_frac: 945. falls below *note min_calls: 946. then
          *note min_calls: 946. are used instead.  This ensures that
          each estimate maintains a reasonable level of accuracy.  The
          default value of *note min_calls: 946. is ‘16 * dim’.

      -- Variable: size_t min_calls_per_bisection

          This parameter specifies the minimum number of function calls
          required to proceed with a bisection step.  When a recursive
          step has fewer calls available than *note
          min_calls_per_bisection: 947. it performs a plain Monte Carlo
          estimate of the current sub-region and terminates its branch
          of the recursion.  The default value of this parameter is ‘32
          * min_calls’.

      -- Variable: double alpha

          This parameter controls how the estimated variances for the
          two sub-regions of a bisection are combined when allocating
          points.  With recursive sampling the overall variance should
          scale better than 1/N, since the values from the sub-regions
          will be obtained using a procedure which explicitly minimizes
          their variance.  To accommodate this behavior the MISER
          algorithm allows the total variance to depend on a scaling
          parameter \alpha,

               \Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.

          The authors of the original paper describing MISER recommend
          the value \alpha = 2 as a good choice, obtained from numerical
          experiments, and this is used as the default value in this
          implementation.

      -- Variable: double dither

          This parameter introduces a random fractional variation of
          size *note dither: 949. into each bisection, which can be used
          to break the symmetry of integrands which are concentrated
          near the exact center of the hypercubic integration region.
          The default value of dither is zero, so no variation is
          introduced.  If needed, a typical value of *note dither: 949.
          is 0.1.

   ---------- Footnotes ----------

   (1) (1) The previous method of accessing these fields directly
through the *note gsl_monte_miser_state: 93d. struct is now deprecated.


File: gsl-ref.info,  Node: VEGAS,  Next: Examples<20>,  Prev: MISER,  Up: Monte Carlo Integration

27.4 VEGAS
==========

The VEGAS algorithm of Lepage is based on importance sampling.  It
samples points from the probability distribution described by the
function |f|, so that the points are concentrated in the regions that
make the largest contribution to the integral.

In general, if the Monte Carlo integral of f is sampled with points
distributed according to a probability distribution described by the
function g, we obtain an estimate E_g(f; N),

     E_g(f; N) = E(f/g; N)

with a corresponding variance,

     \Var_g(f; N) = \Var(f/g; N)

If the probability distribution is chosen as g = |f|/I(|f|) then it can
be shown that the variance V_g(f; N) vanishes, and the error in the
estimate will be zero.  In practice it is not possible to sample from
the exact distribution g for an arbitrary function, so importance
sampling algorithms aim to produce efficient approximations to the
desired distribution.

The VEGAS algorithm approximates the exact distribution by making a
number of passes over the integration region while histogramming the
function f.  Each histogram is used to define a sampling distribution
for the next pass.  Asymptotically this procedure converges to the
desired distribution.  In order to avoid the number of histogram bins
growing like K^d the probability distribution is approximated by a
separable function: g(x_1, x_2, \ldots) = g_1(x_1) g_2(x_2) \ldots so
that the number of bins required is only Kd.  This is equivalent to
locating the peaks of the function from the projections of the integrand
onto the coordinate axes.  The efficiency of VEGAS depends on the
validity of this assumption.  It is most efficient when the peaks of the
integrand are well-localized.  If an integrand can be rewritten in a
form which is approximately separable this will increase the efficiency
of integration with VEGAS.

VEGAS incorporates a number of additional features, and combines both
stratified sampling and importance sampling.  The integration region is
divided into a number of “boxes”, with each box getting a fixed number
of points (the goal is 2).  Each box can then have a fractional number
of bins, but if the ratio of bins-per-box is less than two, Vegas
switches to a kind variance reduction (rather than importance sampling).

 -- Type: gsl_monte_vegas_state

     This workspace is used for VEGAS Monte Carlo integration

 -- Function: *note gsl_monte_vegas_state: 94b. *gsl_monte_vegas_alloc
          (size_t dim)

     This function allocates and initializes a workspace for Monte Carlo
     integration in *note dim: 94c. dimensions.  The workspace is used
     to maintain the state of the integration.

 -- Function: int gsl_monte_vegas_init (gsl_monte_vegas_state *s)

     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_vegas_integrate (gsl_monte_function *f,
          double xl[], double xu[], size_t dim, size_t calls, gsl_rng
          *r, gsl_monte_vegas_state *s, double *result, double *abserr)

     This routines uses the VEGAS Monte Carlo algorithm to integrate the
     function *note f: 94e. over the *note dim: 94e.-dimensional
     hypercubic region defined by the lower and upper limits in the
     arrays *note xl: 94e. and *note xu: 94e, each of size *note dim:
     94e.  The integration uses a fixed number of function calls *note
     calls: 94e, and obtains random sampling points using the random
     number generator *note r: 94e.  A previously allocated workspace
     *note s: 94e. must be supplied.  The result of the integration is
     returned in *note result: 94e, with an estimated absolute error
     *note abserr: 94e.  The result and its error estimate are based on
     a weighted average of independent samples.  The chi-squared per
     degree of freedom for the weighted average is returned via the
     state struct component, ‘s->chisq’, and must be consistent with 1
     for the weighted average to be reliable.

 -- Function: void gsl_monte_vegas_free (gsl_monte_vegas_state *s)

     This function frees the memory associated with the integrator state
     *note s: 94f.

The VEGAS algorithm computes a number of independent estimates of the
integral internally, according to the ‘iterations’ parameter described
below, and returns their weighted average.  Random sampling of the
integrand can occasionally produce an estimate where the error is zero,
particularly if the function is constant in some regions.  An estimate
with zero error causes the weighted average to break down and must be
handled separately.  In the original Fortran implementations of VEGAS
the error estimate is made non-zero by substituting a small value
(typically ‘1e-30’).  The implementation in GSL differs from this and
avoids the use of an arbitrary constant—it either assigns the value a
weight which is the average weight of the preceding estimates or
discards it according to the following procedure,

   * current estimate has zero error, weighted average has finite error

     The current estimate is assigned a weight which is the average
     weight of the preceding estimates.

   * current estimate has finite error, previous estimates had zero
     error

     The previous estimates are discarded and the weighted averaging
     procedure begins with the current estimate.

   * current estimate has zero error, previous estimates had zero error

     The estimates are averaged using the arithmetic mean, but no error
     is computed.

The convergence of the algorithm can be tested using the overall
chi-squared value of the results, which is available from the following
function:

 -- Function: double gsl_monte_vegas_chisq (const gsl_monte_vegas_state
          *s)

     This function returns the chi-squared per degree of freedom for the
     weighted estimate of the integral.  The returned value should be
     close to 1.  A value which differs significantly from 1 indicates
     that the values from different iterations are inconsistent.  In
     this case the weighted error will be under-estimated, and further
     iterations of the algorithm are needed to obtain reliable results.

 -- Function: void gsl_monte_vegas_runval (const gsl_monte_vegas_state
          *s, double *result, double *sigma)

     This function returns the raw (unaveraged) values of the integral
     *note result: 951. and its error *note sigma: 951. from the most
     recent iteration of the algorithm.

The VEGAS algorithm is highly configurable.  Several parameters can be
changed using the following two functions.

 -- Function: void gsl_monte_vegas_params_get (const
          gsl_monte_vegas_state *s, gsl_monte_vegas_params *params)

     This function copies the parameters of the integrator state into
     the user-supplied *note params: 952. structure.

 -- Function: void gsl_monte_vegas_params_set (gsl_monte_vegas_state *s,
          const gsl_monte_vegas_params *params)

     This function sets the integrator parameters based on values
     provided in the *note params: 953. structure.

Typically the values of the parameters are first read using *note
gsl_monte_vegas_params_get(): 952, the necessary changes are made to the
fields of the ‘params’ structure, and the values are copied back into
the integrator state using *note gsl_monte_vegas_params_set(): 953.  The
functions use the *note gsl_monte_vegas_params: 954. structure which
contains the following fields:

 -- Type: gsl_monte_vegas_params

      -- Variable: double alpha

          The parameter *note alpha: 955. controls the stiffness of the
          rebinning algorithm.  It is typically set between one and two.
          A value of zero prevents rebinning of the grid.  The default
          value is 1.5.

      -- Variable: size_t iterations

          The number of iterations to perform for each call to the
          routine.  The default value is 5 iterations.

      -- Variable: int stage

          Setting this determines the `stage' of the calculation.
          Normally, ‘stage = 0’ which begins with a new uniform grid and
          empty weighted average.  Calling VEGAS with ‘stage = 1’
          retains the grid from the previous run but discards the
          weighted average, so that one can “tune” the grid using a
          relatively small number of points and then do a large run with
          ‘stage = 1’ on the optimized grid.  Setting ‘stage = 2’ keeps
          the grid and the weighted average from the previous run, but
          may increase (or decrease) the number of histogram bins in the
          grid depending on the number of calls available.  Choosing
          ‘stage = 3’ enters at the main loop, so that nothing is
          changed, and is equivalent to performing additional iterations
          in a previous call.

      -- Variable: int mode

          The possible choices are ‘GSL_VEGAS_MODE_IMPORTANCE’,
          ‘GSL_VEGAS_MODE_STRATIFIED’, ‘GSL_VEGAS_MODE_IMPORTANCE_ONLY’.
          This determines whether VEGAS will use importance sampling or
          stratified sampling, or whether it can pick on its own.  In
          low dimensions VEGAS uses strict stratified sampling (more
          precisely, stratified sampling is chosen if there are fewer
          than 2 bins per box).

      -- Variable: int verbose
      -- Variable: FILE *ostream

          These parameters set the level of information printed by
          VEGAS. All information is written to the stream *note ostream:
          95a.  The default setting of *note verbose: 959. is ‘-1’,
          which turns off all output.  A *note verbose: 959. value of
          ‘0’ prints summary information about the weighted average and
          final result, while a value of ‘1’ also displays the grid
          coordinates.  A value of ‘2’ prints information from the
          rebinning procedure for each iteration.

The above fields and the ‘chisq’ value can also be accessed directly in
the *note gsl_monte_vegas_state: 94b. but such use is deprecated.


File: gsl-ref.info,  Node: Examples<20>,  Next: References and Further Reading<20>,  Prev: VEGAS,  Up: Monte Carlo Integration

27.5 Examples
=============

The example program below uses the Monte Carlo routines to estimate the
value of the following 3-dimensional integral from the theory of random
walks,

     I = \int_{-pi}^{+pi} {dk_x/(2 pi)}
         \int_{-pi}^{+pi} {dk_y/(2 pi)}
         \int_{-pi}^{+pi} {dk_z/(2 pi)}
          1 / (1 - cos(k_x)cos(k_y)cos(k_z)).

The analytic value of this integral can be shown to be I =
\Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859....  The integral gives
the mean time spent at the origin by a random walk on a body-centered
cubic lattice in three dimensions.

For simplicity we will compute the integral over the region (0,0,0) to
(\pi,\pi,\pi) and multiply by 8 to obtain the full result.  The integral
is slowly varying in the middle of the region but has integrable
singularities at the corners (0,0,0), (0,\pi,\pi), (\pi,0,\pi) and
(\pi,\pi,0).  The Monte Carlo routines only select points which are
strictly within the integration region and so no special measures are
needed to avoid these singularities.

     #include <stdlib.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_monte.h>
     #include <gsl/gsl_monte_plain.h>
     #include <gsl/gsl_monte_miser.h>
     #include <gsl/gsl_monte_vegas.h>

     /* Computation of the integral,

           I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))

        over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
        is Gamma(1/4)^4/(4 pi^3).  This example is taken from
        C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
        Volume 1", Section 1.1, p21, which cites the original
        paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
        1800 (1977) */

     /* For simplicity we compute the integral over the region
        (0,0,0) -> (pi,pi,pi) and multiply by 8 */

     double exact = 1.3932039296856768591842462603255;

     double
     g (double *k, size_t dim, void *params)
     {
       (void)(dim); /* avoid unused parameter warnings */
       (void)(params);
       double A = 1.0 / (M_PI * M_PI * M_PI);
       return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
     }

     void
     display_results (char *title, double result, double error)
     {
       printf ("%s ==================\n", title);
       printf ("result = % .6f\n", result);
       printf ("sigma  = % .6f\n", error);
       printf ("exact  = % .6f\n", exact);
       printf ("error  = % .6f = %.2g sigma\n", result - exact,
               fabs (result - exact) / error);
     }

     int
     main (void)
     {
       double res, err;

       double xl[3] = { 0, 0, 0 };
       double xu[3] = { M_PI, M_PI, M_PI };

       const gsl_rng_type *T;
       gsl_rng *r;

       gsl_monte_function G = { &g, 3, 0 };

       size_t calls = 500000;

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       {
         gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
         gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
                                    &res, &err);
         gsl_monte_plain_free (s);

         display_results ("plain", res, err);
       }

       {
         gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
         gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
                                    &res, &err);
         gsl_monte_miser_free (s);

         display_results ("miser", res, err);
       }

       {
         gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);

         gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
                                    &res, &err);
         display_results ("vegas warm-up", res, err);

         printf ("converging...\n");

         do
           {
             gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
                                        &res, &err);
             printf ("result = % .6f sigma = % .6f "
                     "chisq/dof = %.1f\n", res, err, gsl_monte_vegas_chisq (s));
           }
         while (fabs (gsl_monte_vegas_chisq (s) - 1.0) > 0.5);

         display_results ("vegas final", res, err);

         gsl_monte_vegas_free (s);
       }

       gsl_rng_free (r);

       return 0;
     }

With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 1%.  The estimated error ‘sigma’ is roughly
consistent with the actual error–the computed result differs from the
true result by about 1.4 standard deviations:

     plain ==================
     result =  1.412209
     sigma  =  0.013436
     exact  =  1.393204
     error  =  0.019005 = 1.4 sigma

The MISER algorithm reduces the error by a factor of four, and also
correctly estimates the error:

     miser ==================
     result =  1.391322
     sigma  =  0.003461
     exact  =  1.393204
     error  = -0.001882 = 0.54 sigma

In the case of the VEGAS algorithm the program uses an initial warm-up
run of 10,000 function calls to prepare, or “warm up”, the grid.  This
is followed by a main run with five iterations of 100,000 function
calls.  The chi-squared per degree of freedom for the five iterations
are checked for consistency with 1, and the run is repeated if the
results have not converged.  In this case the estimates are consistent
on the first pass:

     vegas warm-up ==================
     result =  1.392673
     sigma  =  0.003410
     exact  =  1.393204
     error  = -0.000531 = 0.16 sigma
     converging...
     result =  1.393281 sigma =  0.000362 chisq/dof = 1.5
     vegas final ==================
     result =  1.393281
     sigma  =  0.000362
     exact  =  1.393204
     error  =  0.000077 = 0.21 sigma

If the value of ‘chisq’ had differed significantly from 1 it would
indicate inconsistent results, with a correspondingly underestimated
error.  The final estimate from VEGAS (using a similar number of
function calls) is significantly more accurate than the other two
algorithms.


File: gsl-ref.info,  Node: References and Further Reading<20>,  Prev: Examples<20>,  Up: Monte Carlo Integration

27.6 References and Further Reading
===================================

The MISER algorithm is described in the following article by Press and
Farrar,

   * W.H. Press, G.R. Farrar, `Recursive Stratified Sampling for
     Multidimensional Monte Carlo Integration', Computers in Physics, v4
     (1990), pp190–195.

The VEGAS algorithm is described in the following papers,

   * G.P. Lepage, `A New Algorithm for Adaptive Multidimensional
     Integration', Journal of Computational Physics 27, 192–203, (1978)

   * G.P. Lepage, `VEGAS: An Adaptive Multi-dimensional Integration
     Program', Cornell preprint CLNS 80-447, March 1980


File: gsl-ref.info,  Node: Simulated Annealing,  Next: Ordinary Differential Equations,  Prev: Monte Carlo Integration,  Up: Top

28 Simulated Annealing
**********************

Stochastic search techniques are used when the structure of a space is
not well understood or is not smooth, so that techniques like Newton’s
method (which requires calculating Jacobian derivative matrices) cannot
be used.  In particular, these techniques are frequently used to solve
combinatorial optimization problems, such as the traveling salesman
problem.

The goal is to find a point in the space at which a real valued `energy
function' (or `cost function') is minimized.  Simulated annealing is a
minimization technique which has given good results in avoiding local
minima; it is based on the idea of taking a random walk through the
space at successively lower temperatures, where the probability of
taking a step is given by a Boltzmann distribution.

The functions described in this chapter are declared in the header file
‘gsl_siman.h’.

* Menu:

* Simulated Annealing algorithm::
* Simulated Annealing functions::
* Examples: Examples<21>.
* References and Further Reading: References and Further Reading<21>.


File: gsl-ref.info,  Node: Simulated Annealing algorithm,  Next: Simulated Annealing functions,  Up: Simulated Annealing

28.1 Simulated Annealing algorithm
==================================

The simulated annealing algorithm takes random walks through the problem
space, looking for points with low energies; in these random walks, the
probability of taking a step is determined by the Boltzmann
distribution,

     p = e^{-(E_{i+1} - E_i)/(kT)}

if E_{i+1} > E_i, and p = 1 when E_{i+1} \le E_i.

In other words, a step will occur if the new energy is lower.  If the
new energy is higher, the transition can still occur, and its likelihood
is proportional to the temperature T and inversely proportional to the
energy difference E_{i+1} - E_i.

The temperature T is initially set to a high value, and a random walk is
carried out at that temperature.  Then the temperature is lowered very
slightly according to a `cooling schedule', for example: T \rightarrow
T/\mu_T where \mu_T is slightly greater than 1.

The slight probability of taking a step that gives higher energy is what
allows simulated annealing to frequently get out of local minima.


File: gsl-ref.info,  Node: Simulated Annealing functions,  Next: Examples<21>,  Prev: Simulated Annealing algorithm,  Up: Simulated Annealing

28.2 Simulated Annealing functions
==================================

 -- Function: void gsl_siman_solve (const gsl_rng *r, void *x0_p,
          gsl_siman_Efunc_t Ef, gsl_siman_step_t take_step,
          gsl_siman_metric_t distance, gsl_siman_print_t print_position,
          gsl_siman_copy_t copyfunc, gsl_siman_copy_construct_t
          copy_constructor, gsl_siman_destroy_t destructor, size_t
          element_size, gsl_siman_params_t params)

     This function performs a simulated annealing search through a given
     space.  The space is specified by providing the functions *note Ef:
     961. and *note distance: 961.  The simulated annealing steps are
     generated using the random number generator *note r: 961. and the
     function *note take_step: 961.

     The starting configuration of the system should be given by *note
     x0_p: 961.  The routine offers two modes for updating
     configurations, a fixed-size mode and a variable-size mode.  In the
     fixed-size mode the configuration is stored as a single block of
     memory of size *note element_size: 961.  Copies of this
     configuration are created, copied and destroyed internally using
     the standard library functions ‘malloc()’, ‘memcpy()’ and ‘free()’.
     The function pointers *note copyfunc: 961, *note copy_constructor:
     961. and *note destructor: 961. should be null pointers in
     fixed-size mode.  In the variable-size mode the functions *note
     copyfunc: 961, *note copy_constructor: 961. and *note destructor:
     961. are used to create, copy and destroy configurations
     internally.  The variable *note element_size: 961. should be zero
     in the variable-size mode.

     The *note params: 961. structure (described below) controls the run
     by providing the temperature schedule and other tunable parameters
     to the algorithm.

     On exit the best result achieved during the search is placed in
     *note x0_p: 961.  If the annealing process has been successful this
     should be a good approximation to the optimal point in the space.

     If the function pointer *note print_position: 961. is not null, a
     debugging log will be printed to ‘stdout’ with the following
     columns:

          #-iter  #-evals  temperature  position  energy  best_energy

     and the output of the function *note print_position: 961. itself.
     If *note print_position: 961. is null then no information is
     printed.

The simulated annealing routines require several user-specified
functions to define the configuration space and energy function.  The
prototypes for these functions are given below.

 -- Type: gsl_siman_Efunc_t

     This function type should return the energy of a configuration
     ‘xp’:

          double (*gsl_siman_Efunc_t) (void *xp)

 -- Type: gsl_siman_step_t

     This function type should modify the configuration ‘xp’ using a
     random step taken from the generator ‘r’, up to a maximum distance
     of ‘step_size’:

          void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
                                    double step_size)

 -- Type: gsl_siman_metric_t

     This function type should return the distance between two
     configurations ‘xp’ and ‘yp’:

          double (*gsl_siman_metric_t) (void *xp, void *yp)

 -- Type: gsl_siman_print_t

     This function type should print the contents of the configuration
     ‘xp’:

          void (*gsl_siman_print_t) (void *xp)

 -- Type: gsl_siman_copy_t

     This function type should copy the configuration ‘source’ into
     ‘dest’:

          void (*gsl_siman_copy_t) (void *source, void *dest)

 -- Type: gsl_siman_copy_construct_t

     This function type should create a new copy of the configuration
     ‘xp’:

          void * (*gsl_siman_copy_construct_t) (void *xp)

 -- Type: gsl_siman_destroy_t

     This function type should destroy the configuration ‘xp’, freeing
     its memory:

          void (*gsl_siman_destroy_t) (void *xp)

 -- Type: gsl_siman_params_t

     These are the parameters that control a run of *note
     gsl_siman_solve(): 961.  This structure contains all the
     information needed to control the search, beyond the energy
     function, the step function and the initial guess.

     ‘int n_tries’                                 The number of points to try for each step.
                                                   
                                                   
     ‘int iters_fixed_T’                           The number of iterations at each temperature.
                                                   
                                                   
     ‘double step_size’                            The maximum step size in the random walk.
                                                   
                                                   
     ‘double k, t_initial, mu_t, t_min’            The parameters of the Boltzmann distribution and cooling
                                                   schedule.
                                                   


File: gsl-ref.info,  Node: Examples<21>,  Next: References and Further Reading<21>,  Prev: Simulated Annealing functions,  Up: Simulated Annealing

28.3 Examples
=============

The simulated annealing package is clumsy, and it has to be because it
is written in C, for C callers, and tries to be polymorphic at the same
time.  But here we provide some examples which can be pasted into your
application with little change and should make things easier.

* Menu:

* Trivial example::
* Traveling Salesman Problem::


File: gsl-ref.info,  Node: Trivial example,  Next: Traveling Salesman Problem,  Up: Examples<21>

28.3.1 Trivial example
----------------------

The first example, in one dimensional Cartesian space, sets up an energy
function which is a damped sine wave; this has many local minima, but
only one global minimum, somewhere between 1.0 and 1.5.  The initial
guess given is 15.5, which is several local minima away from the global
minimum.

     #include <math.h>
     #include <stdlib.h>
     #include <string.h>
     #include <gsl/gsl_siman.h>

     /* set up parameters for this simulated annealing run */

     /* how many points do we try before stepping */
     #define N_TRIES 200

     /* how many iterations for each T? */
     #define ITERS_FIXED_T 1000

     /* max step size in random walk */
     #define STEP_SIZE 1.0

     /* Boltzmann constant */
     #define K 1.0

     /* initial temperature */
     #define T_INITIAL 0.008

     /* damping factor for temperature */
     #define MU_T 1.003
     #define T_MIN 2.0e-6

     gsl_siman_params_t params
       = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
          K, T_INITIAL, MU_T, T_MIN};

     /* now some functions to test in one dimension */
     double E1(void *xp)
     {
       double x = * ((double *) xp);

       return exp(-pow((x-1.0),2.0))*sin(8*x);
     }

     double M1(void *xp, void *yp)
     {
       double x = *((double *) xp);
       double y = *((double *) yp);

       return fabs(x - y);
     }

     void S1(const gsl_rng * r, void *xp, double step_size)
     {
       double old_x = *((double *) xp);
       double new_x;

       double u = gsl_rng_uniform(r);
       new_x = u * 2 * step_size - step_size + old_x;

       memcpy(xp, &new_x, sizeof(new_x));
     }

     void P1(void *xp)
     {
       printf ("%12g", *((double *) xp));
     }

     int
     main(void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       double x_initial = 15.5;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc(T);

       gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
                       NULL, NULL, NULL,
                       sizeof(double), params);

       gsl_rng_free (r);
       return 0;
     }

Fig.  %s is generated by running ‘siman_test’ in the following way:

     $ ./siman_test | awk '!/^#/ {print $1, $4}'
      | graph -y 1.34 1.4 -W0 -X generation -Y position
      | plot -Tps > siman-test.eps

Fig.  %s is generated by running ‘siman_test’ in the following way:

     $ ./siman_test | awk '!/^#/ {print $1, $5}'
      | graph -y -0.88 -0.83 -W0 -X generation -Y energy
      | plot -Tps > siman-energy.eps

[gsl-ref-figures/siman-test]
Figure: Example of a simulated annealing run: at higher temperatures
(early in the plot) you see that the solution can fluctuate, but at
lower temperatures it converges.

[gsl-ref-figures/siman-energy]
Figure: Simulated annealing energy vs generation


File: gsl-ref.info,  Node: Traveling Salesman Problem,  Prev: Trivial example,  Up: Examples<21>

28.3.2 Traveling Salesman Problem
---------------------------------

The TSP (`Traveling Salesman Problem') is the classic combinatorial
optimization problem.  I have provided a very simple version of it,
based on the coordinates of twelve cities in the southwestern United
States.  This should maybe be called the `Flying Salesman Problem',
since I am using the great-circle distance between cities, rather than
the driving distance.  Also: I assume the earth is a sphere, so I don’t
use geoid distances.

The *note gsl_siman_solve(): 961. routine finds a route which is 3490.62
Kilometers long; this is confirmed by an exhaustive search of all
possible routes with the same initial city.

The full code is given below.

     /* siman/siman_tsp.c
      *
      * Copyright (C) 1996, 1997, 1998, 1999, 2000 Mark Galassi
      *
      * This program is free software; you can redistribute it and/or modify
      * it under the terms of the GNU General Public License as published by
      * the Free Software Foundation; either version 3 of the License, or (at
      * your option) any later version.
      *
      * This program is distributed in the hope that it will be useful, but
      * WITHOUT ANY WARRANTY; without even the implied warranty of
      * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
      * General Public License for more details.
      *
      * You should have received a copy of the GNU General Public License
      * along with this program; if not, write to the Free Software
      * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
      */

     #include <config.h>
     #include <math.h>
     #include <string.h>
     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_siman.h>
     #include <gsl/gsl_ieee_utils.h>

     /* set up parameters for this simulated annealing run */

     #define N_TRIES 200             /* how many points do we try before stepping */
     #define ITERS_FIXED_T 2000      /* how many iterations for each T? */
     #define STEP_SIZE 1.0           /* max step size in random walk */
     #define K 1.0                   /* Boltzmann constant */
     #define T_INITIAL 5000.0        /* initial temperature */
     #define MU_T 1.002              /* damping factor for temperature */
     #define T_MIN 5.0e-1

     gsl_siman_params_t params = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
                                  K, T_INITIAL, MU_T, T_MIN};

     struct s_tsp_city {
       const char * name;
       double lat, longitude;        /* coordinates */
     };
     typedef struct s_tsp_city Stsp_city;

     void prepare_distance_matrix(void);
     void exhaustive_search(void);
     void print_distance_matrix(void);
     double city_distance(Stsp_city c1, Stsp_city c2);
     double Etsp(void *xp);
     double Mtsp(void *xp, void *yp);
     void Stsp(const gsl_rng * r, void *xp, double step_size);
     void Ptsp(void *xp);

     /* in this table, latitude and longitude are obtained from the US
        Census Bureau, at http://www.census.gov/cgi-bin/gazetteer */

     Stsp_city cities[] = {{"Santa Fe",    35.68,   105.95},
                           {"Phoenix",     33.54,   112.07},
                           {"Albuquerque", 35.12,   106.62},
                           {"Clovis",      34.41,   103.20},
                           {"Durango",     37.29,   107.87},
                           {"Dallas",      32.79,    96.77},
                           {"Tesuque",     35.77,   105.92},
                           {"Grants",      35.15,   107.84},
                           {"Los Alamos",  35.89,   106.28},
                           {"Las Cruces",  32.34,   106.76},
                           {"Cortez",      37.35,   108.58},
                           {"Gallup",      35.52,   108.74}};

     #define N_CITIES (sizeof(cities)/sizeof(Stsp_city))

     double distance_matrix[N_CITIES][N_CITIES];

     /* distance between two cities */
     double city_distance(Stsp_city c1, Stsp_city c2)
     {
       const double earth_radius = 6375.000; /* 6000KM approximately */
       /* sin and cos of lat and long; must convert to radians */
       double sla1 = sin(c1.lat*M_PI/180), cla1 = cos(c1.lat*M_PI/180),
         slo1 = sin(c1.longitude*M_PI/180), clo1 = cos(c1.longitude*M_PI/180);
       double sla2 = sin(c2.lat*M_PI/180), cla2 = cos(c2.lat*M_PI/180),
         slo2 = sin(c2.longitude*M_PI/180), clo2 = cos(c2.longitude*M_PI/180);

       double x1 = cla1*clo1;
       double x2 = cla2*clo2;

       double y1 = cla1*slo1;
       double y2 = cla2*slo2;

       double z1 = sla1;
       double z2 = sla2;

       double dot_product = x1*x2 + y1*y2 + z1*z2;

       double angle = acos(dot_product);

       /* distance is the angle (in radians) times the earth radius */
       return angle*earth_radius;
     }

     /* energy for the travelling salesman problem */
     double Etsp(void *xp)
     {
       /* an array of N_CITIES integers describing the order */
       int *route = (int *) xp;
       double E = 0;
       unsigned int i;

       for (i = 0; i < N_CITIES; ++i) {
         /* use the distance_matrix to optimize this calculation; it had
            better be allocated!! */
         E += distance_matrix[route[i]][route[(i + 1) % N_CITIES]];
       }

       return E;
     }

     double Mtsp(void *xp, void *yp)
     {
       int *route1 = (int *) xp, *route2 = (int *) yp;
       double distance = 0;
       unsigned int i;

       for (i = 0; i < N_CITIES; ++i) {
         distance += ((route1[i] == route2[i]) ? 0 : 1);
       }

       return distance;
     }

     /* take a step through the TSP space */
     void Stsp(const gsl_rng * r, void *xp, double step_size)
     {
       int x1, x2, dummy;
       int *route = (int *) xp;

       step_size = 0 ; /* prevent warnings about unused parameter */

       /* pick the two cities to swap in the matrix; we leave the first
          city fixed */
       x1 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
       do {
         x2 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
       } while (x2 == x1);

       dummy = route[x1];
       route[x1] = route[x2];
       route[x2] = dummy;
     }

     void Ptsp(void *xp)
     {
       unsigned int i;
       int *route = (int *) xp;
       printf("  [");
       for (i = 0; i < N_CITIES; ++i) {
         printf(" %d ", route[i]);
       }
       printf("]  ");
     }

     int main(void)
     {
       int x_initial[N_CITIES];
       unsigned int i;

       const gsl_rng * r = gsl_rng_alloc (gsl_rng_env_setup()) ;

       gsl_ieee_env_setup ();

       prepare_distance_matrix();

       /* set up a trivial initial route */
       printf("# initial order of cities:\n");
       for (i = 0; i < N_CITIES; ++i) {
         printf("# \"%s\"\n", cities[i].name);
         x_initial[i] = i;
       }

       printf("# distance matrix is:\n");
       print_distance_matrix();

       printf("# initial coordinates of cities (longitude and latitude)\n");
       /* this can be plotted with */
       /* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "   " $3}' | xyplot -ps -d "xy" > c.eps */
       for (i = 0; i < N_CITIES+1; ++i) {
         printf("###initial_city_coord: %g %g \"%s\"\n",
                -cities[x_initial[i % N_CITIES]].longitude,
                cities[x_initial[i % N_CITIES]].lat,
                cities[x_initial[i % N_CITIES]].name);
       }

     /*   exhaustive_search(); */

       gsl_siman_solve(r, x_initial, Etsp, Stsp, Mtsp, Ptsp, NULL, NULL, NULL,
                       N_CITIES*sizeof(int), params);

       printf("# final order of cities:\n");
       for (i = 0; i < N_CITIES; ++i) {
         printf("# \"%s\"\n", cities[x_initial[i]].name);
       }

       printf("# final coordinates of cities (longitude and latitude)\n");
       /* this can be plotted with */
       /* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "   " $3}' | xyplot -ps -d "xy" > c.eps */
       for (i = 0; i < N_CITIES+1; ++i) {
         printf("###final_city_coord: %g %g %s\n",
                -cities[x_initial[i % N_CITIES]].longitude,
                cities[x_initial[i % N_CITIES]].lat,
                cities[x_initial[i % N_CITIES]].name);
       }

       printf("# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif /* 0 */
       fflush(stdout);

       return 0;
     }

     void prepare_distance_matrix()
     {
       unsigned int i, j;
       double dist;

       for (i = 0; i < N_CITIES; ++i) {
         for (j = 0; j < N_CITIES; ++j) {
           if (i == j) {
             dist = 0;
           } else {
             dist = city_distance(cities[i], cities[j]);
           }
           distance_matrix[i][j] = dist;
         }
       }
     }

     void print_distance_matrix()
     {
       unsigned int i, j;

       for (i = 0; i < N_CITIES; ++i) {
         printf("# ");
         for (j = 0; j < N_CITIES; ++j) {
           printf("%15.8f   ", distance_matrix[i][j]);
         }
         printf("\n");
       }
     }

     /* [only works for 12] search the entire space for solutions */
     static double best_E = 1.0e100, second_E = 1.0e100, third_E = 1.0e100;
     static int best_route[N_CITIES];
     static int second_route[N_CITIES];
     static int third_route[N_CITIES];
     static void do_all_perms(int *route, int n);

     void exhaustive_search()
     {
       static int initial_route[N_CITIES] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11};
       printf("\n# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif
       fflush(stdout);
       do_all_perms(initial_route, 1);
       printf("\n# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif /* 0 */
       fflush(stdout);

       printf("# exhaustive best route: ");
       Ptsp(best_route);
       printf("\n# its energy is: %g\n", best_E);

       printf("# exhaustive second_best route: ");
       Ptsp(second_route);
       printf("\n# its energy is: %g\n", second_E);

       printf("# exhaustive third_best route: ");
       Ptsp(third_route);
       printf("\n# its energy is: %g\n", third_E);
     }

     /* James Theiler's recursive algorithm for generating all routes */
     static void do_all_perms(int *route, int n)
     {
       if (n == (N_CITIES-1)) {
         /* do it! calculate the energy/cost for that route */
         double E;
         E = Etsp(route);            /* TSP energy function */
         /* now save the best 3 energies and routes */
         if (E < best_E) {
           third_E = second_E;
           memcpy(third_route, second_route, N_CITIES*sizeof(*route));
           second_E = best_E;
           memcpy(second_route, best_route, N_CITIES*sizeof(*route));
           best_E = E;
           memcpy(best_route, route, N_CITIES*sizeof(*route));
         } else if (E < second_E) {
           third_E = second_E;
           memcpy(third_route, second_route, N_CITIES*sizeof(*route));
           second_E = E;
           memcpy(second_route, route, N_CITIES*sizeof(*route));
         } else if (E < third_E) {
           third_E = E;
           memcpy(route, third_route, N_CITIES*sizeof(*route));
         }
       } else {
         int new_route[N_CITIES];
         unsigned int j;
         int swap_tmp;
         memcpy(new_route, route, N_CITIES*sizeof(*route));
         for (j = n; j < N_CITIES; ++j) {
           swap_tmp = new_route[j];
           new_route[j] = new_route[n];
           new_route[n] = swap_tmp;
           do_all_perms(new_route, n+1);
         }
       }
     }

Below are some plots generated in the following way:

     $ ./siman_tsp > tsp.output
     $ grep -v "^#" tsp.output
      | awk '{print $1, $NF}'
      | graph -y 3300 6500 -W0 -X generation -Y distance
         -L "TSP - 12 southwest cities"
      | plot -Tps > 12-cities.eps
     $ grep initial_city_coord tsp.output
       | awk '{print $2, $3}'
       | graph -X "longitude (- means west)" -Y "latitude"
          -L "TSP - initial-order" -f 0.03 -S 1 0.1
       | plot -Tps > initial-route.eps
     $ grep final_city_coord tsp.output
       | awk '{print $2, $3}'
       | graph -X "longitude (- means west)" -Y "latitude"
          -L "TSP - final-order" -f 0.03 -S 1 0.1
       | plot -Tps > final-route.eps

This is the output showing the initial order of the cities; longitude is
negative, since it is west and I want the plot to look like a map:

     # initial coordinates of cities (longitude and latitude)
     ###initial_city_coord: -105.95 35.68 Santa Fe
     ###initial_city_coord: -112.07 33.54 Phoenix
     ###initial_city_coord: -106.62 35.12 Albuquerque
     ###initial_city_coord: -103.2 34.41 Clovis
     ###initial_city_coord: -107.87 37.29 Durango
     ###initial_city_coord: -96.77 32.79 Dallas
     ###initial_city_coord: -105.92 35.77 Tesuque
     ###initial_city_coord: -107.84 35.15 Grants
     ###initial_city_coord: -106.28 35.89 Los Alamos
     ###initial_city_coord: -106.76 32.34 Las Cruces
     ###initial_city_coord: -108.58 37.35 Cortez
     ###initial_city_coord: -108.74 35.52 Gallup
     ###initial_city_coord: -105.95 35.68 Santa Fe

The optimal route turns out to be:

     # final coordinates of cities (longitude and latitude)
     ###final_city_coord: -105.95 35.68 Santa Fe
     ###final_city_coord: -103.2 34.41 Clovis
     ###final_city_coord: -96.77 32.79 Dallas
     ###final_city_coord: -106.76 32.34 Las Cruces
     ###final_city_coord: -112.07 33.54 Phoenix
     ###final_city_coord: -108.74 35.52 Gallup
     ###final_city_coord: -108.58 37.35 Cortez
     ###final_city_coord: -107.87 37.29 Durango
     ###final_city_coord: -107.84 35.15 Grants
     ###final_city_coord: -106.62 35.12 Albuquerque
     ###final_city_coord: -106.28 35.89 Los Alamos
     ###final_city_coord: -105.92 35.77 Tesuque
     ###final_city_coord: -105.95 35.68 Santa Fe

[gsl-ref-figures/siman-initial-route]
Figure: Initial route for the 12 southwestern cities Flying Salesman
Problem.

[gsl-ref-figures/siman-final-route]
Figure: Final (optimal) route for the 12 southwestern cities Flying
Salesman Problem.

Here’s a plot of the cost function (energy) versus generation (point in
the calculation at which a new temperature is set) for this problem:

[gsl-ref-figures/siman-12-cities]
Figure: Example of a simulated annealing run for the 12 southwestern
cities Flying Salesman Problem.


File: gsl-ref.info,  Node: References and Further Reading<21>,  Prev: Examples<21>,  Up: Simulated Annealing

28.4 References and Further Reading
===================================

Further information is available in the following book,

   * `Modern Heuristic Techniques for Combinatorial Problems', Colin R.
     Reeves (ed.), McGraw-Hill, 1995 (ISBN 0-07-709239-2).


File: gsl-ref.info,  Node: Ordinary Differential Equations,  Next: Interpolation,  Prev: Simulated Annealing,  Up: Top

29 Ordinary Differential Equations
**********************************

This chapter describes functions for solving ordinary differential
equation (ODE) initial value problems.  The library provides a variety
of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines,
and higher-level components for adaptive step-size control.  The
components can be combined by the user to achieve the desired solution,
with full access to any intermediate steps.  A driver object can be used
as a high level wrapper for easy use of low level functions.

These functions are declared in the header file ‘gsl_odeiv2.h’.  This is
a new interface in version 1.15 and uses the prefix ‘gsl_odeiv2’ for all
functions.  It is recommended over the previous ‘gsl_odeiv’
implementation defined in ‘gsl_odeiv.h’ The old interface has been
retained under the original name for backwards compatibility.

* Menu:

* Defining the ODE System::
* Stepping Functions::
* Adaptive Step-size Control::
* Evolution::
* Driver::
* Examples: Examples<22>.
* References and Further Reading: References and Further Reading<22>.


File: gsl-ref.info,  Node: Defining the ODE System,  Next: Stepping Functions,  Up: Ordinary Differential Equations

29.1 Defining the ODE System
============================

The routines solve the general n-dimensional first-order system,

     dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))

for i = 1, \dots, n.  The stepping functions rely on the vector of
derivatives f_i and the Jacobian matrix,

     J_{ij} = df_i(t,y(t)) / dy_j

A system of equations is defined using the *note gsl_odeiv2_system: 973.
datatype.

 -- Type: gsl_odeiv2_system

     This data type defines a general ODE system with arbitrary
     parameters.

     ‘int (* function) (double t, const double y[], double dydt[], void
     * params)’

          This function should store the vector elements f_i(t,y,params)
          in the array ‘dydt’, for arguments (‘t’, ‘y’) and parameters
          ‘params’.

          The function should return ‘GSL_SUCCESS’ if the calculation
          was completed successfully.  Any other return value indicates
          an error.  A special return value ‘GSL_EBADFUNC’ causes
          ‘gsl_odeiv2’ routines to immediately stop and return.  If
          ‘function’ is modified (for example contents of ‘params’), the
          user must call an appropriate reset function (*note
          gsl_odeiv2_driver_reset(): 974, *note
          gsl_odeiv2_evolve_reset(): 975. or *note
          gsl_odeiv2_step_reset(): 976.) before continuing.  Use return
          values distinct from standard GSL error codes to distinguish
          your function as the source of the error.

     ‘int (* jacobian) (double t, const double y[], double * dfdy,
     double dfdt[], void * params)’

          This function should store the vector of derivative elements

               df_i(t,y,params)/dt

          in the array ‘dfdt’ and the Jacobian matrix J_{ij} in the
          array ‘dfdy’, regarded as a row-ordered matrix ‘J(i,j) =
          dfdy[i * dimension + j]’ where ‘dimension’ is the dimension of
          the system.

          Not all of the stepper algorithms of ‘gsl_odeiv2’ make use of
          the Jacobian matrix, so it may not be necessary to provide
          this function (the ‘jacobian’ element of the struct can be
          replaced by a null pointer for those algorithms).

          The function should return ‘GSL_SUCCESS’ if the calculation
          was completed successfully.  Any other return value indicates
          an error.  A special return value ‘GSL_EBADFUNC’ causes
          ‘gsl_odeiv2’ routines to immediately stop and return.  If
          ‘jacobian’ is modified (for example contents of ‘params’), the
          user must call an appropriate reset function (*note
          gsl_odeiv2_driver_reset(): 974, *note
          gsl_odeiv2_evolve_reset(): 975. or *note
          gsl_odeiv2_step_reset(): 976.) before continuing.  Use return
          values distinct from standard GSL error codes to distinguish
          your function as the source of the error.

     ‘size_t dimension’

          This is the dimension of the system of equations.

     ‘void * params’

          This is a pointer to the arbitrary parameters of the system.


File: gsl-ref.info,  Node: Stepping Functions,  Next: Adaptive Step-size Control,  Prev: Defining the ODE System,  Up: Ordinary Differential Equations

29.2 Stepping Functions
=======================

The lowest level components are the `stepping functions' which advance a
solution from time t to t+h for a fixed step-size h and estimate the
resulting local error.

 -- Type: gsl_odeiv2_step

     This contains internal parameters for a stepping function.

 -- Function: *note gsl_odeiv2_step: 978. *gsl_odeiv2_step_alloc (const
          gsl_odeiv2_step_type *T, size_t dim)

     This function returns a pointer to a newly allocated instance of a
     stepping function of type *note T: 979. for a system of *note dim:
     979. dimensions.  Please note that if you use a stepper method that
     requires access to a driver object, it is advisable to use a driver
     allocation method, which automatically allocates a stepper, too.

 -- Function: int gsl_odeiv2_step_reset (gsl_odeiv2_step *s)

     This function resets the stepping function *note s: 976.  It should
     be used whenever the next use of *note s: 976. will not be a
     continuation of a previous step.

 -- Function: void gsl_odeiv2_step_free (gsl_odeiv2_step *s)

     This function frees all the memory associated with the stepping
     function *note s: 97a.

 -- Function: const char *gsl_odeiv2_step_name (const gsl_odeiv2_step
          *s)

     This function returns a pointer to the name of the stepping
     function.  For example:

          printf ("step method is '%s'\n", gsl_odeiv2_step_name (s));

     would print something like ‘step method is 'rkf45'’.

 -- Function: unsigned int gsl_odeiv2_step_order (const gsl_odeiv2_step
          *s)

     This function returns the order of the stepping function on the
     previous step.  The order can vary if the stepping function itself
     is adaptive.

 -- Function: int gsl_odeiv2_step_set_driver (gsl_odeiv2_step *s, const
          gsl_odeiv2_driver *d)

     This function sets a pointer of the driver object *note d: 97d. for
     stepper *note s: 97d, to allow the stepper to access control (and
     evolve) object through the driver object.  This is a requirement
     for some steppers, to get the desired error level for internal
     iteration of stepper.  Allocation of a driver object calls this
     function automatically.

 -- Function: int gsl_odeiv2_step_apply (gsl_odeiv2_step *s, double t,
          double h, double y[], double yerr[], const double dydt_in[],
          double dydt_out[], const gsl_odeiv2_system *sys)

     This function applies the stepping function *note s: 97e. to the
     system of equations defined by *note sys: 97e, using the step-size
     *note h: 97e. to advance the system from time *note t: 97e. and
     state *note y: 97e. to time *note t: 97e. + *note h: 97e.  The new
     state of the system is stored in *note y: 97e. on output, with an
     estimate of the absolute error in each component stored in *note
     yerr: 97e.  If the argument *note dydt_in: 97e. is not null it
     should point an array containing the derivatives for the system at
     time *note t: 97e. on input.  This is optional as the derivatives
     will be computed internally if they are not provided, but allows
     the reuse of existing derivative information.  On output the new
     derivatives of the system at time *note t: 97e. + *note h: 97e.
     will be stored in *note dydt_out: 97e. if it is not null.

     The stepping function returns ‘GSL_FAILURE’ if it is unable to
     compute the requested step.  Also, if the user-supplied functions
     defined in the system *note sys: 97e. return a status other than
     ‘GSL_SUCCESS’ the step will be aborted.  In that case, the elements
     of *note y: 97e. will be restored to their pre-step values and the
     error code from the user-supplied function will be returned.
     Failure may be due to a singularity in the system or too large
     step-size *note h: 97e.  In that case the step should be attempted
     again with a smaller step-size, e.g.  *note h: 97e. / 2.

     If the driver object is not appropriately set via *note
     gsl_odeiv2_step_set_driver(): 97d. for those steppers that need it,
     the stepping function returns ‘GSL_EFAULT’.  If the user-supplied
     functions defined in the system *note sys: 97e. returns
     ‘GSL_EBADFUNC’, the function returns immediately with the same
     return code.  In this case the user must call *note
     gsl_odeiv2_step_reset(): 976. before calling this function again.

The following algorithms are available.  Please note that algorithms
which use step doubling for error estimation apply the more accurate
values from two half steps instead of values from a single step for the
new state ‘y’.

 -- Type: gsl_odeiv2_step_type

      -- Variable: *note gsl_odeiv2_step_type: 97f. *gsl_odeiv2_step_rk2

          Explicit embedded Runge-Kutta (2, 3) method.

      -- Variable: *note gsl_odeiv2_step_type: 97f. *gsl_odeiv2_step_rk4

          Explicit 4th order (classical) Runge-Kutta.  Error estimation
          is carried out by the step doubling method.  For more
          efficient estimate of the error, use the embedded methods
          described below.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_rkf45

          Explicit embedded Runge-Kutta-Fehlberg (4, 5) method.  This
          method is a good general-purpose integrator.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_rkck

          Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_rk8pd

          Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_rk1imp

          Implicit Gaussian first order Runge-Kutta.  Also known as
          implicit Euler or backward Euler method.  Error estimation is
          carried out by the step doubling method.  This algorithm
          requires the Jacobian and access to the driver object via
          *note gsl_odeiv2_step_set_driver(): 97d.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_rk2imp

          Implicit Gaussian second order Runge-Kutta.  Also known as
          implicit mid-point rule.  Error estimation is carried out by
          the step doubling method.  This stepper requires the Jacobian
          and access to the driver object via *note
          gsl_odeiv2_step_set_driver(): 97d.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_rk4imp

          Implicit Gaussian 4th order Runge-Kutta.  Error estimation is
          carried out by the step doubling method.  This algorithm
          requires the Jacobian and access to the driver object via
          *note gsl_odeiv2_step_set_driver(): 97d.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_bsimp

          Implicit Bulirsch-Stoer method of Bader and Deuflhard.  The
          method is generally suitable for stiff problems.  This stepper
          requires the Jacobian.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_msadams

          A variable-coefficient linear multistep Adams method in
          Nordsieck form.  This stepper uses explicit Adams-Bashforth
          (predictor) and implicit Adams-Moulton (corrector) methods in
          P(EC)^m functional iteration mode.  Method order varies
          dynamically between 1 and 12.  This stepper requires the
          access to the driver object via *note
          gsl_odeiv2_step_set_driver(): 97d.

      -- Variable: *note gsl_odeiv2_step_type: 97f.
               *gsl_odeiv2_step_msbdf

          A variable-coefficient linear multistep backward
          differentiation formula (BDF) method in Nordsieck form.  This
          stepper uses the explicit BDF formula as predictor and
          implicit BDF formula as corrector.  A modified Newton
          iteration method is used to solve the system of non-linear
          equations.  Method order varies dynamically between 1 and 5.
          The method is generally suitable for stiff problems.  This
          stepper requires the Jacobian and the access to the driver
          object via *note gsl_odeiv2_step_set_driver(): 97d.


File: gsl-ref.info,  Node: Adaptive Step-size Control,  Next: Evolution,  Prev: Stepping Functions,  Up: Ordinary Differential Equations

29.3 Adaptive Step-size Control
===============================

The control function examines the proposed change to the solution
produced by a stepping function and attempts to determine the optimal
step-size for a user-specified level of error.

 -- Type: gsl_odeiv2_control

     This is a workspace for controlling step size.

 -- Type: gsl_odeiv2_control_type

     This specifies the control type.

 -- Function: *note gsl_odeiv2_control: 98c.
          *gsl_odeiv2_control_standard_new (double eps_abs, double
          eps_rel, double a_y, double a_dydt)

     The standard control object is a four parameter heuristic based on
     absolute and relative errors *note eps_abs: 98e. and *note eps_rel:
     98e, and scaling factors *note a_y: 98e. and *note a_dydt: 98e. for
     the system state y(t) and derivatives y'(t) respectively.

     The step-size adjustment procedure for this method begins by
     computing the desired error level D_i for each component,

          D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)

     and comparing it with the observed error E_i = |yerr_i|.  If the
     observed error ‘E’ exceeds the desired error level ‘D’ by more than
     10% for any component then the method reduces the step-size by an
     appropriate factor,

          h_{new} = h_{old} * S * (E/D)^{-1/q}

     where q is the consistency order of the method (e.g.  q=4 for 4(5)
     embedded RK), and S is a safety factor of 0.9.  The ratio E/D is
     taken to be the maximum of the ratios E_i/D_i.

     If the observed error E is less than 50% of the desired error level
     ‘D’ for the maximum ratio E_i/D_i then the algorithm takes the
     opportunity to increase the step-size to bring the error in line
     with the desired level,

          h_{new} = h_{old} * S * (E/D)^{-1/(q+1)}

     This encompasses all the standard error scaling methods.  To avoid
     uncontrolled changes in the stepsize, the overall scaling factor is
     limited to the range 1/5 to 5.

 -- Function: *note gsl_odeiv2_control: 98c. *gsl_odeiv2_control_y_new
          (double eps_abs, double eps_rel)

     This function creates a new control object which will keep the
     local error on each step within an absolute error of *note eps_abs:
     98f. and relative error of *note eps_rel: 98f. with respect to the
     solution y_i(t).  This is equivalent to the standard control object
     with ‘a_y’ = 1 and ‘a_dydt’ = 0.

 -- Function: *note gsl_odeiv2_control: 98c. *gsl_odeiv2_control_yp_new
          (double eps_abs, double eps_rel)

     This function creates a new control object which will keep the
     local error on each step within an absolute error of *note eps_abs:
     990. and relative error of *note eps_rel: 990. with respect to the
     derivatives of the solution y'_i(t).  This is equivalent to the
     standard control object with ‘a_y’ = 0 and ‘a_dydt’ = 1.

 -- Function: *note gsl_odeiv2_control: 98c.
          *gsl_odeiv2_control_scaled_new (double eps_abs, double
          eps_rel, double a_y, double a_dydt, const double scale_abs[],
          size_t dim)

     This function creates a new control object which uses the same
     algorithm as *note gsl_odeiv2_control_standard_new(): 98e. but with
     an absolute error which is scaled for each component by the array
     *note scale_abs: 991.  The formula for D_i for this control object
     is,

          D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)

     where s_i is the i-th component of the array *note scale_abs: 991.
     The same error control heuristic is used by the Matlab ODE suite.

 -- Function: *note gsl_odeiv2_control: 98c. *gsl_odeiv2_control_alloc
          (const gsl_odeiv2_control_type *T)

     This function returns a pointer to a newly allocated instance of a
     control function of type *note T: 992.  This function is only
     needed for defining new types of control functions.  For most
     purposes the standard control functions described above should be
     sufficient.

 -- Function: int gsl_odeiv2_control_init (gsl_odeiv2_control *c, double
          eps_abs, double eps_rel, double a_y, double a_dydt)

     This function initializes the control function *note c: 993. with
     the parameters *note eps_abs: 993. (absolute error), *note eps_rel:
     993. (relative error), *note a_y: 993. (scaling factor for y) and
     *note a_dydt: 993. (scaling factor for derivatives).

 -- Function: void gsl_odeiv2_control_free (gsl_odeiv2_control *c)

     This function frees all the memory associated with the control
     function *note c: 994.

 -- Function: int gsl_odeiv2_control_hadjust (gsl_odeiv2_control *c,
          gsl_odeiv2_step *s, const double y[], const double yerr[],
          const double dydt[], double *h)

     This function adjusts the step-size *note h: 995. using the control
     function *note c: 995, and the current values of *note y: 995,
     *note yerr: 995. and *note dydt: 995.  The stepping function ‘step’
     is also needed to determine the order of the method.  If the error
     in the y-values *note yerr: 995. is found to be too large then the
     step-size *note h: 995. is reduced and the function returns
     ‘GSL_ODEIV_HADJ_DEC’.  If the error is sufficiently small then
     *note h: 995. may be increased and ‘GSL_ODEIV_HADJ_INC’ is
     returned.  The function returns ‘GSL_ODEIV_HADJ_NIL’ if the
     step-size is unchanged.  The goal of the function is to estimate
     the largest step-size which satisfies the user-specified accuracy
     requirements for the current point.

 -- Function: const char *gsl_odeiv2_control_name (const
          gsl_odeiv2_control *c)

     This function returns a pointer to the name of the control
     function.  For example:

          printf ("control method is '%s'\n", gsl_odeiv2_control_name (c));

     would print something like ‘control method is 'standard'’

 -- Function: int gsl_odeiv2_control_errlevel (gsl_odeiv2_control *c,
          const double y, const double dydt, const double h, const
          size_t ind, double *errlev)

     This function calculates the desired error level of the *note ind:
     997.-th component to *note errlev: 997.  It requires the value
     (*note y: 997.) and value of the derivative (*note dydt: 997.) of
     the component, and the current step size *note h: 997.

 -- Function: int gsl_odeiv2_control_set_driver (gsl_odeiv2_control *c,
          const gsl_odeiv2_driver *d)

     This function sets a pointer of the driver object *note d: 998. for
     control object *note c: 998.


File: gsl-ref.info,  Node: Evolution,  Next: Driver,  Prev: Adaptive Step-size Control,  Up: Ordinary Differential Equations

29.4 Evolution
==============

The evolution function combines the results of a stepping function and
control function to reliably advance the solution forward one step using
an acceptable step-size.

 -- Type: gsl_odeiv2_evolve

     This workspace contains parameters for controlling the evolution
     function

 -- Function: *note gsl_odeiv2_evolve: 99a. *gsl_odeiv2_evolve_alloc
          (size_t dim)

     This function returns a pointer to a newly allocated instance of an
     evolution function for a system of *note dim: 99b. dimensions.

 -- Function: int gsl_odeiv2_evolve_apply (gsl_odeiv2_evolve *e,
          gsl_odeiv2_control *con, gsl_odeiv2_step *step, const
          gsl_odeiv2_system *sys, double *t, double t1, double *h,
          double y[])

     This function advances the system (*note e: 99c, *note sys: 99c.)
     from time *note t: 99c. and position *note y: 99c. using the
     stepping function *note step: 99c.  The new time and position are
     stored in *note t: 99c. and *note y: 99c. on output.

     The initial step-size is taken as *note h: 99c.  The control
     function *note con: 99c. is applied to check whether the local
     error estimated by the stepping function *note step: 99c. using
     step-size *note h: 99c. exceeds the required error tolerance.  If
     the error is too high, the step is retried by calling *note step:
     99c. with a decreased step-size.  This process is continued until
     an acceptable step-size is found.  An estimate of the local error
     for the step can be obtained from the components of the array
     ‘e->yerr[]’.

     If the user-supplied functions defined in the system *note sys:
     99c. returns ‘GSL_EBADFUNC’, the function returns immediately with
     the same return code.  In this case the user must call *note
     gsl_odeiv2_step_reset(): 976. and *note gsl_odeiv2_evolve_reset():
     975. before calling this function again.

     Otherwise, if the user-supplied functions defined in the system
     *note sys: 99c. or the stepping function *note step: 99c. return a
     status other than ‘GSL_SUCCESS’, the step is retried with a
     decreased step-size.  If the step-size decreases below machine
     precision, a status of ‘GSL_FAILURE’ is returned if the user
     functions returned ‘GSL_SUCCESS’.  Otherwise the value returned by
     user function is returned.  If no acceptable step can be made,
     *note t: 99c. and *note y: 99c. will be restored to their pre-step
     values and *note h: 99c. contains the final attempted step-size.

     If the step is successful the function returns a suggested
     step-size for the next step in *note h: 99c.  The maximum time
     *note t1: 99c. is guaranteed not to be exceeded by the time-step.
     On the final time-step the value of *note t: 99c. will be set to
     *note t1: 99c. exactly.

 -- Function: int gsl_odeiv2_evolve_apply_fixed_step (gsl_odeiv2_evolve
          *e, gsl_odeiv2_control *con, gsl_odeiv2_step *step, const
          gsl_odeiv2_system *sys, double *t, const double h, double y[])

     This function advances the ODE-system (*note e: 99d, *note sys:
     99d, *note con: 99d.) from time *note t: 99d. and position *note y:
     99d. using the stepping function *note step: 99d. by a specified
     step size *note h: 99d.  If the local error estimated by the
     stepping function exceeds the desired error level, the step is not
     taken and the function returns ‘GSL_FAILURE’.  Otherwise the value
     returned by user function is returned.

 -- Function: int gsl_odeiv2_evolve_reset (gsl_odeiv2_evolve *e)

     This function resets the evolution function *note e: 975.  It
     should be used whenever the next use of *note e: 975. will not be a
     continuation of a previous step.

 -- Function: void gsl_odeiv2_evolve_free (gsl_odeiv2_evolve *e)

     This function frees all the memory associated with the evolution
     function *note e: 99e.

 -- Function: int gsl_odeiv2_evolve_set_driver (gsl_odeiv2_evolve *e,
          const gsl_odeiv2_driver *d)

     This function sets a pointer of the driver object *note d: 99f. for
     evolve object *note e: 99f.

If a system has discontinuous changes in the derivatives at known
points, it is advisable to evolve the system between each discontinuity
in sequence.  For example, if a step-change in an external driving force
occurs at times t_a, t_b and t_c then evolution should be carried out
over the ranges (t_0,t_a), (t_a,t_b), (t_b,t_c), and (t_c,t_1)
separately and not directly over the range (t_0,t_1).


File: gsl-ref.info,  Node: Driver,  Next: Examples<22>,  Prev: Evolution,  Up: Ordinary Differential Equations

29.5 Driver
===========

The driver object is a high level wrapper that combines the evolution,
control and stepper objects for easy use.

 -- Function: gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_y_new (const
          gsl_odeiv2_system *sys, const gsl_odeiv2_step_type *T, const
          double hstart, const double epsabs, const double epsrel)
 -- Function: gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_yp_new (const
          gsl_odeiv2_system *sys, const gsl_odeiv2_step_type *T, const
          double hstart, const double epsabs, const double epsrel)
 -- Function: gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_standard_new
          (const gsl_odeiv2_system *sys, const gsl_odeiv2_step_type *T,
          const double hstart, const double epsabs, const double epsrel,
          const double a_y, const double a_dydt)
 -- Function: gsl_odeiv2_driver *gsl_odeiv2_driver_alloc_scaled_new
          (const gsl_odeiv2_system *sys, const gsl_odeiv2_step_type *T,
          const double hstart, const double epsabs, const double epsrel,
          const double a_y, const double a_dydt, const double
          scale_abs[])

     These functions return a pointer to a newly allocated instance of a
     driver object.  The functions automatically allocate and initialise
     the evolve, control and stepper objects for ODE system *note sys:
     9a4. using stepper type *note T: 9a4.  The initial step size is
     given in *note hstart: 9a4.  The rest of the arguments follow the
     syntax and semantics of the control functions with same name
     (‘gsl_odeiv2_control_*_new’).

 -- Function: int gsl_odeiv2_driver_set_hmin (gsl_odeiv2_driver *d,
          const double hmin)

     The function sets a minimum for allowed step size *note hmin: 9a5.
     for driver *note d: 9a5.  Default value is 0.

 -- Function: int gsl_odeiv2_driver_set_hmax (gsl_odeiv2_driver *d,
          const double hmax)

     The function sets a maximum for allowed step size *note hmax: 9a6.
     for driver *note d: 9a6.  Default value is ‘GSL_DBL_MAX’.

 -- Function: int gsl_odeiv2_driver_set_nmax (gsl_odeiv2_driver *d,
          const unsigned long int nmax)

     The function sets a maximum for allowed number of steps *note nmax:
     9a7. for driver *note d: 9a7.  Default value of 0 sets no limit for
     steps.

 -- Function: int gsl_odeiv2_driver_apply (gsl_odeiv2_driver *d, double
          *t, const double t1, double y[])

     This function evolves the driver system *note d: 9a8. from *note t:
     9a8. to *note t1: 9a8.  Initially vector *note y: 9a8. should
     contain the values of dependent variables at point *note t: 9a8.
     If the function is unable to complete the calculation, an error
     code from *note gsl_odeiv2_evolve_apply(): 99c. is returned, and
     *note t: 9a8. and *note y: 9a8. contain the values from last
     successful step.

     If maximum number of steps is reached, a value of ‘GSL_EMAXITER’ is
     returned.  If the step size drops below minimum value, the function
     returns with ‘GSL_ENOPROG’.  If the user-supplied functions defined
     in the system ‘sys’ returns ‘GSL_EBADFUNC’, the function returns
     immediately with the same return code.  In this case the user must
     call *note gsl_odeiv2_driver_reset(): 974. before calling this
     function again.

 -- Function: int gsl_odeiv2_driver_apply_fixed_step (gsl_odeiv2_driver
          *d, double *t, const double h, const unsigned long int n,
          double y[])

     This function evolves the driver system *note d: 9a9. from *note t:
     9a9. with *note n: 9a9. steps of size *note h: 9a9.  If the
     function is unable to complete the calculation, an error code from
     *note gsl_odeiv2_evolve_apply_fixed_step(): 99d. is returned, and
     *note t: 9a9. and *note y: 9a9. contain the values from last
     successful step.

 -- Function: int gsl_odeiv2_driver_reset (gsl_odeiv2_driver *d)

     This function resets the evolution and stepper objects.

 -- Function: int gsl_odeiv2_driver_reset_hstart (gsl_odeiv2_driver *d,
          const double hstart)

     The routine resets the evolution and stepper objects and sets new
     initial step size to *note hstart: 9aa.  This function can be used
     e.g.  to change the direction of integration.

 -- Function: int gsl_odeiv2_driver_free (gsl_odeiv2_driver *d)

     This function frees the driver object, and the related evolution,
     stepper and control objects.


File: gsl-ref.info,  Node: Examples<22>,  Next: References and Further Reading<22>,  Prev: Driver,  Up: Ordinary Differential Equations

29.6 Examples
=============

The following program solves the second-order nonlinear Van der Pol
oscillator equation,

     u''(t) + \mu u'(t) (u(t)^2 - 1) + u(t) = 0

This can be converted into a first order system suitable for use with
the routines described in this chapter by introducing a separate
variable for the velocity, v = u'(t),

     u' = v
     v' = -u + \mu v (1-u^2)

The program begins by defining functions for these derivatives and their
Jacobian.  The main function uses driver level functions to solve the
problem.  The program evolves the solution from (u, v) = (1, 0) at t = 0
to t = 100.  The step-size h is automatically adjusted by the controller
to maintain an absolute accuracy of 10^{-6} in the function values (u,
v).  The loop in the example prints the solution at the points t_i = 1,
2, \dots, 100.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_odeiv2.h>

     int
     func (double t, const double y[], double f[],
           void *params)
     {
       (void)(t); /* avoid unused parameter warning */
       double mu = *(double *)params;
       f[0] = y[1];
       f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
       return GSL_SUCCESS;
     }

     int
     jac (double t, const double y[], double *dfdy,
          double dfdt[], void *params)
     {
       (void)(t); /* avoid unused parameter warning */
       double mu = *(double *)params;
       gsl_matrix_view dfdy_mat
         = gsl_matrix_view_array (dfdy, 2, 2);
       gsl_matrix * m = &dfdy_mat.matrix;
       gsl_matrix_set (m, 0, 0, 0.0);
       gsl_matrix_set (m, 0, 1, 1.0);
       gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
       gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
       dfdt[0] = 0.0;
       dfdt[1] = 0.0;
       return GSL_SUCCESS;
     }

     int
     main (void)
     {
       double mu = 10;
       gsl_odeiv2_system sys = {func, jac, 2, &mu};

       gsl_odeiv2_driver * d =
         gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk8pd,
                                       1e-6, 1e-6, 0.0);
       int i;
       double t = 0.0, t1 = 100.0;
       double y[2] = { 1.0, 0.0 };

       for (i = 1; i <= 100; i++)
         {
           double ti = i * t1 / 100.0;
           int status = gsl_odeiv2_driver_apply (d, &t, ti, y);

           if (status != GSL_SUCCESS)
             {
               printf ("error, return value=%d\n", status);
               break;
             }

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_driver_free (d);
       return 0;
     }

The user can work with the lower level functions directly, as in the
following example.  In this case an intermediate result is printed after
each successful step instead of equidistant time points.

     int
     main (void)
     {
       const gsl_odeiv2_step_type * T
         = gsl_odeiv2_step_rk8pd;

       gsl_odeiv2_step * s
         = gsl_odeiv2_step_alloc (T, 2);
       gsl_odeiv2_control * c
         = gsl_odeiv2_control_y_new (1e-6, 0.0);
       gsl_odeiv2_evolve * e
         = gsl_odeiv2_evolve_alloc (2);

       double mu = 10;
       gsl_odeiv2_system sys = {func, jac, 2, &mu};

       double t = 0.0, t1 = 100.0;
       double h = 1e-6;
       double y[2] = { 1.0, 0.0 };

       while (t < t1)
         {
           int status = gsl_odeiv2_evolve_apply (e, c, s,
                                                &sys,
                                                &t, t1,
                                                &h, y);

           if (status != GSL_SUCCESS)
               break;

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_evolve_free (e);
       gsl_odeiv2_control_free (c);
       gsl_odeiv2_step_free (s);
       return 0;
     }

For functions with multiple parameters, the appropriate information can
be passed in through the ‘params’ argument in *note gsl_odeiv2_system:
973. definition (‘mu’ in this example) by using a pointer to a struct.

[gsl-ref-figures/ode-vdp]
Figure: Numerical solution of the Van der Pol oscillator equation using
Prince-Dormand 8th order Runge-Kutta.

It is also possible to work with a non-adaptive integrator, using only
the stepping function itself, *note
gsl_odeiv2_driver_apply_fixed_step(): 9a9. or *note
gsl_odeiv2_evolve_apply_fixed_step(): 99d.  The following program uses
the driver level function, with fourth-order Runge-Kutta stepping
function with a fixed stepsize of 0.001.

     int
     main (void)
     {
       double mu = 10;
       gsl_odeiv2_system sys = { func, jac, 2, &mu };

       gsl_odeiv2_driver *d =
         gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk4,
                                        1e-3, 1e-8, 1e-8);

       double t = 0.0;
       double y[2] = { 1.0, 0.0 };
       int i, s;

       for (i = 0; i < 100; i++)
         {
           s = gsl_odeiv2_driver_apply_fixed_step (d, &t, 1e-3, 1000, y);

           if (s != GSL_SUCCESS)
             {
               printf ("error: driver returned %d\n", s);
               break;
             }

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_driver_free (d);
       return s;
     }


File: gsl-ref.info,  Node: References and Further Reading<22>,  Prev: Examples<22>,  Up: Ordinary Differential Equations

29.7 References and Further Reading
===================================

   * Ascher, U.M., Petzold, L.R., `Computer Methods for Ordinary
     Differential and Differential-Algebraic Equations', SIAM,
     Philadelphia, 1998.

   * Hairer, E., Norsett, S. P., Wanner, G., `Solving Ordinary
     Differential Equations I: Nonstiff Problems', Springer, Berlin,
     1993.

   * Hairer, E., Wanner, G., `Solving Ordinary Differential Equations
     II: Stiff and Differential-Algebraic Problems', Springer, Berlin,
     1996.

Many of the basic Runge-Kutta formulas can be found in the Handbook of
Mathematical Functions,

   * Abramowitz & Stegun (eds.), `Handbook of Mathematical Functions',
     Section 25.5.

The implicit Bulirsch-Stoer algorithm ‘bsimp’ is described in the
following paper,

   * G. Bader and P. Deuflhard, “A Semi-Implicit Mid-Point Rule for
     Stiff Systems of Ordinary Differential Equations.”, Numer.: Math.:
     41, 373–398, 1983.

The Adams and BDF multistep methods ‘msadams’ and ‘msbdf’ are based on
the following articles,

   * G. D. Byrne and A. C. Hindmarsh, “A Polyalgorithm for the Numerical
     Solution of Ordinary Differential Equations.”, ACM Trans.  Math.
     Software, 1, 71–96, 1975.

   * P. N. Brown, G. D. Byrne and A. C. Hindmarsh, “VODE: A
     Variable-coefficient ODE Solver.”, SIAM J. Sci.  Stat.  Comput.
     10, 1038–1051, 1989.

   * A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D.
     E. Shumaker and C. S. Woodward, “SUNDIALS: Suite of Nonlinear and
     Differential/Algebraic Equation Solvers.”, ACM Trans.  Math.
     Software 31, 363–396, 2005.


File: gsl-ref.info,  Node: Interpolation,  Next: Numerical Differentiation,  Prev: Ordinary Differential Equations,  Up: Top

30 Interpolation
****************

This chapter describes functions for performing interpolation.  The
library provides a variety of interpolation methods, including Cubic,
Akima, and Steffen splines.  The interpolation types are
interchangeable, allowing different methods to be used without
recompiling.  Interpolations can be defined for both normal and periodic
boundary conditions.  Additional functions are available for computing
derivatives and integrals of interpolating functions.  Routines are
provided for interpolating both one and two dimensional datasets.

These interpolation methods produce curves that pass through each
datapoint.  To interpolate noisy data with a smoothing curve see *note
Basis Splines: 9b1.

The functions described in this section are declared in the header files
‘gsl_interp.h’ and ‘gsl_spline.h’.

* Menu:

* Introduction to 1D Interpolation::
* 1D Interpolation Functions::
* 1D Interpolation Types::
* 1D Index Look-up and Acceleration::
* 1D Evaluation of Interpolating Functions::
* 1D Higher-level Interface::
* 1D Interpolation Example Programs::
* Introduction to 2D Interpolation::
* 2D Interpolation Functions::
* 2D Interpolation Grids::
* 2D Interpolation Types::
* 2D Evaluation of Interpolating Functions::
* 2D Higher-level Interface::
* 2D Interpolation Example programs::
* References and Further Reading: References and Further Reading<23>.


File: gsl-ref.info,  Node: Introduction to 1D Interpolation,  Next: 1D Interpolation Functions,  Up: Interpolation

30.1 Introduction to 1D Interpolation
=====================================

Given a set of data points (x_1, y_1) \dots (x_n, y_n) the routines
described in this section compute a continuous interpolating function
y(x) such that y(x_i) = y_i.  The interpolation is piecewise smooth, and
its behavior at the end-points is determined by the type of
interpolation used.


File: gsl-ref.info,  Node: 1D Interpolation Functions,  Next: 1D Interpolation Types,  Prev: Introduction to 1D Interpolation,  Up: Interpolation

30.2 1D Interpolation Functions
===============================

The interpolation function for a given dataset is stored in a *note
gsl_interp: 9b4. object.  These are created by the following functions.

 -- Type: gsl_interp

     Workspace for 1D interpolation

 -- Function: *note gsl_interp: 9b4. *gsl_interp_alloc (const
          gsl_interp_type *T, size_t size)

     This function returns a pointer to a newly allocated interpolation
     object of type *note T: 9b5. for *note size: 9b5. data-points.

 -- Function: int gsl_interp_init (gsl_interp *interp, const double
          xa[], const double ya[], size_t size)

     This function initializes the interpolation object *note interp:
     9b6. for the data (*note xa: 9b6, *note ya: 9b6.) where *note xa:
     9b6. and *note ya: 9b6. are arrays of size *note size: 9b6.  The
     interpolation object (*note gsl_interp: 9b4.) does not save the
     data arrays *note xa: 9b6. and *note ya: 9b6. and only stores the
     static state computed from the data.  The *note xa: 9b6. data array
     is always assumed to be strictly ordered, with increasing x values;
     the behavior for other arrangements is not defined.

 -- Function: void gsl_interp_free (gsl_interp *interp)

     This function frees the interpolation object *note interp: 9b7.


File: gsl-ref.info,  Node: 1D Interpolation Types,  Next: 1D Index Look-up and Acceleration,  Prev: 1D Interpolation Functions,  Up: Interpolation

30.3 1D Interpolation Types
===========================

The interpolation library provides the following interpolation types:

 -- Type: gsl_interp_type

      -- Variable: *note gsl_interp_type: 9b9. *gsl_interp_linear

          Linear interpolation.  This interpolation method does not
          require any additional memory.

      -- Variable: *note gsl_interp_type: 9b9. *gsl_interp_polynomial

          Polynomial interpolation.  This method should only be used for
          interpolating small numbers of points because polynomial
          interpolation introduces large oscillations, even for
          well-behaved datasets.  The number of terms in the
          interpolating polynomial is equal to the number of points.

      -- Variable: *note gsl_interp_type: 9b9. *gsl_interp_cspline

          Cubic spline with natural boundary conditions.  The resulting
          curve is piecewise cubic on each interval, with matching first
          and second derivatives at the supplied data-points.  The
          second derivative is chosen to be zero at the first point and
          last point.

      -- Variable: *note gsl_interp_type: 9b9.
               *gsl_interp_cspline_periodic

          Cubic spline with periodic boundary conditions.  The resulting
          curve is piecewise cubic on each interval, with matching first
          and second derivatives at the supplied data-points.  The
          derivatives at the first and last points are also matched.
          Note that the last point in the data must have the same
          y-value as the first point, otherwise the resulting periodic
          interpolation will have a discontinuity at the boundary.

      -- Variable: *note gsl_interp_type: 9b9. *gsl_interp_akima

          Non-rounded Akima spline with natural boundary conditions.
          This method uses the non-rounded corner algorithm of Wodicka.

      -- Variable: *note gsl_interp_type: 9b9.
               *gsl_interp_akima_periodic

          Non-rounded Akima spline with periodic boundary conditions.
          This method uses the non-rounded corner algorithm of Wodicka.

      -- Variable: *note gsl_interp_type: 9b9. *gsl_interp_steffen

          Steffen’s method guarantees the monotonicity of the
          interpolating function between the given data points.
          Therefore, minima and maxima can only occur exactly at the
          data points, and there can never be spurious oscillations
          between data points.  The interpolated function is piecewise
          cubic in each interval.  The resulting curve and its first
          derivative are guaranteed to be continuous, but the second
          derivative may be discontinuous.

The following related functions are available:

 -- Function: const char *gsl_interp_name (const gsl_interp *interp)

     This function returns the name of the interpolation type used by
     *note interp: 9c1.  For example:

          printf ("interp uses '%s' interpolation.\n", gsl_interp_name (interp));

     would print something like:

          interp uses 'cspline' interpolation.

 -- Function: unsigned int gsl_interp_min_size (const gsl_interp
          *interp)
 -- Function: unsigned int gsl_interp_type_min_size (const
          gsl_interp_type *T)

     These functions return the minimum number of points required by the
     interpolation object ‘interp’ or interpolation type *note T: 9c3.
     For example, Akima spline interpolation requires a minimum of 5
     points.


File: gsl-ref.info,  Node: 1D Index Look-up and Acceleration,  Next: 1D Evaluation of Interpolating Functions,  Prev: 1D Interpolation Types,  Up: Interpolation

30.4 1D Index Look-up and Acceleration
======================================

The state of searches can be stored in a *note gsl_interp_accel: 9c5.
object, which is a kind of iterator for interpolation lookups.

 -- Type: gsl_interp_accel

     This workspace stores state variables for interpolation lookups.
     It caches the previous value of an index lookup.  When the
     subsequent interpolation point falls in the same interval its index
     value can be returned immediately.

 -- Function: size_t gsl_interp_bsearch (const double x_array[], double
          x, size_t index_lo, size_t index_hi)

     This function returns the index i of the array *note x_array: 9c6.
     such that ‘x_array[i] <= x < x_array[i+1]’.  The index is searched
     for in the range [*note index_lo: 9c6, *note index_hi: 9c6.].  An
     inline version of this function is used when ‘HAVE_INLINE’ is
     defined.

 -- Function: *note gsl_interp_accel: 9c5. *gsl_interp_accel_alloc
          (void)

     This function returns a pointer to an accelerator object, which is
     a kind of iterator for interpolation lookups.  It tracks the state
     of lookups, thus allowing for application of various acceleration
     strategies.  When multiple interpolants are in use, the same
     accelerator object may be used for all datasets with the same
     domain (‘x_array’), but different accelerators should be used for
     data defined on different domains.

 -- Function: size_t gsl_interp_accel_find (gsl_interp_accel *a, const
          double x_array[], size_t size, double x)

     This function performs a lookup action on the data array *note
     x_array: 9c8. of size *note size: 9c8, using the given accelerator
     *note a: 9c8.  This is how lookups are performed during evaluation
     of an interpolation.  The function returns an index i such that
     ‘x_array[i] <= x < x_array[i+1]’.  An inline version of this
     function is used when ‘HAVE_INLINE’ is defined.

 -- Function: int gsl_interp_accel_reset (gsl_interp_accel *acc);

     This function reinitializes the accelerator object *note acc: 9c9.
     It should be used when the cached information is no longer
     applicable—for example, when switching to a new dataset.

 -- Function: void gsl_interp_accel_free (gsl_interp_accel *acc)

     This function frees the accelerator object *note acc: 9ca.


File: gsl-ref.info,  Node: 1D Evaluation of Interpolating Functions,  Next: 1D Higher-level Interface,  Prev: 1D Index Look-up and Acceleration,  Up: Interpolation

30.5 1D Evaluation of Interpolating Functions
=============================================

 -- Function: double gsl_interp_eval (const gsl_interp *interp, const
          double xa[], const double ya[], double x, gsl_interp_accel
          *acc)
 -- Function: int gsl_interp_eval_e (const gsl_interp *interp, const
          double xa[], const double ya[], double x, gsl_interp_accel
          *acc, double *y)

     These functions return the interpolated value of *note y: 9cd. for
     a given point *note x: 9cd, using the interpolation object *note
     interp: 9cd, data arrays *note xa: 9cd. and *note ya: 9cd. and the
     accelerator *note acc: 9cd.  When *note x: 9cd. is outside the
     range of *note xa: 9cd, the error code *note GSL_EDOM: 28. is
     returned with a value of *note GSL_NAN: 3c. for *note y: 9cd.

 -- Function: double gsl_interp_eval_deriv (const gsl_interp *interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel *acc)
 -- Function: int gsl_interp_eval_deriv_e (const gsl_interp *interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel *acc, double *d)

     These functions return the derivative *note d: 9cf. of an
     interpolated function for a given point *note x: 9cf, using the
     interpolation object *note interp: 9cf, data arrays *note xa: 9cf.
     and *note ya: 9cf. and the accelerator *note acc: 9cf.

 -- Function: double gsl_interp_eval_deriv2 (const gsl_interp *interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel *acc)
 -- Function: int gsl_interp_eval_deriv2_e (const gsl_interp *interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel *acc, double *d2)

     These functions return the second derivative *note d2: 9d1. of an
     interpolated function for a given point *note x: 9d1, using the
     interpolation object *note interp: 9d1, data arrays *note xa: 9d1.
     and *note ya: 9d1. and the accelerator *note acc: 9d1.

 -- Function: double gsl_interp_eval_integ (const gsl_interp *interp,
          const double xa[], const double ya[], double a, double b,
          gsl_interp_accel *acc)
 -- Function: int gsl_interp_eval_integ_e (const gsl_interp *interp,
          const double xa[], const double ya[], double a, double b,
          gsl_interp_accel *acc, double *result)

     These functions return the numerical integral *note result: 9d3. of
     an interpolated function over the range [*note a: 9d3, *note b:
     9d3.], using the interpolation object *note interp: 9d3, data
     arrays *note xa: 9d3. and *note ya: 9d3. and the accelerator *note
     acc: 9d3.


File: gsl-ref.info,  Node: 1D Higher-level Interface,  Next: 1D Interpolation Example Programs,  Prev: 1D Evaluation of Interpolating Functions,  Up: Interpolation

30.6 1D Higher-level Interface
==============================

The functions described in the previous sections required the user to
supply pointers to the x and y arrays on each call.  The following
functions are equivalent to the corresponding *note gsl_interp: 9b4.
functions but maintain a copy of this data in the *note gsl_spline: 9d5.
object.  This removes the need to pass both ‘xa’ and ‘ya’ as arguments
on each evaluation.  These functions are defined in the header file
‘gsl_spline.h’.

 -- Type: gsl_spline

     This workspace provides a higher level interface for the *note
     gsl_interp: 9b4. object

 -- Function: *note gsl_spline: 9d5. *gsl_spline_alloc (const
          gsl_interp_type *T, size_t size)

 -- Function: int gsl_spline_init (gsl_spline *spline, const double
          xa[], const double ya[], size_t size)

 -- Function: void gsl_spline_free (gsl_spline *spline)

 -- Function: const char *gsl_spline_name (const gsl_spline *spline)

 -- Function: unsigned int gsl_spline_min_size (const gsl_spline
          *spline)

 -- Function: double gsl_spline_eval (const gsl_spline *spline, double
          x, gsl_interp_accel *acc)
 -- Function: int gsl_spline_eval_e (const gsl_spline *spline, double x,
          gsl_interp_accel *acc, double *y)

 -- Function: double gsl_spline_eval_deriv (const gsl_spline *spline,
          double x, gsl_interp_accel *acc)
 -- Function: int gsl_spline_eval_deriv_e (const gsl_spline *spline,
          double x, gsl_interp_accel *acc, double *d)

 -- Function: double gsl_spline_eval_deriv2 (const gsl_spline *spline,
          double x, gsl_interp_accel *acc)
 -- Function: int gsl_spline_eval_deriv2_e (const gsl_spline *spline,
          double x, gsl_interp_accel *acc, double *d2)

 -- Function: double gsl_spline_eval_integ (const gsl_spline *spline,
          double a, double b, gsl_interp_accel *acc)
 -- Function: int gsl_spline_eval_integ_e (const gsl_spline *spline,
          double a, double b, gsl_interp_accel *acc, double *result)


File: gsl-ref.info,  Node: 1D Interpolation Example Programs,  Next: Introduction to 2D Interpolation,  Prev: 1D Higher-level Interface,  Up: Interpolation

30.7 1D Interpolation Example Programs
======================================

The following program demonstrates the use of the interpolation and
spline functions.  It computes a cubic spline interpolation of the
10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i +
\cos(i^2) for i = 0 \dots 9.

     #include <stdlib.h>
     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_spline.h>

     int
     main (void)
     {
       int i;
       double xi, yi, x[10], y[10];

       printf ("#m=0,S=17\n");

       for (i = 0; i < 10; i++)
         {
           x[i] = i + 0.5 * sin (i);
           y[i] = i + cos (i * i);
           printf ("%g %g\n", x[i], y[i]);
         }

       printf ("#m=1,S=0\n");

       {
         gsl_interp_accel *acc
           = gsl_interp_accel_alloc ();
         gsl_spline *spline
           = gsl_spline_alloc (gsl_interp_cspline, 10);

         gsl_spline_init (spline, x, y, 10);

         for (xi = x[0]; xi < x[9]; xi += 0.01)
           {
             yi = gsl_spline_eval (spline, xi, acc);
             printf ("%g %g\n", xi, yi);
           }
         gsl_spline_free (spline);
         gsl_interp_accel_free (acc);
       }
       return 0;
     }

The output is designed to be used with the GNU plotutils ‘graph’
program:

     $ ./a.out > interp.dat
     $ graph -T ps < interp.dat > interp.ps

[gsl-ref-figures/interp]
Figure: Cubic spline interpolation

Fig.  %s shows a smooth interpolation of the original points.  The
interpolation method can be changed simply by varying the first argument
of *note gsl_spline_alloc(): 9d6.

The next program demonstrates a periodic cubic spline with 4 data
points.  Note that the first and last points must be supplied with the
same y-value for a periodic spline.

     #include <stdlib.h>
     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_spline.h>

     int
     main (void)
     {
       int N = 4;
       double x[4] = {0.00, 0.10,  0.27,  0.30};
       double y[4] = {0.15, 0.70, -0.10,  0.15};
                  /* Note: y[0] == y[3] for periodic data */

       gsl_interp_accel *acc = gsl_interp_accel_alloc ();
       const gsl_interp_type *t = gsl_interp_cspline_periodic;
       gsl_spline *spline = gsl_spline_alloc (t, N);

       int i; double xi, yi;

       printf ("#m=0,S=5\n");
       for (i = 0; i < N; i++)
         {
           printf ("%g %g\n", x[i], y[i]);
         }

       printf ("#m=1,S=0\n");
       gsl_spline_init (spline, x, y, N);

       for (i = 0; i <= 100; i++)
         {
           xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
           yi = gsl_spline_eval (spline, xi, acc);
           printf ("%g %g\n", xi, yi);
         }

       gsl_spline_free (spline);
       gsl_interp_accel_free (acc);
       return 0;
     }

The output can be plotted with GNU ‘graph’:

     $ ./a.out > interp.dat
     $ graph -T ps < interp.dat > interp.ps

[gsl-ref-figures/interpp]
Figure: Periodic cubic spline interpolation

Fig.  %s shows a periodic interpolation of the original points.  The
slope of the fitted curve is the same at the beginning and end of the
data, and the second derivative is also.

The next program illustrates the difference between the cubic spline,
Akima, and Steffen interpolation types on a difficult dataset.

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_spline.h>

     int
     main(void)
     {
       size_t i;
       const size_t N = 9;

       /* this dataset is taken from
        * J. M. Hyman, Accurate Monotonicity preserving cubic interpolation,
        * SIAM J. Sci. Stat. Comput. 4, 4, 1983. */
       const double x[] = { 7.99, 8.09, 8.19, 8.7, 9.2,
                            10.0, 12.0, 15.0, 20.0 };
       const double y[] = { 0.0, 2.76429e-5, 4.37498e-2,
                            0.169183, 0.469428, 0.943740,
                            0.998636, 0.999919, 0.999994 };

       gsl_interp_accel *acc = gsl_interp_accel_alloc();
       gsl_spline *spline_cubic = gsl_spline_alloc(gsl_interp_cspline, N);
       gsl_spline *spline_akima = gsl_spline_alloc(gsl_interp_akima, N);
       gsl_spline *spline_steffen = gsl_spline_alloc(gsl_interp_steffen, N);

       gsl_spline_init(spline_cubic, x, y, N);
       gsl_spline_init(spline_akima, x, y, N);
       gsl_spline_init(spline_steffen, x, y, N);

       for (i = 0; i < N; ++i)
         printf("%g %g\n", x[i], y[i]);

       printf("\n\n");

       for (i = 0; i <= 100; ++i)
         {
           double xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
           double yi_cubic = gsl_spline_eval(spline_cubic, xi, acc);
           double yi_akima = gsl_spline_eval(spline_akima, xi, acc);
           double yi_steffen = gsl_spline_eval(spline_steffen, xi, acc);

           printf("%g %g %g %g\n", xi, yi_cubic, yi_akima, yi_steffen);
         }

       gsl_spline_free(spline_cubic);
       gsl_spline_free(spline_akima);
       gsl_spline_free(spline_steffen);
       gsl_interp_accel_free(acc);

       return 0;
     }

[gsl-ref-figures/interp_compare]
Figure: Comparison of different 1D interpolation methods

The output is shown in Fig.  %s.  The cubic method exhibits a local
maxima between the 6th and 7th data points and continues oscillating for
the rest of the data.  Akima also shows a local maxima but recovers and
follows the data well after the 7th grid point.  Steffen preserves
monotonicity in all intervals and does not exhibit oscillations, at the
expense of having a discontinuous second derivative.


File: gsl-ref.info,  Node: Introduction to 2D Interpolation,  Next: 2D Interpolation Functions,  Prev: 1D Interpolation Example Programs,  Up: Interpolation

30.8 Introduction to 2D Interpolation
=====================================

Given a set of x coordinates x_1,...,x_m and a set of y coordinates
y_1,...,y_n, each in increasing order, plus a set of function values
z_{ij} for each grid point (x_i,y_j), the routines described in this
section compute a continuous interpolation function z(x,y) such that
z(x_i,y_j) = z_{ij}.


File: gsl-ref.info,  Node: 2D Interpolation Functions,  Next: 2D Interpolation Grids,  Prev: Introduction to 2D Interpolation,  Up: Interpolation

30.9 2D Interpolation Functions
===============================

The interpolation function for a given dataset is stored in a *note
gsl_interp2d: 9e9. object.  These are created by the following
functions.

 -- Type: gsl_interp2d

     Workspace for 2D interpolation

 -- Function: *note gsl_interp2d: 9e9. *gsl_interp2d_alloc (const
          gsl_interp2d_type *T, const size_t xsize, const size_t ysize)

     This function returns a pointer to a newly allocated interpolation
     object of type *note T: 9ea. for *note xsize: 9ea. grid points in
     the x direction and *note ysize: 9ea. grid points in the y
     direction.

 -- Function: int gsl_interp2d_init (gsl_interp2d *interp, const double
          xa[], const double ya[], const double za[], const size_t
          xsize, const size_t ysize)

     This function initializes the interpolation object *note interp:
     9eb. for the data (*note xa: 9eb, *note ya: 9eb, *note za: 9eb.)
     where *note xa: 9eb. and *note ya: 9eb. are arrays of the x and y
     grid points of size *note xsize: 9eb. and *note ysize: 9eb.
     respectively, and *note za: 9eb. is an array of function values of
     size *note xsize: 9eb. * *note ysize: 9eb.  The interpolation
     object (*note gsl_interp2d: 9e9.) does not save the data arrays
     *note xa: 9eb, *note ya: 9eb, and *note za: 9eb. and only stores
     the static state computed from the data.  The *note xa: 9eb. and
     *note ya: 9eb. data arrays are always assumed to be strictly
     ordered, with increasing x,y values; the behavior for other
     arrangements is not defined.

 -- Function: void gsl_interp2d_free (gsl_interp2d *interp)

     This function frees the interpolation object *note interp: 9ec.


File: gsl-ref.info,  Node: 2D Interpolation Grids,  Next: 2D Interpolation Types,  Prev: 2D Interpolation Functions,  Up: Interpolation

30.10 2D Interpolation Grids
============================

The 2D interpolation routines access the function values z_{ij} with the
following ordering:

     z_{ij} = za[j*xsize + i]

with i = 0,...,xsize-1 and j = 0,...,ysize-1.  However, for ease of use,
the following functions are provided to add and retrieve elements from
the function grid without requiring knowledge of the internal ordering.

 -- Function: int gsl_interp2d_set (const gsl_interp2d *interp, double
          za[], const size_t i, const size_t j, const double z)

     This function sets the value z_{ij} for grid point (*note i: 9ee,
     *note j: 9ee.) of the array *note za: 9ee. to *note z: 9ee.

 -- Function: double gsl_interp2d_get (const gsl_interp2d *interp, const
          double za[], const size_t i, const size_t j)

     This function returns the value z_{ij} for grid point (*note i:
     9ef, *note j: 9ef.) stored in the array *note za: 9ef.

 -- Function: size_t gsl_interp2d_idx (const gsl_interp2d *interp, const
          size_t i, const size_t j)

     This function returns the index corresponding to the grid point
     (*note i: 9f0, *note j: 9f0.).  The index is given by j*xsize + i.


File: gsl-ref.info,  Node: 2D Interpolation Types,  Next: 2D Evaluation of Interpolating Functions,  Prev: 2D Interpolation Grids,  Up: Interpolation

30.11 2D Interpolation Types
============================

 -- Type: gsl_interp2d_type

     The interpolation library provides the following 2D interpolation
     types:

      -- Variable: *note gsl_interp2d_type: 9f2. *gsl_interp2d_bilinear

          Bilinear interpolation.  This interpolation method does not
          require any additional memory.

      -- Variable: *note gsl_interp2d_type: 9f2. *gsl_interp2d_bicubic

          Bicubic interpolation.

 -- Function: const char *gsl_interp2d_name (const gsl_interp2d *interp)

     This function returns the name of the interpolation type used by
     *note interp: 9f5.  For example:

          printf ("interp uses '%s' interpolation.\n", gsl_interp2d_name (interp));

     would print something like:

          interp uses 'bilinear' interpolation.

 -- Function: unsigned int gsl_interp2d_min_size (const gsl_interp2d
          *interp)
 -- Function: unsigned int gsl_interp2d_type_min_size (const
          gsl_interp2d_type *T)

     These functions return the minimum number of points required by the
     interpolation object ‘interp’ or interpolation type *note T: 9f7.
     For example, bicubic interpolation requires a minimum of 4 points.


File: gsl-ref.info,  Node: 2D Evaluation of Interpolating Functions,  Next: 2D Higher-level Interface,  Prev: 2D Interpolation Types,  Up: Interpolation

30.12 2D Evaluation of Interpolating Functions
==============================================

 -- Function: double gsl_interp2d_eval (const gsl_interp2d *interp,
          const double xa[], const double ya[], const double za[], const
          double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_interp2d_eval_e (const gsl_interp2d *interp, const
          double xa[], const double ya[], const double za[], const
          double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *z)

     These functions return the interpolated value of *note z: 9fa. for
     a given point (*note x: 9fa, *note y: 9fa.), using the
     interpolation object *note interp: 9fa, data arrays *note xa: 9fa,
     *note ya: 9fa, and *note za: 9fa. and the accelerators *note xacc:
     9fa. and *note yacc: 9fa.  When *note x: 9fa. is outside the range
     of *note xa: 9fa. or *note y: 9fa. is outside the range of *note
     ya: 9fa, the error code *note GSL_EDOM: 28. is returned.

 -- Function: double gsl_interp2d_eval_extrap (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_interp2d_eval_extrap_e (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *z)

     These functions return the interpolated value of *note z: 9fc. for
     a given point (*note x: 9fc, *note y: 9fc.), using the
     interpolation object *note interp: 9fc, data arrays *note xa: 9fc,
     *note ya: 9fc, and *note za: 9fc. and the accelerators *note xacc:
     9fc. and *note yacc: 9fc.  The functions perform no bounds
     checking, so when *note x: 9fc. is outside the range of *note xa:
     9fc. or *note y: 9fc. is outside the range of *note ya: 9fc,
     extrapolation is performed.

 -- Function: double gsl_interp2d_eval_deriv_x (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_interp2d_eval_deriv_x_e (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *d)

     These functions return the interpolated value *note d: 9fe. =
     \partial z / \partial x for a given point (*note x: 9fe, *note y:
     9fe.), using the interpolation object *note interp: 9fe, data
     arrays *note xa: 9fe, *note ya: 9fe, and *note za: 9fe. and the
     accelerators *note xacc: 9fe. and *note yacc: 9fe.  When *note x:
     9fe. is outside the range of *note xa: 9fe. or *note y: 9fe. is
     outside the range of *note ya: 9fe, the error code *note GSL_EDOM:
     28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_y (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_interp2d_eval_deriv_y_e (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *d)

     These functions return the interpolated value *note d: a00. =
     \partial z / \partial y for a given point (*note x: a00, *note y:
     a00.), using the interpolation object *note interp: a00, data
     arrays *note xa: a00, *note ya: a00, and *note za: a00. and the
     accelerators *note xacc: a00. and *note yacc: a00.  When *note x:
     a00. is outside the range of *note xa: a00. or *note y: a00. is
     outside the range of *note ya: a00, the error code *note GSL_EDOM:
     28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_xx (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_interp2d_eval_deriv_xx_e (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *d)

     These functions return the interpolated value *note d: a02. =
     \partial^2 z / \partial x^2 for a given point (*note x: a02, *note
     y: a02.), using the interpolation object *note interp: a02, data
     arrays *note xa: a02, *note ya: a02, and *note za: a02. and the
     accelerators *note xacc: a02. and *note yacc: a02.  When *note x:
     a02. is outside the range of *note xa: a02. or *note y: a02. is
     outside the range of *note ya: a02, the error code *note GSL_EDOM:
     28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_yy (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_interp2d_eval_deriv_yy_e (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *d)

     These functions return the interpolated value *note d: a04. =
     \partial^2 z / \partial y^2 for a given point (*note x: a04, *note
     y: a04.), using the interpolation object *note interp: a04, data
     arrays *note xa: a04, *note ya: a04, and *note za: a04. and the
     accelerators *note xacc: a04. and *note yacc: a04.  When *note x:
     a04. is outside the range of *note xa: a04. or *note y: a04. is
     outside the range of *note ya: a04, the error code *note GSL_EDOM:
     28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_xy (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_interp2d_eval_deriv_xy_e (const gsl_interp2d
          *interp, const double xa[], const double ya[], const double
          za[], const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *d)

     These functions return the interpolated value *note d: a06. =
     \partial^2 z / \partial x \partial y for a given point (*note x:
     a06, *note y: a06.), using the interpolation object *note interp:
     a06, data arrays *note xa: a06, *note ya: a06, and *note za: a06.
     and the accelerators *note xacc: a06. and *note yacc: a06.  When
     *note x: a06. is outside the range of *note xa: a06. or *note y:
     a06. is outside the range of *note ya: a06, the error code *note
     GSL_EDOM: 28. is returned.


File: gsl-ref.info,  Node: 2D Higher-level Interface,  Next: 2D Interpolation Example programs,  Prev: 2D Evaluation of Interpolating Functions,  Up: Interpolation

30.13 2D Higher-level Interface
===============================

The functions described in the previous sections required the user to
supply pointers to the x, y, and z arrays on each call.  The following
functions are equivalent to the corresponding ‘gsl_interp2d’ functions
but maintain a copy of this data in the *note gsl_spline2d: a08. object.
This removes the need to pass ‘xa’, ‘ya’, and ‘za’ as arguments on each
evaluation.  These functions are defined in the header file
‘gsl_spline2d.h’.

 -- Type: gsl_spline2d

     This workspace provides a higher level interface for the *note
     gsl_interp2d: 9e9. object

 -- Function: *note gsl_spline2d: a08. *gsl_spline2d_alloc (const
          gsl_interp2d_type *T, size_t xsize, size_t ysize)

 -- Function: int gsl_spline2d_init (gsl_spline2d *spline, const double
          xa[], const double ya[], const double za[], size_t xsize,
          size_t ysize)

 -- Function: void gsl_spline2d_free (gsl_spline2d *spline)

 -- Function: const char *gsl_spline2d_name (const gsl_spline2d *spline)

 -- Function: unsigned int gsl_spline2d_min_size (const gsl_spline2d
          *spline)

 -- Function: double gsl_spline2d_eval (const gsl_spline2d *spline,
          const double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc)
 -- Function: int gsl_spline2d_eval_e (const gsl_spline2d *spline, const
          double x, const double y, gsl_interp_accel *xacc,
          gsl_interp_accel *yacc, double *z)

 -- Function: double gsl_spline2d_eval_extrap (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc)
 -- Function: int gsl_spline2d_eval_extrap_e (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc, double *z)

 -- Function: double gsl_spline2d_eval_deriv_x (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc)
 -- Function: int gsl_spline2d_eval_deriv_x_e (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc, double *d)

 -- Function: double gsl_spline2d_eval_deriv_y (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc)
 -- Function: int gsl_spline2d_eval_deriv_y_e (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc, double *d)

 -- Function: double gsl_spline2d_eval_deriv_xx (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc)
 -- Function: int gsl_spline2d_eval_deriv_xx_e (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc, double *d)

 -- Function: double gsl_spline2d_eval_deriv_yy (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc)
 -- Function: int gsl_spline2d_eval_deriv_yy_e (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc, double *d)

 -- Function: double gsl_spline2d_eval_deriv_xy (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc)
 -- Function: int gsl_spline2d_eval_deriv_xy_e (const gsl_spline2d
          *spline, const double x, const double y, gsl_interp_accel
          *xacc, gsl_interp_accel *yacc, double *d)

 -- Function: int gsl_spline2d_set (const gsl_spline2d *spline, double
          za[], const size_t i, const size_t j, const double z)

 -- Function: double gsl_spline2d_get (const gsl_spline2d *spline, const
          double za[], const size_t i, const size_t j)

     This function returns the value z_{ij} for grid point (*note i:
     a1d, *note j: a1d.) stored in the array *note za: a1d.


File: gsl-ref.info,  Node: 2D Interpolation Example programs,  Next: References and Further Reading<23>,  Prev: 2D Higher-level Interface,  Up: Interpolation

30.14 2D Interpolation Example programs
=======================================

The following example performs bilinear interpolation on the unit
square, using z values of (0,1,0.5,1) going clockwise around the square.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_interp2d.h>
     #include <gsl/gsl_spline2d.h>

     int
     main()
     {
       const gsl_interp2d_type *T = gsl_interp2d_bilinear;
       const size_t N = 100;             /* number of points to interpolate */
       const double xa[] = { 0.0, 1.0 }; /* define unit square */
       const double ya[] = { 0.0, 1.0 };
       const size_t nx = sizeof(xa) / sizeof(double); /* x grid points */
       const size_t ny = sizeof(ya) / sizeof(double); /* y grid points */
       double *za = malloc(nx * ny * sizeof(double));
       gsl_spline2d *spline = gsl_spline2d_alloc(T, nx, ny);
       gsl_interp_accel *xacc = gsl_interp_accel_alloc();
       gsl_interp_accel *yacc = gsl_interp_accel_alloc();
       size_t i, j;

       /* set z grid values */
       gsl_spline2d_set(spline, za, 0, 0, 0.0);
       gsl_spline2d_set(spline, za, 0, 1, 1.0);
       gsl_spline2d_set(spline, za, 1, 1, 0.5);
       gsl_spline2d_set(spline, za, 1, 0, 1.0);

       /* initialize interpolation */
       gsl_spline2d_init(spline, xa, ya, za, nx, ny);

       /* interpolate N values in x and y and print out grid for plotting */
       for (i = 0; i < N; ++i)
         {
           double xi = i / (N - 1.0);

           for (j = 0; j < N; ++j)
             {
               double yj = j / (N - 1.0);
               double zij = gsl_spline2d_eval(spline, xi, yj, xacc, yacc);

               printf("%f %f %f\n", xi, yj, zij);
             }
           printf("\n");
         }

       gsl_spline2d_free(spline);
       gsl_interp_accel_free(xacc);
       gsl_interp_accel_free(yacc);
       free(za);

       return 0;
     }

The results of the interpolation are shown in Fig.  %s, where the
corners are labeled with their fixed z values.

[gsl-ref-figures/interp2d]
Figure: 2D interpolation example


File: gsl-ref.info,  Node: References and Further Reading<23>,  Prev: 2D Interpolation Example programs,  Up: Interpolation

30.15 References and Further Reading
====================================

Descriptions of the interpolation algorithms and further references can
be found in the following publications:

   * C.W. Ueberhuber, `Numerical Computation (Volume 1), Chapter 9
     “Interpolation”', Springer (1997), ISBN 3-540-62058-3.

   * D.M. Young, R.T. Gregory, `A Survey of Numerical Mathematics
     (Volume 1), Chapter 6.8', Dover (1988), ISBN 0-486-65691-8.

   * M. Steffen, `A simple method for monotonic interpolation in one
     dimension', Astron.  Astrophys.  239, 443-450, 1990.


File: gsl-ref.info,  Node: Numerical Differentiation,  Next: Chebyshev Approximations,  Prev: Interpolation,  Up: Top

31 Numerical Differentiation
****************************

The functions described in this chapter compute numerical derivatives by
finite differencing.  An adaptive algorithm is used to find the best
choice of finite difference and to estimate the error in the derivative.
These functions are declared in the header file ‘gsl_deriv.h’.

* Menu:

* Functions::
* Examples: Examples<23>.
* References and Further Reading: References and Further Reading<24>.


File: gsl-ref.info,  Node: Functions,  Next: Examples<23>,  Up: Numerical Differentiation

31.1 Functions
==============

 -- Function: int gsl_deriv_central (const gsl_function *f, double x,
          double h, double *result, double *abserr)

     This function computes the numerical derivative of the function
     *note f: a24. at the point *note x: a24. using an adaptive central
     difference algorithm with a step-size of *note h: a24.  The
     derivative is returned in *note result: a24. and an estimate of its
     absolute error is returned in *note abserr: a24.

     The initial value of *note h: a24. is used to estimate an optimal
     step-size, based on the scaling of the truncation error and
     round-off error in the derivative calculation.  The derivative is
     computed using a 5-point rule for equally spaced abscissae at x -
     h, x - h/2, x, x + h/2, x+h, with an error estimate taken from the
     difference between the 5-point rule and the corresponding 3-point
     rule x-h, x, x+h.  Note that the value of the function at x does
     not contribute to the derivative calculation, so only 4-points are
     actually used.

 -- Function: int gsl_deriv_forward (const gsl_function *f, double x,
          double h, double *result, double *abserr)

     This function computes the numerical derivative of the function
     *note f: a25. at the point *note x: a25. using an adaptive forward
     difference algorithm with a step-size of *note h: a25.  The
     function is evaluated only at points greater than *note x: a25, and
     never at *note x: a25. itself.  The derivative is returned in *note
     result: a25. and an estimate of its absolute error is returned in
     *note abserr: a25.  This function should be used if f(x) has a
     discontinuity at *note x: a25, or is undefined for values less than
     *note x: a25.

     The initial value of *note h: a25. is used to estimate an optimal
     step-size, based on the scaling of the truncation error and
     round-off error in the derivative calculation.  The derivative at x
     is computed using an “open” 4-point rule for equally spaced
     abscissae at x+h/4, x + h/2, x + 3h/4, x+h, with an error estimate
     taken from the difference between the 4-point rule and the
     corresponding 2-point rule x+h/2, x+h.

 -- Function: int gsl_deriv_backward (const gsl_function *f, double x,
          double h, double *result, double *abserr)

     This function computes the numerical derivative of the function
     *note f: a26. at the point *note x: a26. using an adaptive backward
     difference algorithm with a step-size of *note h: a26.  The
     function is evaluated only at points less than *note x: a26, and
     never at *note x: a26. itself.  The derivative is returned in *note
     result: a26. and an estimate of its absolute error is returned in
     *note abserr: a26.  This function should be used if f(x) has a
     discontinuity at *note x: a26, or is undefined for values greater
     than *note x: a26.

     This function is equivalent to calling *note gsl_deriv_forward():
     a25. with a negative step-size.


File: gsl-ref.info,  Node: Examples<23>,  Next: References and Further Reading<24>,  Prev: Functions,  Up: Numerical Differentiation

31.2 Examples
=============

The following code estimates the derivative of the function f(x) =
x^{3/2} at x = 2 and at x = 0.  The function f(x) is undefined for x < 0
so the derivative at x=0 is computed using *note gsl_deriv_forward():
a25.

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_deriv.h>

     double f (double x, void * params)
     {
       (void)(params); /* avoid unused parameter warning */
       return pow (x, 1.5);
     }

     int
     main (void)
     {
       gsl_function F;
       double result, abserr;

       F.function = &f;
       F.params = 0;

       printf ("f(x) = x^(3/2)\n");

       gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
       printf ("x = 2.0\n");
       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
       printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));

       gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
       printf ("x = 0.0\n");
       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
       printf ("exact = %.10f\n", 0.0);

       return 0;
     }

Here is the output of the program,

     f(x) = x^(3/2)
     x = 2.0
     f'(x) = 2.1213203120 +/- 0.0000005006
     exact = 2.1213203436

     x = 0.0
     f'(x) = 0.0000000160 +/- 0.0000000339
     exact = 0.0000000000


File: gsl-ref.info,  Node: References and Further Reading<24>,  Prev: Examples<23>,  Up: Numerical Differentiation

31.3 References and Further Reading
===================================

The algorithms used by these functions are described in the following
sources:

   * Abramowitz and Stegun, `Handbook of Mathematical Functions',
     Section 25.3.4, and Table 25.5 (Coefficients for Differentiation).

   * S.D. Conte and Carl de Boor, `Elementary Numerical Analysis: An
     Algorithmic Approach', McGraw-Hill, 1972.


File: gsl-ref.info,  Node: Chebyshev Approximations,  Next: Series Acceleration,  Prev: Numerical Differentiation,  Up: Top

32 Chebyshev Approximations
***************************

This chapter describes routines for computing Chebyshev approximations
to univariate functions.  A Chebyshev approximation is a truncation of
the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials
T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on
the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}.  The
first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2
x^2 - 1.  For further information see Abramowitz & Stegun, Chapter 22.

The functions described in this chapter are declared in the header file
‘gsl_chebyshev.h’.

* Menu:

* Definitions::
* Creation and Calculation of Chebyshev Series::
* Auxiliary Functions::
* Chebyshev Series Evaluation::
* Derivatives and Integrals::
* Examples: Examples<24>.
* References and Further Reading: References and Further Reading<25>.


File: gsl-ref.info,  Node: Definitions,  Next: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations

32.1 Definitions
================

 -- Type: gsl_cheb_series

     A Chebyshev series is stored using the following structure:

          typedef struct
          {
            double * c;   /* coefficients  c[0] .. c[order] */
            int order;    /* order of expansion             */
            double a;     /* lower interval point           */
            double b;     /* upper interval point           */
            ...
          } gsl_cheb_series

The approximation is made over the range [a,b] using ‘order’ + 1 terms,
including the coefficient c[0].  The series is computed using the
following convention,

     f(x) = (c_0 / 2) + \sum_{n=1} c_n T_n(x)

which is needed when accessing the coefficients directly.


File: gsl-ref.info,  Node: Creation and Calculation of Chebyshev Series,  Next: Auxiliary Functions,  Prev: Definitions,  Up: Chebyshev Approximations

32.2 Creation and Calculation of Chebyshev Series
=================================================

 -- Function: *note gsl_cheb_series: a2c. *gsl_cheb_alloc (const size_t
          n)

     This function allocates space for a Chebyshev series of order *note
     n: a2e. and returns a pointer to a new *note gsl_cheb_series: a2c.
     struct.

 -- Function: void gsl_cheb_free (gsl_cheb_series *cs)

     This function frees a previously allocated Chebyshev series *note
     cs: a2f.

 -- Function: int gsl_cheb_init (gsl_cheb_series *cs, const gsl_function
          *f, const double a, const double b)

     This function computes the Chebyshev approximation *note cs: a30.
     for the function *note f: a30. over the range (a,b) to the
     previously specified order.  The computation of the Chebyshev
     approximation is an O(n^2) process, and requires n function
     evaluations.


File: gsl-ref.info,  Node: Auxiliary Functions,  Next: Chebyshev Series Evaluation,  Prev: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations

32.3 Auxiliary Functions
========================

The following functions provide information about an existing Chebyshev
series.

 -- Function: size_t gsl_cheb_order (const gsl_cheb_series *cs)

     This function returns the order of Chebyshev series *note cs: a32.

 -- Function: size_t gsl_cheb_size (const gsl_cheb_series *cs)
 -- Function: double *gsl_cheb_coeffs (const gsl_cheb_series *cs)

     These functions return the size of the Chebyshev coefficient array
     ‘c[]’ and a pointer to its location in memory for the Chebyshev
     series *note cs: a34.


File: gsl-ref.info,  Node: Chebyshev Series Evaluation,  Next: Derivatives and Integrals,  Prev: Auxiliary Functions,  Up: Chebyshev Approximations

32.4 Chebyshev Series Evaluation
================================

 -- Function: double gsl_cheb_eval (const gsl_cheb_series *cs, double x)

     This function evaluates the Chebyshev series *note cs: a36. at a
     given point *note x: a36.

 -- Function: int gsl_cheb_eval_err (const gsl_cheb_series *cs, const
          double x, double *result, double *abserr)

     This function computes the Chebyshev series *note cs: a37. at a
     given point *note x: a37, estimating both the series *note result:
     a37. and its absolute error *note abserr: a37.  The error estimate
     is made from the first neglected term in the series.

 -- Function: double gsl_cheb_eval_n (const gsl_cheb_series *cs, size_t
          order, double x)

     This function evaluates the Chebyshev series *note cs: a38. at a
     given point *note x: a38, to (at most) the given order *note order:
     a38.

 -- Function: int gsl_cheb_eval_n_err (const gsl_cheb_series *cs, const
          size_t order, const double x, double *result, double *abserr)

     This function evaluates a Chebyshev series *note cs: a39. at a
     given point *note x: a39, estimating both the series *note result:
     a39. and its absolute error *note abserr: a39, to (at most) the
     given order *note order: a39.  The error estimate is made from the
     first neglected term in the series.


File: gsl-ref.info,  Node: Derivatives and Integrals,  Next: Examples<24>,  Prev: Chebyshev Series Evaluation,  Up: Chebyshev Approximations

32.5 Derivatives and Integrals
==============================

The following functions allow a Chebyshev series to be differentiated or
integrated, producing a new Chebyshev series.  Note that the error
estimate produced by evaluating the derivative series will be
underestimated due to the contribution of higher order terms being
neglected.

 -- Function: int gsl_cheb_calc_deriv (gsl_cheb_series *deriv, const
          gsl_cheb_series *cs)

     This function computes the derivative of the series *note cs: a3b,
     storing the derivative coefficients in the previously allocated
     *note deriv: a3b.  The two series *note cs: a3b. and *note deriv:
     a3b. must have been allocated with the same order.

 -- Function: int gsl_cheb_calc_integ (gsl_cheb_series *integ, const
          gsl_cheb_series *cs)

     This function computes the integral of the series *note cs: a3c,
     storing the integral coefficients in the previously allocated *note
     integ: a3c.  The two series *note cs: a3c. and *note integ: a3c.
     must have been allocated with the same order.  The lower limit of
     the integration is taken to be the left hand end of the range ‘a’.


File: gsl-ref.info,  Node: Examples<24>,  Next: References and Further Reading<25>,  Prev: Derivatives and Integrals,  Up: Chebyshev Approximations

32.6 Examples
=============

The following example program computes Chebyshev approximations to a
step function.  This is an extremely difficult approximation to make,
due to the discontinuity, and was chosen as an example where
approximation error is visible.  For smooth functions the Chebyshev
approximation converges extremely rapidly and errors would not be
visible.

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_chebyshev.h>

     double
     f (double x, void *p)
     {
       (void)(p); /* avoid unused parameter warning */

       if (x < 0.5)
         return 0.25;
       else
         return 0.75;
     }

     int
     main (void)
     {
       int i, n = 10000;

       gsl_cheb_series *cs = gsl_cheb_alloc (40);

       gsl_function F;

       F.function = f;
       F.params = 0;

       gsl_cheb_init (cs, &F, 0.0, 1.0);

       for (i = 0; i < n; i++)
         {
           double x = i / (double)n;
           double r10 = gsl_cheb_eval_n (cs, 10, x);
           double r40 = gsl_cheb_eval (cs, x);
           printf ("%g %g %g %g\n",
                   x, GSL_FN_EVAL (&F, x), r10, r40);
         }

       gsl_cheb_free (cs);

       return 0;
     }

Fig.  %s shows output from the program with the original function, 10-th
order approximation and 40-th order approximation, all sampled at
intervals of 0.001 in x.

[gsl-ref-figures/cheb]
Figure: Chebyshev approximations to a step function


File: gsl-ref.info,  Node: References and Further Reading<25>,  Prev: Examples<24>,  Up: Chebyshev Approximations

32.7 References and Further Reading
===================================

The following paper describes the use of Chebyshev series,

   * R. Broucke, “Ten Subroutines for the Manipulation of Chebyshev
     Series [C1] (Algorithm 446)”.  `Communications of the ACM' 16(4),
     254–256 (1973)


File: gsl-ref.info,  Node: Series Acceleration,  Next: Wavelet Transforms,  Prev: Chebyshev Approximations,  Up: Top

33 Series Acceleration
**********************

The functions described in this chapter accelerate the convergence of a
series using the Levin u-transform.  This method takes a small number of
terms from the start of a series and uses a systematic approximation to
compute an extrapolated value and an estimate of its error.  The
u-transform works for both convergent and divergent series, including
asymptotic series.

These functions are declared in the header file ‘gsl_sum.h’.

* Menu:

* Acceleration functions::
* Acceleration functions without error estimation::
* Examples: Examples<25>.
* References and Further Reading: References and Further Reading<26>.


File: gsl-ref.info,  Node: Acceleration functions,  Next: Acceleration functions without error estimation,  Up: Series Acceleration

33.1 Acceleration functions
===========================

The following functions compute the full Levin u-transform of a series
with its error estimate.  The error estimate is computed by propagating
rounding errors from each term through to the final extrapolation.

These functions are intended for summing analytic series where each term
is known to high accuracy, and the rounding errors are assumed to
originate from finite precision.  They are taken to be relative errors
of order ‘GSL_DBL_EPSILON’ for each term.

The calculation of the error in the extrapolated value is an O(N^2)
process, which is expensive in time and memory.  A faster but less
reliable method which estimates the error from the convergence of the
extrapolated value is described in the next section.  For the method
described here a full table of intermediate values and derivatives
through to O(N) must be computed and stored, but this does give a
reliable error estimate.

 -- Type: gsl_sum_levin_u_workspace

     Workspace for a Leven u-transform.

 -- Function: *note gsl_sum_levin_u_workspace: a43.
          *gsl_sum_levin_u_alloc (size_t n)

     This function allocates a workspace for a Levin u-transform of
     *note n: a44. terms.  The size of the workspace is O(2n^2 + 3n).

 -- Function: void gsl_sum_levin_u_free (gsl_sum_levin_u_workspace *w)

     This function frees the memory associated with the workspace *note
     w: a45.

 -- Function: int gsl_sum_levin_u_accel (const double *array, size_t
          array_size, gsl_sum_levin_u_workspace *w, double *sum_accel,
          double *abserr)

     This function takes the terms of a series in *note array: a46. of
     size *note array_size: a46. and computes the extrapolated limit of
     the series using a Levin u-transform.  Additional working space
     must be provided in *note w: a46.  The extrapolated sum is stored
     in *note sum_accel: a46, with an estimate of the absolute error
     stored in *note abserr: a46.  The actual term-by-term sum is
     returned in ‘w->sum_plain’.  The algorithm calculates the
     truncation error (the difference between two successive
     extrapolations) and round-off error (propagated from the individual
     terms) to choose an optimal number of terms for the extrapolation.
     All the terms of the series passed in through *note array: a46.
     should be non-zero.


File: gsl-ref.info,  Node: Acceleration functions without error estimation,  Next: Examples<25>,  Prev: Acceleration functions,  Up: Series Acceleration

33.2 Acceleration functions without error estimation
====================================================

The functions described in this section compute the Levin u-transform of
series and attempt to estimate the error from the “truncation error” in
the extrapolation, the difference between the final two approximations.
Using this method avoids the need to compute an intermediate table of
derivatives because the error is estimated from the behavior of the
extrapolated value itself.  Consequently this algorithm is an O(N)
process and only requires O(N) terms of storage.  If the series
converges sufficiently fast then this procedure can be acceptable.  It
is appropriate to use this method when there is a need to compute many
extrapolations of series with similar convergence properties at
high-speed.  For example, when numerically integrating a function
defined by a parameterized series where the parameter varies only
slightly.  A reliable error estimate should be computed first using the
full algorithm described above in order to verify the consistency of the
results.

 -- Type: gsl_sum_levin_utrunc_workspace

     Workspace for a Levin u-transform without error estimation

 -- Function: *note gsl_sum_levin_utrunc_workspace: a48.
          *gsl_sum_levin_utrunc_alloc (size_t n)

     This function allocates a workspace for a Levin u-transform of
     *note n: a49. terms, without error estimation.  The size of the
     workspace is O(3n).

 -- Function: void gsl_sum_levin_utrunc_free
          (gsl_sum_levin_utrunc_workspace *w)

     This function frees the memory associated with the workspace *note
     w: a4a.

 -- Function: int gsl_sum_levin_utrunc_accel (const double *array,
          size_t array_size, gsl_sum_levin_utrunc_workspace *w, double
          *sum_accel, double *abserr_trunc)

     This function takes the terms of a series in *note array: a4b. of
     size *note array_size: a4b. and computes the extrapolated limit of
     the series using a Levin u-transform.  Additional working space
     must be provided in *note w: a4b.  The extrapolated sum is stored
     in *note sum_accel: a4b.  The actual term-by-term sum is returned
     in ‘w->sum_plain’.  The algorithm terminates when the difference
     between two successive extrapolations reaches a minimum or is
     sufficiently small.  The difference between these two values is
     used as estimate of the error and is stored in *note abserr_trunc:
     a4b.  To improve the reliability of the algorithm the extrapolated
     values are replaced by moving averages when calculating the
     truncation error, smoothing out any fluctuations.


File: gsl-ref.info,  Node: Examples<25>,  Next: References and Further Reading<26>,  Prev: Acceleration functions without error estimation,  Up: Series Acceleration

33.3 Examples
=============

The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using
the series,

     \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + \dots

After ‘N’ terms the error in the sum is O(1/N), making direct summation
of the series converge slowly.

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_sum.h>

     #define N 20

     int
     main (void)
     {
       double t[N];
       double sum_accel, err;
       double sum = 0;
       int n;

       gsl_sum_levin_u_workspace * w
         = gsl_sum_levin_u_alloc (N);

       const double zeta_2 = M_PI * M_PI / 6.0;

       /* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */

       for (n = 0; n < N; n++)
         {
           double np1 = n + 1.0;
           t[n] = 1.0 / (np1 * np1);
           sum += t[n];
         }

       gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);

       printf ("term-by-term sum = % .16f using %d terms\n",
               sum, N);

       printf ("term-by-term sum = % .16f using %zu terms\n",
               w->sum_plain, w->terms_used);

       printf ("exact value      = % .16f\n", zeta_2);
       printf ("accelerated sum  = % .16f using %zu terms\n",
               sum_accel, w->terms_used);

       printf ("estimated error  = % .16f\n", err);
       printf ("actual error     = % .16f\n",
               sum_accel - zeta_2);

       gsl_sum_levin_u_free (w);
       return 0;
     }

The output below shows that the Levin u-transform is able to obtain an
estimate of the sum to 1 part in 10^{10} using the first eleven terms of
the series.  The error estimate returned by the function is also
accurate, giving the correct number of significant digits.

     term-by-term sum =  1.5961632439130233 using 20 terms
     term-by-term sum =  1.5759958390005426 using 13 terms
     exact value      =  1.6449340668482264
     accelerated sum  =  1.6449340669228176 using 13 terms
     estimated error  =  0.0000000000888360
     actual error     =  0.0000000000745912

Note that a direct summation of this series would require 10^{10} terms
to achieve the same precision as the accelerated sum does in 13 terms.


File: gsl-ref.info,  Node: References and Further Reading<26>,  Prev: Examples<25>,  Up: Series Acceleration

33.4 References and Further Reading
===================================

The algorithms used by these functions are described in the following
papers,

   * T. Fessler, W.F. Ford, D.A. Smith, HURRY: An acceleration algorithm
     for scalar sequences and series `ACM Transactions on Mathematical
     Software', 9(3):346–354, 1983.  and Algorithm 602 9(3):355–357,
     1983.

The theory of the u-transform was presented by Levin,

   * D. Levin, Development of Non-Linear Transformations for Improving
     Convergence of Sequences, `Intern.: J.: Computer Math.'
     B3:371–388, 1973.

A review paper on the Levin Transform is available online,

   * Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
     ‘http://arxiv.org/abs/math/0005209’


File: gsl-ref.info,  Node: Wavelet Transforms,  Next: Discrete Hankel Transforms,  Prev: Series Acceleration,  Up: Top

34 Wavelet Transforms
*********************

This chapter describes functions for performing Discrete Wavelet
Transforms (DWTs).  The library includes wavelets for real data in both
one and two dimensions.  The wavelet functions are declared in the
header files ‘gsl_wavelet.h’ and ‘gsl_wavelet2d.h’.

* Menu:

* Definitions: Definitions<2>.
* Initialization::
* Transform Functions::
* Examples: Examples<26>.
* References and Further Reading: References and Further Reading<27>.


File: gsl-ref.info,  Node: Definitions<2>,  Next: Initialization,  Up: Wavelet Transforms

34.1 Definitions
================

The continuous wavelet transform and its inverse are defined by the
relations,

     w(s, \tau) = \int_{-\infty}^\infty f(t) * \psi^*_{s,\tau}(t) dt

and,

     f(t) = \int_0^\infty ds \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau

where the basis functions \psi_{s,\tau} are obtained by scaling and
translation from a single function, referred to as the `mother wavelet'.

The discrete version of the wavelet transform acts on equally-spaced
samples, with fixed scaling and translation steps (s, \tau).  The
frequency and time axes are sampled `dyadically' on scales of 2^j
through a level parameter j.

The resulting family of functions \{\psi_{j,n}\} constitutes an
orthonormal basis for square-integrable signals.  The discrete wavelet
transform is an O(N) algorithm, and is also referred to as the `fast
wavelet transform'.


File: gsl-ref.info,  Node: Initialization,  Next: Transform Functions,  Prev: Definitions<2>,  Up: Wavelet Transforms

34.2 Initialization
===================

 -- Type: gsl_wavelet

     This structure contains the filter coefficients defining the
     wavelet and any associated offset parameters.

 -- Function: *note gsl_wavelet: a52. *gsl_wavelet_alloc (const
          gsl_wavelet_type *T, size_t k)

     This function allocates and initializes a wavelet object of type
     *note T: a53.  The parameter *note k: a53. selects the specific
     member of the wavelet family.  A null pointer is returned if
     insufficient memory is available or if a unsupported member is
     selected.

The following wavelet types are implemented:

 -- Type: gsl_wavelet_type

      -- Variable: *note gsl_wavelet_type: a54. *gsl_wavelet_daubechies
      -- Variable: *note gsl_wavelet_type: a54.
               *gsl_wavelet_daubechies_centered

          This is the Daubechies wavelet family of maximum phase with
          k/2 vanishing moments.  The implemented wavelets are k=4, 6,
          \dots, 20, with ‘k’ even.

      -- Variable: *note gsl_wavelet_type: a54. *gsl_wavelet_haar
      -- Variable: *note gsl_wavelet_type: a54.
               *gsl_wavelet_haar_centered

          This is the Haar wavelet.  The only valid choice of k for the
          Haar wavelet is k=2.

      -- Variable: *note gsl_wavelet_type: a54. *gsl_wavelet_bspline
      -- Variable: *note gsl_wavelet_type: a54.
               *gsl_wavelet_bspline_centered

          This is the biorthogonal B-spline wavelet family of order
          (i,j).  The implemented values of k = 100*i + j are 103, 105,
          202, 204, 206, 208, 301, 303, 305 307, 309.

The centered forms of the wavelets align the coefficients of the various
sub-bands on edges.  Thus the resulting visualization of the
coefficients of the wavelet transform in the phase plane is easier to
understand.

 -- Function: const char *gsl_wavelet_name (const gsl_wavelet *w)

     This function returns a pointer to the name of the wavelet family
     for *note w: a5b.

 -- Function: void gsl_wavelet_free (gsl_wavelet *w)

     This function frees the wavelet object *note w: a5c.

 -- Type: gsl_wavelet_workspace

     This structure contains scratch space of the same size as the input
     data and is used to hold intermediate results during the transform.

 -- Function: *note gsl_wavelet_workspace: a5d.
          *gsl_wavelet_workspace_alloc (size_t n)

     This function allocates a workspace for the discrete wavelet
     transform.  To perform a one-dimensional transform on *note n: a5e.
     elements, a workspace of size *note n: a5e. must be provided.  For
     two-dimensional transforms of *note n: a5e.-by-*note n: a5e.
     matrices it is sufficient to allocate a workspace of size *note n:
     a5e, since the transform operates on individual rows and columns.
     A null pointer is returned if insufficient memory is available.

 -- Function: void gsl_wavelet_workspace_free (gsl_wavelet_workspace
          *work)

     This function frees the allocated workspace *note work: a5f.


File: gsl-ref.info,  Node: Transform Functions,  Next: Examples<26>,  Prev: Initialization,  Up: Wavelet Transforms

34.3 Transform Functions
========================

This sections describes the actual functions performing the discrete
wavelet transform.  Note that the transforms use periodic boundary
conditions.  If the signal is not periodic in the sample length then
spurious coefficients will appear at the beginning and end of each level
of the transform.

* Menu:

* Wavelet transforms in one dimension::
* Wavelet transforms in two dimension::


File: gsl-ref.info,  Node: Wavelet transforms in one dimension,  Next: Wavelet transforms in two dimension,  Up: Transform Functions

34.3.1 Wavelet transforms in one dimension
------------------------------------------

 -- Function: int gsl_wavelet_transform (const gsl_wavelet *w, double
          *data, size_t stride, size_t n, gsl_wavelet_direction dir,
          gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet_transform_forward (const gsl_wavelet *w,
          double *data, size_t stride, size_t n, gsl_wavelet_workspace
          *work)
 -- Function: int gsl_wavelet_transform_inverse (const gsl_wavelet *w,
          double *data, size_t stride, size_t n, gsl_wavelet_workspace
          *work)

     These functions compute in-place forward and inverse discrete
     wavelet transforms of length *note n: a64. with stride *note
     stride: a64. on the array *note data: a64.  The length of the
     transform *note n: a64. is restricted to powers of two.  For the
     ‘transform’ version of the function the argument ‘dir’ can be
     either ‘forward’ (+1) or ‘backward’ (-1).  A workspace *note work:
     a64. of length *note n: a64. must be provided.

     For the forward transform, the elements of the original array are
     replaced by the discrete wavelet transform f_i \rightarrow w_{j,k}
     in a packed triangular storage layout, where ‘j’ is the index of
     the level j = 0 \dots J-1 and ‘k’ is the index of the coefficient
     within each level, k = 0 \dots 2^j - 1.  The total number of levels
     is J = \log_2(n).  The output data has the following form,

          (s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0},\cdots, d_{j,k},\cdots, d_{J-1,2^{J-1} - 1})

     where the first element is the smoothing coefficient s_{-1,0},
     followed by the detail coefficients d_{j,k} for each level j.  The
     backward transform inverts these coefficients to obtain the
     original data.

     These functions return a status of ‘GSL_SUCCESS’ upon successful
     completion.  *note GSL_EINVAL: 2b. is returned if *note n: a64. is
     not an integer power of 2 or if insufficient workspace is provided.


File: gsl-ref.info,  Node: Wavelet transforms in two dimension,  Prev: Wavelet transforms in one dimension,  Up: Transform Functions

34.3.2 Wavelet transforms in two dimension
------------------------------------------

The library provides functions to perform two-dimensional discrete
wavelet transforms on square matrices.  The matrix dimensions must be an
integer power of two.  There are two possible orderings of the rows and
columns in the two-dimensional wavelet transform, referred to as the
“standard” and “non-standard” forms.

The “standard” transform performs a complete discrete wavelet transform
on the rows of the matrix, followed by a separate complete discrete
wavelet transform on the columns of the resulting row-transformed
matrix.  This procedure uses the same ordering as a two-dimensional
Fourier transform.

The “non-standard” transform is performed in interleaved passes on the
rows and columns of the matrix for each level of the transform.  The
first level of the transform is applied to the matrix rows, and then to
the matrix columns.  This procedure is then repeated across the rows and
columns of the data for the subsequent levels of the transform, until
the full discrete wavelet transform is complete.  The non-standard form
of the discrete wavelet transform is typically used in image analysis.

The functions described in this section are declared in the header file
‘gsl_wavelet2d.h’.

 -- Function: int gsl_wavelet2d_transform (const gsl_wavelet *w, double
          *data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_direction dir, gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_transform_forward (const gsl_wavelet *w,
          double *data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_transform_inverse (const gsl_wavelet *w,
          double *data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace *work)

     These functions compute two-dimensional in-place forward and
     inverse discrete wavelet transforms in standard form on the array
     *note data: a68. stored in row-major form with dimensions *note
     size1: a68. and *note size2: a68. and physical row length *note
     tda: a68.  The dimensions must be equal (square matrix) and are
     restricted to powers of two.  For the ‘transform’ version of the
     function the argument ‘dir’ can be either ‘forward’ (+1) or
     ‘backward’ (-1).  A workspace *note work: a68. of the appropriate
     size must be provided.  On exit, the appropriate elements of the
     array *note data: a68. are replaced by their two-dimensional
     wavelet transform.

     The functions return a status of ‘GSL_SUCCESS’ upon successful
     completion.  *note GSL_EINVAL: 2b. is returned if *note size1: a68.
     and *note size2: a68. are not equal and integer powers of 2, or if
     insufficient workspace is provided.

 -- Function: int gsl_wavelet2d_transform_matrix (const gsl_wavelet *w,
          gsl_matrix *m, gsl_wavelet_direction dir,
          gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_transform_matrix_forward (const
          gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_transform_matrix_inverse (const
          gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_workspace *work)

     These functions compute the two-dimensional in-place wavelet
     transform on a matrix *note m: a6b.

 -- Function: int gsl_wavelet2d_nstransform (const gsl_wavelet *w,
          double *data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_direction dir, gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_nstransform_forward (const gsl_wavelet
          *w, double *data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_nstransform_inverse (const gsl_wavelet
          *w, double *data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace *work)

     These functions compute the two-dimensional wavelet transform in
     non-standard form.

 -- Function: int gsl_wavelet2d_nstransform_matrix (const gsl_wavelet
          *w, gsl_matrix *m, gsl_wavelet_direction dir,
          gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_nstransform_matrix_forward (const
          gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_workspace *work)
 -- Function: int gsl_wavelet2d_nstransform_matrix_inverse (const
          gsl_wavelet *w, gsl_matrix *m, gsl_wavelet_workspace *work)

     These functions compute the non-standard form of the
     two-dimensional in-place wavelet transform on a matrix *note m:
     a71.


File: gsl-ref.info,  Node: Examples<26>,  Next: References and Further Reading<27>,  Prev: Transform Functions,  Up: Wavelet Transforms

34.4 Examples
=============

The following program demonstrates the use of the one-dimensional
wavelet transform functions.  It computes an approximation to an input
signal (of length 256) using the 20 largest components of the wavelet
transform, while setting the others to zero.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_wavelet.h>

     int
     main (int argc, char **argv)
     {
       (void)(argc); /* avoid unused parameter warning */
       int i, n = 256, nc = 20;
       double *orig_data = malloc (n * sizeof (double));
       double *data = malloc (n * sizeof (double));
       double *abscoeff = malloc (n * sizeof (double));
       size_t *p = malloc (n * sizeof (size_t));

       FILE * f;
       gsl_wavelet *w;
       gsl_wavelet_workspace *work;

       w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
       work = gsl_wavelet_workspace_alloc (n);

       f = fopen (argv[1], "r");
       for (i = 0; i < n; i++)
         {
           fscanf (f, "%lg", &orig_data[i]);
           data[i] = orig_data[i];
         }
       fclose (f);

       gsl_wavelet_transform_forward (w, data, 1, n, work);

       for (i = 0; i < n; i++)
         {
           abscoeff[i] = fabs (data[i]);
         }

       gsl_sort_index (p, abscoeff, 1, n);

       for (i = 0; (i + nc) < n; i++)
         data[p[i]] = 0;

       gsl_wavelet_transform_inverse (w, data, 1, n, work);

       for (i = 0; i < n; i++)
         {
           printf ("%g %g\n", orig_data[i], data[i]);
         }

       gsl_wavelet_free (w);
       gsl_wavelet_workspace_free (work);

       free (data);
       free (orig_data);
       free (abscoeff);
       free (p);
       return 0;
     }

The output can be used with the GNU plotutils ‘graph’ program:

     $ ./a.out ecg.dat > dwt.txt
     $ graph -T ps -x 0 256 32 -h 0.3 -a dwt.txt > dwt.ps

Fig.  %s shows an original and compressed version of a sample ECG
recording from the MIT-BIH Arrhythmia Database, part of the PhysioNet
archive of public-domain of medical datasets.

[gsl-ref-figures/dwt]
Figure: Original (upper) and wavelet-compressed (lower) ECG signals,
using the 20 largest components of the Daubechies(4) discrete wavelet
transform.


File: gsl-ref.info,  Node: References and Further Reading<27>,  Prev: Examples<26>,  Up: Wavelet Transforms

34.5 References and Further Reading
===================================

The mathematical background to wavelet transforms is covered in the
original lectures by Daubechies,

   * Ingrid Daubechies.  Ten Lectures on Wavelets.  `CBMS-NSF Regional
     Conference Series in Applied Mathematics' (1992), SIAM, ISBN
     0898712742.

An easy to read introduction to the subject with an emphasis on the
application of the wavelet transform in various branches of science is,

   * Paul S. Addison.  `The Illustrated Wavelet Transform Handbook'.
     Institute of Physics Publishing (2002), ISBN 0750306920.

For extensive coverage of signal analysis by wavelets, wavelet packets
and local cosine bases see,

   * S. G. Mallat.  `A wavelet tour of signal processing' (Second
     edition).  Academic Press (1999), ISBN 012466606X.

The concept of multiresolution analysis underlying the wavelet transform
is described in,

   * S. G. Mallat.  Multiresolution Approximations and Wavelet
     Orthonormal Bases of L^2(R). `Transactions of the American
     Mathematical Society', 315(1), 1989, 69–87.

   * S. G. Mallat.  A Theory for Multiresolution Signal
     Decomposition—The Wavelet Representation.  `IEEE Transactions on
     Pattern Analysis and Machine Intelligence', 11, 1989, 674–693.

The coefficients for the individual wavelet families implemented by the
library can be found in the following papers,

   * I. Daubechies.  Orthonormal Bases of Compactly Supported Wavelets.
     `Communications on Pure and Applied Mathematics', 41 (1988)
     909–996.

   * A. Cohen, I. Daubechies, and J.-C. Feauveau.  Biorthogonal Bases of
     Compactly Supported Wavelets.  `Communications on Pure and Applied
     Mathematics', 45 (1992) 485–560.

The PhysioNet archive of physiological datasets can be found online at
‘http://www.physionet.org/’ and is described in the following paper,

   * Goldberger et al.  PhysioBank, PhysioToolkit, and PhysioNet:
     Components of a New Research Resource for Complex Physiologic
     Signals.  `Circulation' 101(23):e215-e220 2000.


File: gsl-ref.info,  Node: Discrete Hankel Transforms,  Next: One Dimensional Root-Finding,  Prev: Wavelet Transforms,  Up: Top

35 Discrete Hankel Transforms
*****************************

This chapter describes functions for performing Discrete Hankel
Transforms (DHTs).  The functions are declared in the header file
‘gsl_dht.h’.

* Menu:

* Definitions: Definitions<3>.
* Functions: Functions<2>.
* References and Further Reading: References and Further Reading<28>.


File: gsl-ref.info,  Node: Definitions<3>,  Next: Functions<2>,  Up: Discrete Hankel Transforms

35.1 Definitions
================

The discrete Hankel transform acts on a vector of sampled data, where
the samples are assumed to have been taken at points related to the
zeros of a Bessel function of fixed order; compare this to the case of
the discrete Fourier transform, where samples are taken at points
related to the zeroes of the sine or cosine function.

Starting with its definition, the Hankel transform (or Bessel transform)
of order \nu of a function f with \nu > -1/2 is defined as (see Johnson,
1987 and Lemoine, 1994)

     F_\nu(u) = \int_0^\infty f(t) J_\nu(u t) t dt

If the integral exists, F_\nu is called the Hankel transformation of f.
The reverse transform is given by

     f(t) = \int_0^\infty F_\nu(u) J_\nu(u t) u du

where \int_0^\infty f(t) t^{1/2} dt must exist and be absolutely
convergent, and where f(t) satisfies Dirichlet’s conditions (of limited
total fluctuations) in the interval [0,\infty].

Now the discrete Hankel transform works on a discrete function f, which
is sampled on points n=1...M located at positions
t_n=(j_{\nu,n}/j_{\nu,M}) X in real space and at u_n=j_{\nu,n}/X in
reciprocal space.  Here, j_{\nu,m} are the m-th zeros of the Bessel
function J_\nu(x) arranged in ascending order.  Moreover, the discrete
functions are assumed to be band limited, so f(t_n)=0 and F(u_n)=0 for
n>M. Accordingly, the function f is defined on the interval [0,X].

Following the work of Johnson, 1987 and Lemoine, 1994, the discrete
Hankel transform is given by

     F_\nu(u_m) = (2 X^2 / j_(\nu,M)^2)
           \sum_{k=1}^{M-1} f(j_(\nu,k) X/j_(\nu,M))
               (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,k))^2).

It is this discrete expression which defines the discrete Hankel
transform calculated by GSL. In GSL, forward and backward transforms are
defined equally and calculate F_\nu(u_m).  Following Johnson, the
backward transform reads

     f(t_k) = (2 / X^2)
           \sum_{m=1}^{M-1} F(j_(\nu,m)/X)
               (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,m))^2).

Obviously, using the forward transform instead of the backward transform
gives an additional factor X^4/j_{\nu,M}^2=t_m^2/u_m^2.

The kernel in the summation above defines the matrix of the \nu-Hankel
transform of size M-1.  The coefficients of this matrix, being dependent
on \nu and M, must be precomputed and stored; the *note gsl_dht: a78.
object encapsulates this data.  The allocation function *note
gsl_dht_alloc(): a79. returns a *note gsl_dht: a78. object which must be
properly initialized with *note gsl_dht_init(): a7a. before it can be
used to perform transforms on data sample vectors, for fixed \nu and M,
using the *note gsl_dht_apply(): a7b. function.  The implementation
allows to define the length X of the fundamental interval, for
convenience, while discrete Hankel transforms are often defined on the
unit interval instead of [0,X].

Notice that by assumption f(t) vanishes at the endpoints of the
interval, consistent with the inversion formula and the sampling formula
given above.  Therefore, this transform corresponds to an orthogonal
expansion in eigenfunctions of the Dirichlet problem for the Bessel
differential equation.


File: gsl-ref.info,  Node: Functions<2>,  Next: References and Further Reading<28>,  Prev: Definitions<3>,  Up: Discrete Hankel Transforms

35.2 Functions
==============

 -- Type: gsl_dht

     Workspace for computing discrete Hankel transforms

 -- Function: *note gsl_dht: a78. *gsl_dht_alloc (size_t size)

     This function allocates a Discrete Hankel transform object of size
     *note size: a79.

 -- Function: int gsl_dht_init (gsl_dht *t, double nu, double xmax)

     This function initializes the transform *note t: a7a. for the given
     values of *note nu: a7a. and *note xmax: a7a.

 -- Function: *note gsl_dht: a78. *gsl_dht_new (size_t size, double nu,
          double xmax)

     This function allocates a Discrete Hankel transform object of size
     *note size: a7d. and initializes it for the given values of *note
     nu: a7d. and *note xmax: a7d.

 -- Function: void gsl_dht_free (gsl_dht *t)

     This function frees the transform *note t: a7e.

 -- Function: int gsl_dht_apply (const gsl_dht *t, double *f_in, double
          *f_out)

     This function applies the transform *note t: a7b. to the array
     *note f_in: a7b. whose size is equal to the size of the transform.
     The result is stored in the array *note f_out: a7b. which must be
     of the same length.

     Applying this function to its output gives the original data
     multiplied by (X^2/j_{\nu,M})^2, up to numerical errors.

 -- Function: double gsl_dht_x_sample (const gsl_dht *t, int n)

     This function returns the value of the *note n: a7f.-th sample
     point in the unit interval, {({j_{\nu,n+1}} / {j_{\nu,M}}}) X.
     These are the points where the function f(t) is assumed to be
     sampled.

 -- Function: double gsl_dht_k_sample (const gsl_dht *t, int n)

     This function returns the value of the *note n: a80.-th sample
     point in “k-space”, {{j_{\nu,n+1}} / X}.


File: gsl-ref.info,  Node: References and Further Reading<28>,  Prev: Functions<2>,  Up: Discrete Hankel Transforms

35.3 References and Further Reading
===================================

The algorithms used by these functions are described in the following
papers,

   * 
       8. Fisk Johnson, Comp.: Phys.: Comm.: 43, 181 (1987).

   * 
       4. Lemoine, J. Chem.: Phys.: 101, 3936 (1994).


File: gsl-ref.info,  Node: One Dimensional Root-Finding,  Next: One Dimensional Minimization,  Prev: Discrete Hankel Transforms,  Up: Top

36 One Dimensional Root-Finding
*******************************

This chapter describes routines for finding roots of arbitrary
one-dimensional functions.  The library provides low level components
for a variety of iterative solvers and convergence tests.  These can be
combined by the user to achieve the desired solution, with full access
to the intermediate steps of the iteration.  Each class of methods uses
the same framework, so that you can switch between solvers at runtime
without needing to recompile your program.  Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs.

The header file ‘gsl_roots.h’ contains prototypes for the root finding
functions and related declarations.

* Menu:

* Overview::
* Caveats::
* Initializing the Solver::
* Providing the function to solve::
* Search Bounds and Guesses::
* Iteration::
* Search Stopping Parameters::
* Root Bracketing Algorithms::
* Root Finding Algorithms using Derivatives::
* Examples: Examples<27>.
* References and Further Reading: References and Further Reading<29>.


File: gsl-ref.info,  Node: Overview,  Next: Caveats,  Up: One Dimensional Root-Finding

36.1 Overview
=============

One-dimensional root finding algorithms can be divided into two classes,
`root bracketing' and `root polishing'.  Algorithms which proceed by
bracketing a root are guaranteed to converge.  Bracketing algorithms
begin with a bounded region known to contain a root.  The size of this
bounded region is reduced, iteratively, until it encloses the root to a
desired tolerance.  This provides a rigorous error estimate for the
location of the root.

The technique of `root polishing' attempts to improve an initial guess
to the root.  These algorithms converge only if started “close enough”
to a root, and sacrifice a rigorous error bound for speed.  By
approximating the behavior of a function in the vicinity of a root they
attempt to find a higher order improvement of an initial guess.  When
the behavior of the function is compatible with the algorithm and a good
initial guess is available a polishing algorithm can provide rapid
convergence.

In GSL both types of algorithm are available in similar frameworks.  The
user provides a high-level driver for the algorithms, and the library
provides the individual functions necessary for each of the steps.
There are three main phases of the iteration.  The steps are,

   * initialize solver state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

The state for bracketing solvers is held in a *note gsl_root_fsolver:
a85. struct.  The updating procedure uses only function evaluations (not
derivatives).  The state for root polishing solvers is held in a *note
gsl_root_fdfsolver: a86. struct.  The updates require both the function
and its derivative (hence the name ‘fdf’) to be supplied by the user.


File: gsl-ref.info,  Node: Caveats,  Next: Initializing the Solver,  Prev: Overview,  Up: One Dimensional Root-Finding

36.2 Caveats
============

Note that root finding functions can only search for one root at a time.
When there are several roots in the search area, the first root to be
found will be returned; however it is difficult to predict which of the
roots this will be.  `In most cases, no error will be reported if you
try to find a root in an area where there is more than one.'

Care must be taken when a function may have a multiple root (such as
f(x) = (x-x_0)^2 or f(x) = (x-x_0)^3.  It is not possible to use
root-bracketing algorithms on even-multiplicity roots.  For these
algorithms the initial interval must contain a zero-crossing, where the
function is negative at one end of the interval and positive at the
other end.  Roots with even-multiplicity do not cross zero, but only
touch it instantaneously.  Algorithms based on root bracketing will
still work for odd-multiplicity roots (e.g.  cubic, quintic, …).  Root
polishing algorithms generally work with higher multiplicity roots, but
at a reduced rate of convergence.  In these cases the `Steffenson
algorithm' can be used to accelerate the convergence of multiple roots.

While it is not absolutely required that f have a root within the search
region, numerical root finding functions should not be used haphazardly
to check for the `existence' of roots.  There are better ways to do
this.  Because it is easy to create situations where numerical root
finders can fail, it is a bad idea to throw a root finder at a function
you do not know much about.  In general it is best to examine the
function visually by plotting before searching for a root.


File: gsl-ref.info,  Node: Initializing the Solver,  Next: Providing the function to solve,  Prev: Caveats,  Up: One Dimensional Root-Finding

36.3 Initializing the Solver
============================

 -- Type: gsl_root_fsolver

     This is a workspace for finding roots using methods which do not
     require derivatives.

 -- Type: gsl_root_fdfsolver

     This is a workspace for finding roots using methods which require
     derivatives.

 -- Function: *note gsl_root_fsolver: a85. *gsl_root_fsolver_alloc
          (const gsl_root_fsolver_type *T)

     This function returns a pointer to a newly allocated instance of a
     solver of type *note T: a89.  For example, the following code
     creates an instance of a bisection solver:

          const gsl_root_fsolver_type * T = gsl_root_fsolver_bisection;
          gsl_root_fsolver * s = gsl_root_fsolver_alloc (T);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: *note gsl_root_fdfsolver: a86. *gsl_root_fdfsolver_alloc
          (const gsl_root_fdfsolver_type *T)

     This function returns a pointer to a newly allocated instance of a
     derivative-based solver of type *note T: a8a.  For example, the
     following code creates an instance of a Newton-Raphson solver:

          const gsl_root_fdfsolver_type * T = gsl_root_fdfsolver_newton;
          gsl_root_fdfsolver * s = gsl_root_fdfsolver_alloc (T);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_root_fsolver_set (gsl_root_fsolver *s,
          gsl_function *f, double x_lower, double x_upper)

     This function initializes, or reinitializes, an existing solver
     *note s: a8b. to use the function *note f: a8b. and the initial
     search interval [*note x_lower: a8b, *note x_upper: a8b.].

 -- Function: int gsl_root_fdfsolver_set (gsl_root_fdfsolver *s,
          gsl_function_fdf *fdf, double root)

     This function initializes, or reinitializes, an existing solver
     *note s: a8c. to use the function and derivative *note fdf: a8c.
     and the initial guess *note root: a8c.

 -- Function: void gsl_root_fsolver_free (gsl_root_fsolver *s)
 -- Function: void gsl_root_fdfsolver_free (gsl_root_fdfsolver *s)

     These functions free all the memory associated with the solver
     *note s: a8e.

 -- Function: const char *gsl_root_fsolver_name (const gsl_root_fsolver
          *s)
 -- Function: const char *gsl_root_fdfsolver_name (const
          gsl_root_fdfsolver *s)

     These functions return a pointer to the name of the solver.  For
     example:

          printf ("s is a '%s' solver\n", gsl_root_fsolver_name (s));

     would print something like ‘s is a 'bisection' solver’.


File: gsl-ref.info,  Node: Providing the function to solve,  Next: Search Bounds and Guesses,  Prev: Initializing the Solver,  Up: One Dimensional Root-Finding

36.4 Providing the function to solve
====================================

You must provide a continuous function of one variable for the root
finders to operate on, and, sometimes, its first derivative.  In order
to allow for general parameters the functions are defined by the
following data types:

 -- Type: gsl_function

     This data type defines a general function with parameters.

     ‘double (* function) (double x, void * params)’

          this function should return the value f(x,params) for argument
          ‘x’ and parameters ‘params’

     ‘void * params’

          a pointer to the parameters of the function

Here is an example for the general quadratic function,

     f(x) = a x^2 + b x + c

with a = 3, b = 2, c = 1.  The following code defines a *note
gsl_function: a93. ‘F’ which you could pass to a root finder as a
function pointer:

     struct my_f_params { double a; double b; double c; };

     double
     my_f (double x, void * p)
       {
         struct my_f_params * params = (struct my_f_params *)p;
         double a = (params->a);
         double b = (params->b);
         double c = (params->c);

         return  (a * x + b) * x + c;
       }

     gsl_function F;
     struct my_f_params params = { 3.0, 2.0, 1.0 };

     F.function = &my_f;
     F.params = &params;

The function f(x) can be evaluated using the macro ‘GSL_FN_EVAL(&F,x)’
defined in ‘gsl_math.h’.

 -- Type: gsl_function_fdf

     This data type defines a general function with parameters and its
     first derivative.

     ‘double (* f) (double x, void * params)’

          this function should return the value of f(x,params) for
          argument ‘x’ and parameters ‘params’

     ‘double (* df) (double x, void * params)’

          this function should return the value of the derivative of ‘f’
          with respect to ‘x’, f'(x,params), for argument ‘x’ and
          parameters ‘params’

     ‘void (* fdf) (double x, void * params, double * f, double * df)’

          this function should set the values of the function ‘f’ to
          f(x,params) and its derivative ‘df’ to f'(x,params) for
          argument ‘x’ and parameters ‘params’.  This function provides
          an optimization of the separate functions for f(x) and
          f'(x)—it is always faster to compute the function and its
          derivative at the same time.

     ‘void * params’

          a pointer to the parameters of the function

Here is an example where f(x) = \exp(2x):

     double
     my_f (double x, void * params)
     {
        return exp (2 * x);
     }

     double
     my_df (double x, void * params)
     {
        return 2 * exp (2 * x);
     }

     void
     my_fdf (double x, void * params,
             double * f, double * df)
     {
        double t = exp (2 * x);

        *f = t;
        *df = 2 * t;   /* uses existing value */
     }

     gsl_function_fdf FDF;

     FDF.f = &my_f;
     FDF.df = &my_df;
     FDF.fdf = &my_fdf;
     FDF.params = 0;

The function f(x) can be evaluated using the macro
‘GSL_FN_FDF_EVAL_F(&FDF,x)’ and the derivative f'(x) can be evaluated
using the macro ‘GSL_FN_FDF_EVAL_DF(&FDF,x)’.  Both the function y =
f(x) and its derivative dy = f'(x) can be evaluated at the same time
using the macro ‘GSL_FN_FDF_EVAL_F_DF(&FDF,x,y,dy)’.  The macro stores
f(x) in its ‘y’ argument and f'(x) in its ‘dy’ argument—both of these
should be pointers to ‘double’.


File: gsl-ref.info,  Node: Search Bounds and Guesses,  Next: Iteration,  Prev: Providing the function to solve,  Up: One Dimensional Root-Finding

36.5 Search Bounds and Guesses
==============================

You provide either search bounds or an initial guess; this section
explains how search bounds and guesses work and how function arguments
control them.

A guess is simply an x value which is iterated until it is within the
desired precision of a root.  It takes the form of a ‘double’.

Search bounds are the endpoints of an interval which is iterated until
the length of the interval is smaller than the requested precision.  The
interval is defined by two values, the lower limit and the upper limit.
Whether the endpoints are intended to be included in the interval or not
depends on the context in which the interval is used.


File: gsl-ref.info,  Node: Iteration,  Next: Search Stopping Parameters,  Prev: Search Bounds and Guesses,  Up: One Dimensional Root-Finding

36.6 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

 -- Function: int gsl_root_fsolver_iterate (gsl_root_fsolver *s)
 -- Function: int gsl_root_fdfsolver_iterate (gsl_root_fdfsolver *s)

     These functions perform a single iteration of the solver *note s:
     a98.  If the iteration encounters an unexpected problem then an
     error code will be returned,

     ‘GSL_EBADFUNC’

          the iteration encountered a singular point where the function
          or its derivative evaluated to ‘Inf’ or ‘NaN’.

     ‘GSL_EZERODIV’

          the derivative of the function vanished at the iteration
          point, preventing the algorithm from continuing without a
          division by zero.

The solver maintains a current best estimate of the root at all times.
The bracketing solvers also keep track of the current best interval
bounding the root.  This information can be accessed with the following
auxiliary functions,

 -- Function: double gsl_root_fsolver_root (const gsl_root_fsolver *s)
 -- Function: double gsl_root_fdfsolver_root (const gsl_root_fdfsolver
          *s)

     These functions return the current estimate of the root for the
     solver *note s: a9a.

 -- Function: double gsl_root_fsolver_x_lower (const gsl_root_fsolver
          *s)
 -- Function: double gsl_root_fsolver_x_upper (const gsl_root_fsolver
          *s)

     These functions return the current bracketing interval for the
     solver *note s: a9c.


File: gsl-ref.info,  Node: Search Stopping Parameters,  Next: Root Bracketing Algorithms,  Prev: Iteration,  Up: One Dimensional Root-Finding

36.7 Search Stopping Parameters
===============================

A root finding procedure should stop when one of the following
conditions is true:

   * A root has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result in
several standard ways.

 -- Function: int gsl_root_test_interval (double x_lower, double
          x_upper, double epsabs, double epsrel)

     This function tests for the convergence of the interval [*note
     x_lower: a9e, *note x_upper: a9e.] with absolute error *note
     epsabs: a9e. and relative error *note epsrel: a9e.  The test
     returns ‘GSL_SUCCESS’ if the following condition is achieved,

          |a - b| < epsabs + epsrel min(|a|,|b|)

     when the interval x = [a,b] does not include the origin.  If the
     interval includes the origin then \min(|a|,|b|) is replaced by zero
     (which is the minimum value of |x| over the interval).  This
     ensures that the relative error is accurately estimated for roots
     close to the origin.

     This condition on the interval also implies that any estimate of
     the root r in the interval satisfies the same condition with
     respect to the true root r^*,

          |r - r^*| < epsabs + epsrel r^*

     assuming that the true root r^* is contained within the interval.

 -- Function: int gsl_root_test_delta (double x1, double x0, double
          epsabs, double epsrel)

     This function tests for the convergence of the sequence *note x0:
     a9f, *note x1: a9f. with absolute error *note epsabs: a9f. and
     relative error *note epsrel: a9f.  The test returns ‘GSL_SUCCESS’
     if the following condition is achieved,

          |x_1 - x_0| < epsabs + epsrel |x_1|

     and returns ‘GSL_CONTINUE’ otherwise.

 -- Function: int gsl_root_test_residual (double f, double epsabs)

     This function tests the residual value *note f: aa0. against the
     absolute error bound *note epsabs: aa0.  The test returns
     ‘GSL_SUCCESS’ if the following condition is achieved,

          |f| < epsabs

     and returns ‘GSL_CONTINUE’ otherwise.  This criterion is suitable
     for situations where the precise location of the root, x, is
     unimportant provided a value can be found where the residual,
     |f(x)|, is small enough.


File: gsl-ref.info,  Node: Root Bracketing Algorithms,  Next: Root Finding Algorithms using Derivatives,  Prev: Search Stopping Parameters,  Up: One Dimensional Root-Finding

36.8 Root Bracketing Algorithms
===============================

The root bracketing algorithms described in this section require an
initial interval which is guaranteed to contain a root—if a and b are
the endpoints of the interval then f(a) must differ in sign from f(b).
This ensures that the function crosses zero at least once in the
interval.  If a valid initial interval is used then these algorithm
cannot fail, provided the function is well-behaved.

Note that a bracketing algorithm cannot find roots of even degree, since
these do not cross the x-axis.

 -- Type: gsl_root_fsolver_type

      -- Variable: *note gsl_root_fsolver_type: aa2.
               *gsl_root_fsolver_bisection

          The `bisection algorithm' is the simplest method of bracketing
          the roots of a function.  It is the slowest algorithm provided
          by the library, with linear convergence.

          On each iteration, the interval is bisected and the value of
          the function at the midpoint is calculated.  The sign of this
          value is used to determine which half of the interval does not
          contain a root.  That half is discarded to give a new, smaller
          interval containing the root.  This procedure can be continued
          indefinitely until the interval is sufficiently small.

          At any time the current estimate of the root is taken as the
          midpoint of the interval.

      -- Variable: *note gsl_root_fsolver_type: aa2.
               *gsl_root_fsolver_falsepos

          The `false position algorithm' is a method of finding roots
          based on linear interpolation.  Its convergence is linear, but
          it is usually faster than bisection.

          On each iteration a line is drawn between the endpoints
          (a,f(a)) and (b,f(b)) and the point where this line crosses
          the x-axis taken as a “midpoint”.  The value of the function
          at this point is calculated and its sign is used to determine
          which side of the interval does not contain a root.  That side
          is discarded to give a new, smaller interval containing the
          root.  This procedure can be continued indefinitely until the
          interval is sufficiently small.

          The best estimate of the root is taken from the linear
          interpolation of the interval on the current iteration.

      -- Variable: *note gsl_root_fsolver_type: aa2.
               *gsl_root_fsolver_brent

          The `Brent-Dekker method' (referred to here as `Brent’s
          method') combines an interpolation strategy with the bisection
          algorithm.  This produces a fast algorithm which is still
          robust.

          On each iteration Brent’s method approximates the function
          using an interpolating curve.  On the first iteration this is
          a linear interpolation of the two endpoints.  For subsequent
          iterations the algorithm uses an inverse quadratic fit to the
          last three points, for higher accuracy.  The intercept of the
          interpolating curve with the x-axis is taken as a guess for
          the root.  If it lies within the bounds of the current
          interval then the interpolating point is accepted, and used to
          generate a smaller interval.  If the interpolating point is
          not accepted then the algorithm falls back to an ordinary
          bisection step.

          The best estimate of the root is taken from the most recent
          interpolation or bisection.


File: gsl-ref.info,  Node: Root Finding Algorithms using Derivatives,  Next: Examples<27>,  Prev: Root Bracketing Algorithms,  Up: One Dimensional Root-Finding

36.9 Root Finding Algorithms using Derivatives
==============================================

The root polishing algorithms described in this section require an
initial guess for the location of the root.  There is no absolute
guarantee of convergence—the function must be suitable for this
technique and the initial guess must be sufficiently close to the root
for it to work.  When these conditions are satisfied then convergence is
quadratic.

These algorithms make use of both the function and its derivative.

 -- Type: gsl_root_fdfsolver_type

      -- Variable: *note gsl_root_fdfsolver_type: aa7.
               *gsl_root_fdfsolver_newton

          Newton’s Method is the standard root-polishing algorithm.  The
          algorithm begins with an initial guess for the location of the
          root.  On each iteration, a line tangent to the function f is
          drawn at that position.  The point where this line crosses the
          x-axis becomes the new guess.  The iteration is defined by the
          following sequence,

               x_{i+1} = x_i - f(x_i)/f'(x_i)

          Newton’s method converges quadratically for single roots, and
          linearly for multiple roots.

      -- Variable: *note gsl_root_fdfsolver_type: aa7.
               *gsl_root_fdfsolver_secant

          The `secant method' is a simplified version of Newton’s method
          which does not require the computation of the derivative on
          every step.

          On its first iteration the algorithm begins with Newton’s
          method, using the derivative to compute a first step,

               x_1 = x_0 - f(x_0)/f'(x_0)

          Subsequent iterations avoid the evaluation of the derivative
          by replacing it with a numerical estimate, the slope of the
          line through the previous two points,

               x_{i+1} = x_i f(x_i) / f'_{est} where
                f'_{est} = (f(x_i) - f(x_{i-1})/(x_i - x_{i-1})

          When the derivative does not change significantly in the
          vicinity of the root the secant method gives a useful saving.
          Asymptotically the secant method is faster than Newton’s
          method whenever the cost of evaluating the derivative is more
          than 0.44 times the cost of evaluating the function itself.
          As with all methods of computing a numerical derivative the
          estimate can suffer from cancellation errors if the separation
          of the points becomes too small.

          On single roots, the method has a convergence of order (1 +
          \sqrt 5)/2 (approximately 1.62).  It converges linearly for
          multiple roots.

      -- Variable: *note gsl_root_fdfsolver_type: aa7.
               *gsl_root_fdfsolver_steffenson

          The `Steffenson Method' (1) provides the fastest convergence
          of all the routines.  It combines the basic Newton algorithm
          with an Aitken “delta-squared” acceleration.  If the Newton
          iterates are x_i then the acceleration procedure generates a
          new sequence R_i,

               R_i = x_i - (x_{i+1} - x_i)^2 / (x_{i+2} - 2 x_{i+1} + x_{i})

          which converges faster than the original sequence under
          reasonable conditions.  The new sequence requires three terms
          before it can produce its first value so the method returns
          accelerated values on the second and subsequent iterations.
          On the first iteration it returns the ordinary Newton
          estimate.  The Newton iterate is also returned if the
          denominator of the acceleration term ever becomes zero.

          As with all acceleration procedures this method can become
          unstable if the function is not well-behaved.

   ---------- Footnotes ----------

   (1) (1) J.F. Steffensen (1873–1961).  The spelling used in the name
of the function is slightly incorrect, but has been preserved to avoid
incompatibility.


File: gsl-ref.info,  Node: Examples<27>,  Next: References and Further Reading<29>,  Prev: Root Finding Algorithms using Derivatives,  Up: One Dimensional Root-Finding

36.10 Examples
==============

For any root finding algorithm we need to prepare the function to be
solved.  For this example we will use the general quadratic equation
described earlier.  We first need a header file (‘demo_fn.h’) to define
the function parameters,

     struct quadratic_params
       {
         double a, b, c;
       };

     double quadratic (double x, void *params);
     double quadratic_deriv (double x, void *params);
     void quadratic_fdf (double x, void *params,
                         double *y, double *dy);

We place the function definitions in a separate file (‘demo_fn.c’),

     double
     quadratic (double x, void *params)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;
       double c = p->c;

       return (a * x + b) * x + c;
     }

     double
     quadratic_deriv (double x, void *params)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;

       return 2.0 * a * x + b;
     }

     void
     quadratic_fdf (double x, void *params,
                    double *y, double *dy)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;
       double c = p->c;

       *y = (a * x + b) * x + c;
       *dy = 2.0 * a * x + b;
     }

The first program uses the function solver ‘gsl_root_fsolver_brent’ for
Brent’s method and the general quadratic defined above to solve the
following equation,

     x^2 - 5 = 0

with solution x = \sqrt 5 = 2.236068...

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_roots.h>

     #include "demo_fn.h"
     #include "demo_fn.c"

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_root_fsolver_type *T;
       gsl_root_fsolver *s;
       double r = 0, r_expected = sqrt (5.0);
       double x_lo = 0.0, x_hi = 5.0;
       gsl_function F;
       struct quadratic_params params = {1.0, 0.0, -5.0};

       F.function = &quadratic;
       F.params = &params;

       T = gsl_root_fsolver_brent;
       s = gsl_root_fsolver_alloc (T);
       gsl_root_fsolver_set (s, &F, x_lo, x_hi);

       printf ("using %s method\n",
               gsl_root_fsolver_name (s));

       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
               "iter", "lower", "upper", "root",
               "err", "err(est)");

       do
         {
           iter++;
           status = gsl_root_fsolver_iterate (s);
           r = gsl_root_fsolver_root (s);
           x_lo = gsl_root_fsolver_x_lower (s);
           x_hi = gsl_root_fsolver_x_upper (s);
           status = gsl_root_test_interval (x_lo, x_hi,
                                            0, 0.001);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
                   iter, x_lo, x_hi,
                   r, r - r_expected,
                   x_hi - x_lo);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       gsl_root_fsolver_free (s);

       return status;
     }

Here are the results of the iterations:

     $ ./a.out
     using brent method
      iter [    lower,     upper]      root        err  err(est)
         1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
         2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
         3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
         4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
         5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
     Converged:
         6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666

If the program is modified to use the bisection solver instead of
Brent’s method, by changing ‘gsl_root_fsolver_brent’ to
‘gsl_root_fsolver_bisection’ the slower convergence of the Bisection
method can be observed:

     $ ./a.out
     using bisection method
      iter [    lower,     upper]      root        err  err(est)
         1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
         2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
         3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
         4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
         5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
         6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
         7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
         8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
         9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
        10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
        11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
     Converged:
        12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207

The next program solves the same function using a derivative solver
instead.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_roots.h>

     #include "demo_fn.h"
     #include "demo_fn.c"

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_root_fdfsolver_type *T;
       gsl_root_fdfsolver *s;
       double x0, x = 5.0, r_expected = sqrt (5.0);
       gsl_function_fdf FDF;
       struct quadratic_params params = {1.0, 0.0, -5.0};

       FDF.f = &quadratic;
       FDF.df = &quadratic_deriv;
       FDF.fdf = &quadratic_fdf;
       FDF.params = &params;

       T = gsl_root_fdfsolver_newton;
       s = gsl_root_fdfsolver_alloc (T);
       gsl_root_fdfsolver_set (s, &FDF, x);

       printf ("using %s method\n",
               gsl_root_fdfsolver_name (s));

       printf ("%-5s %10s %10s %10s\n",
               "iter", "root", "err", "err(est)");
       do
         {
           iter++;
           status = gsl_root_fdfsolver_iterate (s);
           x0 = x;
           x = gsl_root_fdfsolver_root (s);
           status = gsl_root_test_delta (x, x0, 0, 1e-3);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d %10.7f %+10.7f %10.7f\n",
                   iter, x, x - r_expected, x - x0);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       gsl_root_fdfsolver_free (s);
       return status;
     }

Here are the results for Newton’s method:

     $ ./a.out
     using newton method
     iter        root        err   err(est)
         1  3.0000000 +0.7639320 -2.0000000
         2  2.3333333 +0.0972654 -0.6666667
         3  2.2380952 +0.0020273 -0.0952381
     Converged:
         4  2.2360689 +0.0000009 -0.0020263

Note that the error can be estimated more accurately by taking the
difference between the current iterate and next iterate rather than the
previous iterate.  The other derivative solvers can be investigated by
changing ‘gsl_root_fdfsolver_newton’ to ‘gsl_root_fdfsolver_secant’ or
‘gsl_root_fdfsolver_steffenson’.


File: gsl-ref.info,  Node: References and Further Reading<29>,  Prev: Examples<27>,  Up: One Dimensional Root-Finding

36.11 References and Further Reading
====================================

For information on the Brent-Dekker algorithm see the following two
papers,

   * R. P. Brent, “An algorithm with guaranteed convergence for finding
     a zero of a function”, `Computer Journal', 14 (1971) 422–425

   * J. C. P. Bus and T. J. Dekker, “Two Efficient Algorithms with
     Guaranteed Convergence for Finding a Zero of a Function”, `ACM
     Transactions of Mathematical Software', Vol.: 1 No.: 4 (1975)
     330–345


File: gsl-ref.info,  Node: One Dimensional Minimization,  Next: Multidimensional Root-Finding,  Prev: One Dimensional Root-Finding,  Up: Top

37 One Dimensional Minimization
*******************************

This chapter describes routines for finding minima of arbitrary
one-dimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, with full
access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs.

The header file ‘gsl_min.h’ contains prototypes for the minimization
functions and related declarations.  To use the minimization algorithms
to find the maximum of a function simply invert its sign.

* Menu:

* Overview: Overview<2>.
* Caveats: Caveats<2>.
* Initializing the Minimizer::
* Providing the function to minimize::
* Iteration: Iteration<2>.
* Stopping Parameters::
* Minimization Algorithms::
* Examples: Examples<28>.
* References and Further Reading: References and Further Reading<30>.


File: gsl-ref.info,  Node: Overview<2>,  Next: Caveats<2>,  Up: One Dimensional Minimization

37.1 Overview
=============

The minimization algorithms begin with a bounded region known to contain
a minimum.  The region is described by a lower bound a and an upper
bound b, with an estimate of the location of the minimum x, as shown in
Fig.  %s.

[gsl-ref-figures/min-interval]
Figure: Function with lower and upper bounds with an estimate of the
minimum.

The value of the function at x must be less than the value of the
function at the ends of the interval,

     f(a) > f(x) < f(b)

This condition guarantees that a minimum is contained somewhere within
the interval.  On each iteration a new point x' is selected using one of
the available algorithms.  If the new point is a better estimate of the
minimum, i.e.: where f(x') < f(x), then the current estimate of the
minimum x is updated.  The new point also allows the size of the bounded
interval to be reduced, by choosing the most compact set of points which
satisfies the constraint f(a) > f(x) < f(b).  The interval is reduced
until it encloses the true minimum to a desired tolerance.  This
provides a best estimate of the location of the minimum and a rigorous
error estimate.

Several bracketing algorithms are available within a single framework.
The user provides a high-level driver for the algorithm, and the library
provides the individual functions necessary for each of the steps.
There are three main phases of the iteration.  The steps are,

   * initialize minimizer state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

The state for the minimizers is held in a *note gsl_min_fminimizer: ab1.
struct.  The updating procedure uses only function evaluations (not
derivatives).


File: gsl-ref.info,  Node: Caveats<2>,  Next: Initializing the Minimizer,  Prev: Overview<2>,  Up: One Dimensional Minimization

37.2 Caveats
============

Note that minimization functions can only search for one minimum at a
time.  When there are several minima in the search area, the first
minimum to be found will be returned; however it is difficult to predict
which of the minima this will be.  `In most cases, no error will be
reported if you try to find a minimum in an area where there is more
than one.'

With all minimization algorithms it can be difficult to determine the
location of the minimum to full numerical precision.  The behavior of
the function in the region of the minimum x^* can be approximated by a
Taylor expansion,

     y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2

and the second term of this expansion can be lost when added to the
first term at finite precision.  This magnifies the error in locating
x^*, making it proportional to \sqrt \epsilon (where \epsilon is the
relative accuracy of the floating point numbers).  For functions with
higher order minima, such as x^4, the magnification of the error is
correspondingly worse.  The best that can be achieved is to converge to
the limit of numerical accuracy in the function values, rather than the
location of the minimum itself.


File: gsl-ref.info,  Node: Initializing the Minimizer,  Next: Providing the function to minimize,  Prev: Caveats<2>,  Up: One Dimensional Minimization

37.3 Initializing the Minimizer
===============================

 -- Type: gsl_min_fminimizer

     This is a workspace for minimizing functions.

 -- Function: *note gsl_min_fminimizer: ab1. *gsl_min_fminimizer_alloc
          (const gsl_min_fminimizer_type *T)

     This function returns a pointer to a newly allocated instance of a
     minimizer of type *note T: ab4.  For example, the following code
     creates an instance of a golden section minimizer:

          const gsl_min_fminimizer_type * T = gsl_min_fminimizer_goldensection;
          gsl_min_fminimizer * s = gsl_min_fminimizer_alloc (T);

     If there is insufficient memory to create the minimizer then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_min_fminimizer_set (gsl_min_fminimizer *s,
          gsl_function *f, double x_minimum, double x_lower, double
          x_upper)

     This function sets, or resets, an existing minimizer *note s: ab5.
     to use the function *note f: ab5. and the initial search interval
     [*note x_lower: ab5, *note x_upper: ab5.], with a guess for the
     location of the minimum *note x_minimum: ab5.

     If the interval given does not contain a minimum, then the function
     returns an error code of *note GSL_EINVAL: 2b.

 -- Function: int gsl_min_fminimizer_set_with_values (gsl_min_fminimizer
          *s, gsl_function *f, double x_minimum, double f_minimum,
          double x_lower, double f_lower, double x_upper, double
          f_upper)

     This function is equivalent to *note gsl_min_fminimizer_set(): ab5.
     but uses the values *note f_minimum: ab6, *note f_lower: ab6. and
     *note f_upper: ab6. instead of computing ‘f(x_minimum)’,
     ‘f(x_lower)’ and ‘f(x_upper)’.

 -- Function: void gsl_min_fminimizer_free (gsl_min_fminimizer *s)

     This function frees all the memory associated with the minimizer
     *note s: ab7.

 -- Function: const char *gsl_min_fminimizer_name (const
          gsl_min_fminimizer *s)

     This function returns a pointer to the name of the minimizer.  For
     example:

          printf ("s is a '%s' minimizer\n", gsl_min_fminimizer_name (s));

     would print something like ‘s is a 'brent' minimizer’.


File: gsl-ref.info,  Node: Providing the function to minimize,  Next: Iteration<2>,  Prev: Initializing the Minimizer,  Up: One Dimensional Minimization

37.4 Providing the function to minimize
=======================================

You must provide a continuous function of one variable for the
minimizers to operate on.  In order to allow for general parameters the
functions are defined by a *note gsl_function: a93. data type (*note
Providing the function to solve: a91.).


File: gsl-ref.info,  Node: Iteration<2>,  Next: Stopping Parameters,  Prev: Providing the function to minimize,  Up: One Dimensional Minimization

37.5 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any minimizer of
the corresponding type.  The same functions work for all minimizers so
that different methods can be substituted at runtime without
modifications to the code.

 -- Function: int gsl_min_fminimizer_iterate (gsl_min_fminimizer *s)

     This function performs a single iteration of the minimizer *note s:
     abb.  If the iteration encounters an unexpected problem then an
     error code will be returned,

     ‘GSL_EBADFUNC’

          the iteration encountered a singular point where the function
          evaluated to ‘Inf’ or ‘NaN’.

     ‘GSL_FAILURE’

          the algorithm could not improve the current best approximation
          or bounding interval.

The minimizer maintains a current best estimate of the position of the
minimum at all times, and the current interval bounding the minimum.
This information can be accessed with the following auxiliary functions,

 -- Function: double gsl_min_fminimizer_x_minimum (const
          gsl_min_fminimizer *s)

     This function returns the current estimate of the position of the
     minimum for the minimizer *note s: abc.

 -- Function: double gsl_min_fminimizer_x_upper (const
          gsl_min_fminimizer *s)
 -- Function: double gsl_min_fminimizer_x_lower (const
          gsl_min_fminimizer *s)

     These functions return the current upper and lower bound of the
     interval for the minimizer *note s: abe.

 -- Function: double gsl_min_fminimizer_f_minimum (const
          gsl_min_fminimizer *s)
 -- Function: double gsl_min_fminimizer_f_upper (const
          gsl_min_fminimizer *s)
 -- Function: double gsl_min_fminimizer_f_lower (const
          gsl_min_fminimizer *s)

     These functions return the value of the function at the current
     estimate of the minimum and at the upper and lower bounds of the
     interval for the minimizer *note s: ac1.


File: gsl-ref.info,  Node: Stopping Parameters,  Next: Minimization Algorithms,  Prev: Iteration<2>,  Up: One Dimensional Minimization

37.6 Stopping Parameters
========================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The function
below allows the user to test the precision of the current result.

 -- Function: int gsl_min_test_interval (double x_lower, double x_upper,
          double epsabs, double epsrel)

     This function tests for the convergence of the interval [*note
     x_lower: ac3, *note x_upper: ac3.] with absolute error *note
     epsabs: ac3. and relative error *note epsrel: ac3.  The test
     returns ‘GSL_SUCCESS’ if the following condition is achieved,

          |a - b| < epsabs + epsrel min(|a|,|b|)

     when the interval x = [a,b] does not include the origin.  If the
     interval includes the origin then \min(|a|,|b|) is replaced by zero
     (which is the minimum value of |x| over the interval).  This
     ensures that the relative error is accurately estimated for minima
     close to the origin.

     This condition on the interval also implies that any estimate of
     the minimum x_m in the interval satisfies the same condition with
     respect to the true minimum x_m^*,

          |x_m - x_m^*| < epsabs + epsrel x_m^*

     assuming that the true minimum x_m^* is contained within the
     interval.


File: gsl-ref.info,  Node: Minimization Algorithms,  Next: Examples<28>,  Prev: Stopping Parameters,  Up: One Dimensional Minimization

37.7 Minimization Algorithms
============================

The minimization algorithms described in this section require an initial
interval which is guaranteed to contain a minimum—if a and b are the
endpoints of the interval and x is an estimate of the minimum then f(a)
> f(x) < f(b).  This ensures that the function has at least one minimum
somewhere in the interval.  If a valid initial interval is used then
these algorithm cannot fail, provided the function is well-behaved.

 -- Type: gsl_min_fminimizer_type

      -- Variable: *note gsl_min_fminimizer_type: ac5.
               *gsl_min_fminimizer_goldensection

          The `golden section algorithm' is the simplest method of
          bracketing the minimum of a function.  It is the slowest
          algorithm provided by the library, with linear convergence.

          On each iteration, the algorithm first compares the
          subintervals from the endpoints to the current minimum.  The
          larger subinterval is divided in a golden section (using the
          famous ratio (3-\sqrt 5)/2 \approx 0.3819660 and the value of
          the function at this new point is calculated.  The new value
          is used with the constraint f(a') > f(x') < f(b') to a select
          new interval containing the minimum, by discarding the least
          useful point.  This procedure can be continued indefinitely
          until the interval is sufficiently small.  Choosing the golden
          section as the bisection ratio can be shown to provide the
          fastest convergence for this type of algorithm.

      -- Variable: *note gsl_min_fminimizer_type: ac5.
               *gsl_min_fminimizer_brent

          The `Brent minimization algorithm' combines a parabolic
          interpolation with the golden section algorithm.  This
          produces a fast algorithm which is still robust.

          The outline of the algorithm can be summarized as follows: on
          each iteration Brent’s method approximates the function using
          an interpolating parabola through three existing points.  The
          minimum of the parabola is taken as a guess for the minimum.
          If it lies within the bounds of the current interval then the
          interpolating point is accepted, and used to generate a
          smaller interval.  If the interpolating point is not accepted
          then the algorithm falls back to an ordinary golden section
          step.  The full details of Brent’s method include some
          additional checks to improve convergence.

      -- Variable: *note gsl_min_fminimizer_type: ac5.
               *gsl_min_fminimizer_quad_golden

          This is a variant of Brent’s algorithm which uses the
          safeguarded step-length algorithm of Gill and Murray.


File: gsl-ref.info,  Node: Examples<28>,  Next: References and Further Reading<30>,  Prev: Minimization Algorithms,  Up: One Dimensional Minimization

37.8 Examples
=============

The following program uses the Brent algorithm to find the minimum of
the function f(x) = \cos(x) + 1, which occurs at x = \pi.  The starting
interval is (0,6), with an initial guess for the minimum of 2.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_min.h>

     double fn1 (double x, void * params)
     {
       (void)(params); /* avoid unused parameter warning */
       return cos(x) + 1.0;
     }

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_min_fminimizer_type *T;
       gsl_min_fminimizer *s;
       double m = 2.0, m_expected = M_PI;
       double a = 0.0, b = 6.0;
       gsl_function F;

       F.function = &fn1;
       F.params = 0;

       T = gsl_min_fminimizer_brent;
       s = gsl_min_fminimizer_alloc (T);
       gsl_min_fminimizer_set (s, &F, m, a, b);

       printf ("using %s method\n",
               gsl_min_fminimizer_name (s));

       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
               "iter", "lower", "upper", "min",
               "err", "err(est)");

       printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
               iter, a, b,
               m, m - m_expected, b - a);

       do
         {
           iter++;
           status = gsl_min_fminimizer_iterate (s);

           m = gsl_min_fminimizer_x_minimum (s);
           a = gsl_min_fminimizer_x_lower (s);
           b = gsl_min_fminimizer_x_upper (s);

           status
             = gsl_min_test_interval (a, b, 0.001, 0.0);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d [%.7f, %.7f] "
                   "%.7f %+.7f %.7f\n",
                   iter, a, b,
                   m, m - m_expected, b - a);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       gsl_min_fminimizer_free (s);

       return status;
     }

Here are the results of the minimization procedure.

     using brent method
      iter [    lower,     upper]       min        err  err(est)
         0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
         1 [2.0000000, 6.0000000] 3.5278640 +0.3862713 4.0000000
         2 [2.0000000, 3.5278640] 3.1748217 +0.0332290 1.5278640
         3 [2.0000000, 3.1748217] 3.1264576 -0.0151351 1.1748217
         4 [3.1264576, 3.1748217] 3.1414743 -0.0001183 0.0483641
         5 [3.1414743, 3.1748217] 3.1415930 +0.0000004 0.0333474
     Converged:
         6 [3.1414743, 3.1415930] 3.1415927 +0.0000000 0.0001187


File: gsl-ref.info,  Node: References and Further Reading<30>,  Prev: Examples<28>,  Up: One Dimensional Minimization

37.9 References and Further Reading
===================================

Further information on Brent’s algorithm is available in the following
book,

   * Richard Brent, `Algorithms for minimization without derivatives',
     Prentice-Hall (1973), republished by Dover in paperback (2002),
     ISBN 0-486-41998-3.


File: gsl-ref.info,  Node: Multidimensional Root-Finding,  Next: Multidimensional Minimization,  Prev: One Dimensional Minimization,  Up: Top

38 Multidimensional Root-Finding
********************************

This chapter describes functions for multidimensional root-finding
(solving nonlinear systems with n equations in n unknowns).  The library
provides low level components for a variety of iterative solvers and
convergence tests.  These can be combined by the user to achieve the
desired solution, with full access to the intermediate steps of the
iteration.  Each class of methods uses the same framework, so that you
can switch between solvers at runtime without needing to recompile your
program.  Each instance of a solver keeps track of its own state,
allowing the solvers to be used in multi-threaded programs.  The solvers
are based on the original Fortran library MINPACK.

The header file ‘gsl_multiroots.h’ contains prototypes for the
multidimensional root finding functions and related declarations.

* Menu:

* Overview: Overview<3>.
* Initializing the Solver: Initializing the Solver<2>.
* Providing the function to solve: Providing the function to solve<2>.
* Iteration: Iteration<3>.
* Search Stopping Parameters: Search Stopping Parameters<2>.
* Algorithms using Derivatives::
* Algorithms without Derivatives::
* Examples: Examples<29>.
* References and Further Reading: References and Further Reading<31>.


File: gsl-ref.info,  Node: Overview<3>,  Next: Initializing the Solver<2>,  Up: Multidimensional Root-Finding

38.1 Overview
=============

The problem of multidimensional root finding requires the simultaneous
solution of n equations, f_i, in n variables, x_i,

     f_i (x_1, ..., x_n) = 0    for i = 1 ... n.

In general there are no bracketing methods available for n dimensional
systems, and no way of knowing whether any solutions exist.  All
algorithms proceed from an initial guess using a variant of the Newton
iteration,

     x -> x' = x - J^{-1} f(x)

where x, f are vector quantities and J is the Jacobian matrix J_{ij} =
\partial f_i / \partial x_j.  Additional strategies can be used to
enlarge the region of convergence.  These include requiring a decrease
in the norm |f| on each step proposed by Newton’s method, or taking
steepest-descent steps in the direction of the negative gradient of |f|.

Several root-finding algorithms are available within a single framework.
The user provides a high-level driver for the algorithms, and the
library provides the individual functions necessary for each of the
steps.  There are three main phases of the iteration.  The steps are,

   * initialize solver state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

The evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
n^2 terms of the matrix becomes too expensive.  For these reasons the
algorithms provided by the library are divided into two classes
according to whether the derivatives are available or not.

The state for solvers with an analytic Jacobian matrix is held in a
*note gsl_multiroot_fdfsolver: ace. struct.  The updating procedure
requires both the function and its derivatives to be supplied by the
user.

The state for solvers which do not use an analytic Jacobian matrix is
held in a *note gsl_multiroot_fsolver: acf. struct.  The updating
procedure uses only function evaluations (not derivatives).  The
algorithms estimate the matrix J or J^{-1} by approximate methods.


File: gsl-ref.info,  Node: Initializing the Solver<2>,  Next: Providing the function to solve<2>,  Prev: Overview<3>,  Up: Multidimensional Root-Finding

38.2 Initializing the Solver
============================

The following functions initialize a multidimensional solver, either
with or without derivatives.  The solver itself depends only on the
dimension of the problem and the algorithm and can be reused for
different problems.

 -- Type: gsl_multiroot_fsolver

     This is a workspace for multidimensional root-finding without
     derivatives.

 -- Type: gsl_multiroot_fdfsolver

     This is a workspace for multidimensional root-finding with
     derivatives.

 -- Function: *note gsl_multiroot_fsolver: acf.
          *gsl_multiroot_fsolver_alloc (const gsl_multiroot_fsolver_type
          *T, size_t n)

     This function returns a pointer to a newly allocated instance of a
     solver of type *note T: ad1. for a system of *note n: ad1.
     dimensions.  For example, the following code creates an instance of
     a hybrid solver, to solve a 3-dimensional system of equations:

          const gsl_multiroot_fsolver_type * T = gsl_multiroot_fsolver_hybrid;
          gsl_multiroot_fsolver * s = gsl_multiroot_fsolver_alloc (T, 3);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: *note gsl_multiroot_fdfsolver: ace.
          *gsl_multiroot_fdfsolver_alloc (const
          gsl_multiroot_fdfsolver_type *T, size_t n)

     This function returns a pointer to a newly allocated instance of a
     derivative solver of type *note T: ad2. for a system of *note n:
     ad2. dimensions.  For example, the following code creates an
     instance of a Newton-Raphson solver, for a 2-dimensional system of
     equations:

          const gsl_multiroot_fdfsolver_type * T = gsl_multiroot_fdfsolver_newton;
          gsl_multiroot_fdfsolver * s = gsl_multiroot_fdfsolver_alloc (T, 2);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver *s,
          gsl_multiroot_function *f, const gsl_vector *x)
 -- Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver
          *s, gsl_multiroot_function_fdf *fdf, const gsl_vector *x)

     These functions set, or reset, an existing solver *note s: ad4. to
     use the function ‘f’ or function and derivative *note fdf: ad4, and
     the initial guess *note x: ad4.  Note that the initial position is
     copied from *note x: ad4, this argument is not modified by
     subsequent iterations.

 -- Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver *s)
 -- Function: void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver
          *s)

     These functions free all the memory associated with the solver
     *note s: ad6.

 -- Function: const char *gsl_multiroot_fsolver_name (const
          gsl_multiroot_fsolver *s)
 -- Function: const char *gsl_multiroot_fdfsolver_name (const
          gsl_multiroot_fdfsolver *s)

     These functions return a pointer to the name of the solver.  For
     example:

          printf ("s is a '%s' solver\n", gsl_multiroot_fdfsolver_name (s));

     would print something like ‘s is a 'newton' solver’.


File: gsl-ref.info,  Node: Providing the function to solve<2>,  Next: Iteration<3>,  Prev: Initializing the Solver<2>,  Up: Multidimensional Root-Finding

38.3 Providing the function to solve
====================================

You must provide n functions of n variables for the root finders to
operate on.  In order to allow for general parameters the functions are
defined by the following data types:

 -- Type: gsl_multiroot_function

     This data type defines a general system of functions with
     parameters.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          this function should store the vector result f(x,params) in
          ‘f’ for argument ‘x’ and parameters ‘params’, returning an
          appropriate error code if the function cannot be computed.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’ and ‘f’.

     ‘void * params’

          a pointer to the parameters of the function.

Here is an example using Powell’s test function,

     f_1(x) = A x_0 x_1 - 1,
     f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)

with A = 10^4.  The following code defines a *note
gsl_multiroot_function: ada. system ‘F’ which you could pass to a
solver:

     struct powell_params { double A; };

     int
     powell (gsl_vector * x, void * p, gsl_vector * f) {
        struct powell_params * params
          = (struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);

        gsl_vector_set (f, 0, A * x0 * x1 - 1);
        gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)
                               - (1.0 + 1.0/A)));
        return GSL_SUCCESS
     }

     gsl_multiroot_function F;
     struct powell_params params = { 10000.0 };

     F.f = &powell;
     F.n = 2;
     F.params = &params;

 -- Type: gsl_multiroot_function_fdf

     This data type defines a general system of functions with
     parameters and the corresponding Jacobian matrix of derivatives,

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          this function should store the vector result f(x,params) in
          ‘f’ for argument ‘x’ and parameters ‘params’, returning an
          appropriate error code if the function cannot be computed.

     ‘int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)’

          this function should store the ‘n’-by-‘n’ matrix result

               J_ij = d f_i(x,params) / d x_j

          in ‘J’ for argument ‘x’ and parameters ‘params’, returning an
          appropriate error code if the function cannot be computed.

     ‘int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f,
     gsl_matrix * J)’

          This function should set the values of the ‘f’ and ‘J’ as
          above, for arguments ‘x’ and parameters ‘params’.  This
          function provides an optimization of the separate functions
          for f(x) and J(x)—it is always faster to compute the function
          and its derivative at the same time.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’ and ‘f’.

     ‘void * params’

          a pointer to the parameters of the function.

The example of Powell’s test function defined above can be extended to
include analytic derivatives using the following code:

     int
     powell_df (gsl_vector * x, void * p, gsl_matrix * J)
     {
        struct powell_params * params
          = (struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);
        gsl_matrix_set (J, 0, 0, A * x1);
        gsl_matrix_set (J, 0, 1, A * x0);
        gsl_matrix_set (J, 1, 0, -exp(-x0));
        gsl_matrix_set (J, 1, 1, -exp(-x1));
        return GSL_SUCCESS
     }

     int
     powell_fdf (gsl_vector * x, void * p,
                 gsl_matrix * f, gsl_matrix * J) {
        struct powell_params * params
          = (struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);

        const double u0 = exp(-x0);
        const double u1 = exp(-x1);

        gsl_vector_set (f, 0, A * x0 * x1 - 1);
        gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));

        gsl_matrix_set (J, 0, 0, A * x1);
        gsl_matrix_set (J, 0, 1, A * x0);
        gsl_matrix_set (J, 1, 0, -u0);
        gsl_matrix_set (J, 1, 1, -u1);
        return GSL_SUCCESS
     }

     gsl_multiroot_function_fdf FDF;

     FDF.f = &powell_f;
     FDF.df = &powell_df;
     FDF.fdf = &powell_fdf;
     FDF.n = 2;
     FDF.params = 0;

Note that the function ‘powell_fdf’ is able to reuse existing terms from
the function when calculating the Jacobian, thus saving time.


File: gsl-ref.info,  Node: Iteration<3>,  Next: Search Stopping Parameters<2>,  Prev: Providing the function to solve<2>,  Up: Multidimensional Root-Finding

38.4 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

 -- Function: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver
          *s)
 -- Function: int gsl_multiroot_fdfsolver_iterate
          (gsl_multiroot_fdfsolver *s)

     These functions perform a single iteration of the solver *note s:
     ade.  If the iteration encounters an unexpected problem then an
     error code will be returned,

     ‘GSL_EBADFUNC’

          the iteration encountered a singular point where the function
          or its derivative evaluated to ‘Inf’ or ‘NaN’.

     ‘GSL_ENOPROG’

          the iteration is not making any progress, preventing the
          algorithm from continuing.

The solver maintains a current best estimate of the root ‘s->x’ and its
function value ‘s->f’ at all times.  This information can be accessed
with the following auxiliary functions,

 -- Function: *note gsl_vector: 35f. *gsl_multiroot_fsolver_root (const
          gsl_multiroot_fsolver *s)
 -- Function: *note gsl_vector: 35f. *gsl_multiroot_fdfsolver_root
          (const gsl_multiroot_fdfsolver *s)

     These functions return the current estimate of the root for the
     solver *note s: ae0, given by ‘s->x’.

 -- Function: *note gsl_vector: 35f. *gsl_multiroot_fsolver_f (const
          gsl_multiroot_fsolver *s)
 -- Function: *note gsl_vector: 35f. *gsl_multiroot_fdfsolver_f (const
          gsl_multiroot_fdfsolver *s)

     These functions return the function value f(x) at the current
     estimate of the root for the solver *note s: ae2, given by ‘s->f’.

 -- Function: *note gsl_vector: 35f. *gsl_multiroot_fsolver_dx (const
          gsl_multiroot_fsolver *s)
 -- Function: *note gsl_vector: 35f. *gsl_multiroot_fdfsolver_dx (const
          gsl_multiroot_fdfsolver *s)

     These functions return the last step dx taken by the solver *note
     s: ae4, given by ‘s->dx’.


File: gsl-ref.info,  Node: Search Stopping Parameters<2>,  Next: Algorithms using Derivatives,  Prev: Iteration<3>,  Up: Multidimensional Root-Finding

38.5 Search Stopping Parameters
===============================

A root finding procedure should stop when one of the following
conditions is true:

   * A multidimensional root has been found to within the user-specified
     precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result in
several standard ways.

 -- Function: int gsl_multiroot_test_delta (const gsl_vector *dx, const
          gsl_vector *x, double epsabs, double epsrel)

     This function tests for the convergence of the sequence by
     comparing the last step *note dx: ae6. with the absolute error
     *note epsabs: ae6. and relative error *note epsrel: ae6. to the
     current position *note x: ae6.  The test returns ‘GSL_SUCCESS’ if
     the following condition is achieved,

          |dx_i| < epsabs + epsrel |x_i|

     for each component of *note x: ae6. and returns ‘GSL_CONTINUE’
     otherwise.

 -- Function: int gsl_multiroot_test_residual (const gsl_vector *f,
          double epsabs)

     This function tests the residual value *note f: ae7. against the
     absolute error bound *note epsabs: ae7.  The test returns
     ‘GSL_SUCCESS’ if the following condition is achieved,

          \sum_i |f_i| < epsabs

     and returns ‘GSL_CONTINUE’ otherwise.  This criterion is suitable
     for situations where the precise location of the root, x, is
     unimportant provided a value can be found where the residual is
     small enough.


File: gsl-ref.info,  Node: Algorithms using Derivatives,  Next: Algorithms without Derivatives,  Prev: Search Stopping Parameters<2>,  Up: Multidimensional Root-Finding

38.6 Algorithms using Derivatives
=================================

The root finding algorithms described in this section make use of both
the function and its derivative.  They require an initial guess for the
location of the root, but there is no absolute guarantee of
convergence—the function must be suitable for this technique and the
initial guess must be sufficiently close to the root for it to work.
When the conditions are satisfied then convergence is quadratic.

 -- Type: gsl_multiroot_fdfsolver_type

     The following are available algorithms for minimizing functions
     using derivatives.

      -- Variable: *note gsl_multiroot_fdfsolver_type: ae9.
               *gsl_multiroot_fdfsolver_hybridsj

          This is a modified version of Powell’s Hybrid method as
          implemented in the HYBRJ algorithm in MINPACK. Minpack was
          written by Jorge J. Moré, Burton S. Garbow and Kenneth E.
          Hillstrom.  The Hybrid algorithm retains the fast convergence
          of Newton’s method but will also reduce the residual when
          Newton’s method is unreliable.

          The algorithm uses a generalized trust region to keep each
          step under control.  In order to be accepted a proposed new
          position x' must satisfy the condition |D (x' - x)| < \delta,
          where D is a diagonal scaling matrix and \delta is the size of
          the trust region.  The components of D are computed
          internally, using the column norms of the Jacobian to estimate
          the sensitivity of the residual to each component of x.  This
          improves the behavior of the algorithm for badly scaled
          functions.

          On each iteration the algorithm first determines the standard
          Newton step by solving the system J dx = - f.  If this step
          falls inside the trust region it is used as a trial step in
          the next stage.  If not, the algorithm uses the linear
          combination of the Newton and gradient directions which is
          predicted to minimize the norm of the function while staying
          inside the trust region,

               dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2

          This combination of Newton and gradient directions is referred
          to as a `dogleg step'.

          The proposed step is now tested by evaluating the function at
          the resulting point, x'.  If the step reduces the norm of the
          function sufficiently then it is accepted and size of the
          trust region is increased.  If the proposed step fails to
          improve the solution then the size of the trust region is
          decreased and another trial step is computed.

          The speed of the algorithm is increased by computing the
          changes to the Jacobian approximately, using a rank-1 update.
          If two successive attempts fail to reduce the residual then
          the full Jacobian is recomputed.  The algorithm also monitors
          the progress of the solution and returns an error if several
          steps fail to make any improvement,

          ‘GSL_ENOPROG’

               the iteration is not making any progress, preventing the
               algorithm from continuing.

          ‘GSL_ENOPROGJ’

               re-evaluations of the Jacobian indicate that the
               iteration is not making any progress, preventing the
               algorithm from continuing.

      -- Variable: *note gsl_multiroot_fdfsolver_type: ae9.
               *gsl_multiroot_fdfsolver_hybridj

          This algorithm is an unscaled version of HYBRIDSJ. The steps
          are controlled by a spherical trust region |x' - x| < \delta,
          instead of a generalized region.  This can be useful if the
          generalized region estimated by HYBRIDSJ is inappropriate.

      -- Variable: *note gsl_multiroot_fdfsolver_type: ae9.
               *gsl_multiroot_fdfsolver_newton

          Newton’s Method is the standard root-polishing algorithm.  The
          algorithm begins with an initial guess for the location of the
          solution.  On each iteration a linear approximation to the
          function F is used to estimate the step which will zero all
          the components of the residual.  The iteration is defined by
          the following sequence,

               x -> x' = x - J^{-1} f(x)

          where the Jacobian matrix J is computed from the derivative
          functions provided by ‘f’.  The step dx is obtained by solving
          the linear system,

               J dx = - f(x)

          using LU decomposition.  If the Jacobian matrix is singular,
          an error code of *note GSL_EDOM: 28. is returned.

      -- Variable: *note gsl_multiroot_fdfsolver_type: ae9.
               *gsl_multiroot_fdfsolver_gnewton

          This is a modified version of Newton’s method which attempts
          to improve global convergence by requiring every step to
          reduce the Euclidean norm of the residual, |f(x)|.  If the
          Newton step leads to an increase in the norm then a reduced
          step of relative size,

               t = (\sqrt{1 + 6 r} - 1) / (3 r)

          is proposed, with r being the ratio of norms
          |f(x')|^2/|f(x)|^2.  This procedure is repeated until a
          suitable step size is found.


File: gsl-ref.info,  Node: Algorithms without Derivatives,  Next: Examples<29>,  Prev: Algorithms using Derivatives,  Up: Multidimensional Root-Finding

38.7 Algorithms without Derivatives
===================================

The algorithms described in this section do not require any derivative
information to be supplied by the user.  Any derivatives needed are
approximated by finite differences.  Note that if the
finite-differencing step size chosen by these routines is inappropriate,
an explicit user-supplied numerical derivative can always be used with
the algorithms described in the previous section.

 -- Type: gsl_multiroot_fsolver_type

     The following are available algorithms for minimizing functions
     without derivatives.

      -- Variable: *note gsl_multiroot_fsolver_type: aef.
               *gsl_multiroot_fsolver_hybrids

          This is a version of the Hybrid algorithm which replaces calls
          to the Jacobian function by its finite difference
          approximation.  The finite difference approximation is
          computed using ‘gsl_multiroots_fdjac()’ with a relative step
          size of ‘GSL_SQRT_DBL_EPSILON’.  Note that this step size will
          not be suitable for all problems.

      -- Variable: *note gsl_multiroot_fsolver_type: aef.
               *gsl_multiroot_fsolver_hybrid

          This is a finite difference version of the Hybrid algorithm
          without internal scaling.

      -- Variable: *note gsl_multiroot_fsolver_type: aef.
               *gsl_multiroot_fsolver_dnewton

          The `discrete Newton algorithm' is the simplest method of
          solving a multidimensional system.  It uses the Newton
          iteration

               x -> x - J^{-1} f(x)

          where the Jacobian matrix J is approximated by taking finite
          differences of the function ‘f’.  The approximation scheme
          used by this implementation is,

               J_{ij} = (f_i(x + \delta_j) - f_i(x)) / \delta_j

          where \delta_j is a step of size \sqrt\epsilon |x_j| with
          \epsilon being the machine precision (\epsilon \approx 2.22
          \times 10^{-16}).  The order of convergence of Newton’s
          algorithm is quadratic, but the finite differences require n^2
          function evaluations on each iteration.  The algorithm may
          become unstable if the finite differences are not a good
          approximation to the true derivatives.

      -- Variable: *note gsl_multiroot_fsolver_type: aef.
               *gsl_multiroot_fsolver_broyden

          The `Broyden algorithm' is a version of the discrete Newton
          algorithm which attempts to avoids the expensive update of the
          Jacobian matrix on each iteration.  The changes to the
          Jacobian are also approximated, using a rank-1 update,

               J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df

          where the vectors dx and df are the changes in x and f.  On
          the first iteration the inverse Jacobian is estimated using
          finite differences, as in the discrete Newton algorithm.

          This approximation gives a fast update but is unreliable if
          the changes are not small, and the estimate of the inverse
          Jacobian becomes worse as time passes.  The algorithm has a
          tendency to become unstable unless it starts close to the
          root.  The Jacobian is refreshed if this instability is
          detected (consult the source for details).

          This algorithm is included only for demonstration purposes,
          and is not recommended for serious use.


File: gsl-ref.info,  Node: Examples<29>,  Next: References and Further Reading<31>,  Prev: Algorithms without Derivatives,  Up: Multidimensional Root-Finding

38.8 Examples
=============

The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms.  This first example
demonstrates the HYBRIDS scaled-hybrid algorithm, which does not require
derivatives.  The program solves the Rosenbrock system of equations,

     f_1 (x, y) = a (1 - x)
     f_2 (x, y) = b (y - x^2)

with a = 1, b = 10.  The solution of this system lies at (x,y) = (1,1)
in a narrow valley.

The first stage of the program is to define the system of equations:

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_multiroots.h>

     struct rparams
       {
         double a;
         double b;
       };

     int
     rosenbrock_f (const gsl_vector * x, void *params,
                   gsl_vector * f)
     {
       double a = ((struct rparams *) params)->a;
       double b = ((struct rparams *) params)->b;

       const double x0 = gsl_vector_get (x, 0);
       const double x1 = gsl_vector_get (x, 1);

       const double y0 = a * (1 - x0);
       const double y1 = b * (x1 - x0 * x0);

       gsl_vector_set (f, 0, y0);
       gsl_vector_set (f, 1, y1);

       return GSL_SUCCESS;
     }

The main program begins by creating the function object ‘f’, with the
arguments ‘(x,y)’ and parameters ‘(a,b)’.  The solver ‘s’ is initialized
to use this function, with the ‘gsl_multiroot_fsolver_hybrids’ method:

     int
     main (void)
     {
       const gsl_multiroot_fsolver_type *T;
       gsl_multiroot_fsolver *s;

       int status;
       size_t i, iter = 0;

       const size_t n = 2;
       struct rparams p = {1.0, 10.0};
       gsl_multiroot_function f = {&rosenbrock_f, n, &p};

       double x_init[2] = {-10.0, -5.0};
       gsl_vector *x = gsl_vector_alloc (n);

       gsl_vector_set (x, 0, x_init[0]);
       gsl_vector_set (x, 1, x_init[1]);

       T = gsl_multiroot_fsolver_hybrids;
       s = gsl_multiroot_fsolver_alloc (T, 2);
       gsl_multiroot_fsolver_set (s, &f, x);

       print_state (iter, s);

       do
         {
           iter++;
           status = gsl_multiroot_fsolver_iterate (s);

           print_state (iter, s);

           if (status)   /* check if solver is stuck */
             break;

           status =
             gsl_multiroot_test_residual (s->f, 1e-7);
         }
       while (status == GSL_CONTINUE && iter < 1000);

       printf ("status = %s\n", gsl_strerror (status));

       gsl_multiroot_fsolver_free (s);
       gsl_vector_free (x);
       return 0;
     }

Note that it is important to check the return status of each solver
step, in case the algorithm becomes stuck.  If an error condition is
detected, indicating that the algorithm cannot proceed, then the error
can be reported to the user, a new starting point chosen or a different
algorithm used.

The intermediate state of the solution is displayed by the following
function.  The solver state contains the vector ‘s->x’ which is the
current position, and the vector ‘s->f’ with corresponding function
values:

     int
     print_state (size_t iter, gsl_multiroot_fsolver * s)
     {
       printf ("iter = %3u x = % .3f % .3f "
               "f(x) = % .3e % .3e\n",
               iter,
               gsl_vector_get (s->x, 0),
               gsl_vector_get (s->x, 1),
               gsl_vector_get (s->f, 0),
               gsl_vector_get (s->f, 1));
     }

Here are the results of running the program.  The algorithm is started
at (-10,-5) far from the solution.  Since the solution is hidden in a
narrow valley the earliest steps follow the gradient of the function
downhill, in an attempt to reduce the large value of the residual.  Once
the root has been approximately located, on iteration 8, the Newton
behavior takes over and convergence is very rapid:

     iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
     iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
     iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
     iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
     iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
     iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
     iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00
     iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
     iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
     status = success

Note that the algorithm does not update the location on every iteration.
Some iterations are used to adjust the trust-region parameter, after
trying a step which was found to be divergent, or to recompute the
Jacobian, when poor convergence behavior is detected.

The next example program adds derivative information, in order to
accelerate the solution.  There are two derivative functions
‘rosenbrock_df’ and ‘rosenbrock_fdf’.  The latter computes both the
function and its derivative simultaneously.  This allows the
optimization of any common terms.  For simplicity we substitute calls to
the separate ‘f’ and ‘df’ functions at this point in the code below:

     int
     rosenbrock_df (const gsl_vector * x, void *params,
                    gsl_matrix * J)
     {
       const double a = ((struct rparams *) params)->a;
       const double b = ((struct rparams *) params)->b;

       const double x0 = gsl_vector_get (x, 0);

       const double df00 = -a;
       const double df01 = 0;
       const double df10 = -2 * b  * x0;
       const double df11 = b;

       gsl_matrix_set (J, 0, 0, df00);
       gsl_matrix_set (J, 0, 1, df01);
       gsl_matrix_set (J, 1, 0, df10);
       gsl_matrix_set (J, 1, 1, df11);

       return GSL_SUCCESS;
     }

     int
     rosenbrock_fdf (const gsl_vector * x, void *params,
                     gsl_vector * f, gsl_matrix * J)
     {
       rosenbrock_f (x, params, f);
       rosenbrock_df (x, params, J);

       return GSL_SUCCESS;
     }

The main program now makes calls to the corresponding ‘fdfsolver’
versions of the functions:

     int
     main (void)
     {
       const gsl_multiroot_fdfsolver_type *T;
       gsl_multiroot_fdfsolver *s;

       int status;
       size_t i, iter = 0;

       const size_t n = 2;
       struct rparams p = {1.0, 10.0};
       gsl_multiroot_function_fdf f = {&rosenbrock_f,
                                       &rosenbrock_df,
                                       &rosenbrock_fdf,
                                       n, &p};

       double x_init[2] = {-10.0, -5.0};
       gsl_vector *x = gsl_vector_alloc (n);

       gsl_vector_set (x, 0, x_init[0]);
       gsl_vector_set (x, 1, x_init[1]);

       T = gsl_multiroot_fdfsolver_gnewton;
       s = gsl_multiroot_fdfsolver_alloc (T, n);
       gsl_multiroot_fdfsolver_set (s, &f, x);

       print_state (iter, s);

       do
         {
           iter++;

           status = gsl_multiroot_fdfsolver_iterate (s);

           print_state (iter, s);

           if (status)
             break;

           status = gsl_multiroot_test_residual (s->f, 1e-7);
         }
       while (status == GSL_CONTINUE && iter < 1000);

       printf ("status = %s\n", gsl_strerror (status));

       gsl_multiroot_fdfsolver_free (s);
       gsl_vector_free (x);
       return 0;
     }

The addition of derivative information to the
‘gsl_multiroot_fsolver_hybrids’ solver does not make any significant
difference to its behavior, since it able to approximate the Jacobian
numerically with sufficient accuracy.  To illustrate the behavior of a
different derivative solver we switch to
‘gsl_multiroot_fdfsolver_gnewton’.  This is a traditional Newton solver
with the constraint that it scales back its step if the full step would
lead “uphill”.  Here is the output for the
‘gsl_multiroot_fdfsolver_gnewton’ algorithm:

     iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
     iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
     iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
     iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
     status = success

The convergence is much more rapid, but takes a wide excursion out to
the point (-4.23,-65.3).  This could cause the algorithm to go astray in
a realistic application.  The hybrid algorithm follows the downhill path
to the solution more reliably.


File: gsl-ref.info,  Node: References and Further Reading<31>,  Prev: Examples<29>,  Up: Multidimensional Root-Finding

38.9 References and Further Reading
===================================

The original version of the Hybrid method is described in the following
articles by Powell,

   * M.J.D. Powell, “A Hybrid Method for Nonlinear Equations” (Chap 6, p
     87–114) and “A Fortran Subroutine for Solving systems of Nonlinear
     Algebraic Equations” (Chap 7, p 115–161), in `Numerical Methods for
     Nonlinear Algebraic Equations', P. Rabinowitz, editor.  Gordon and
     Breach, 1970.

The following papers are also relevant to the algorithms described in
this section,

   * J.J. Moré, M.Y. Cosnard, “Numerical Solution of Nonlinear
     Equations”, `ACM Transactions on Mathematical Software', Vol 5, No
     1, (1979), p 64–85

   * C.G. Broyden, “A Class of Methods for Solving Nonlinear
     Simultaneous Equations”, `Mathematics of Computation', Vol 19
     (1965), p 577–593

   * J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained
     Optimization Software”, ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17–41


File: gsl-ref.info,  Node: Multidimensional Minimization,  Next: Linear Least-Squares Fitting,  Prev: Multidimensional Root-Finding,  Up: Top

39 Multidimensional Minimization
********************************

This chapter describes routines for finding minima of arbitrary
multidimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, while providing
full access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs.  The minimization
algorithms can be used to maximize a function by inverting its sign.

The header file ‘gsl_multimin.h’ contains prototypes for the
minimization functions and related declarations.

* Menu:

* Overview: Overview<4>.
* Caveats: Caveats<3>.
* Initializing the Multidimensional Minimizer::
* Providing a function to minimize::
* Iteration: Iteration<4>.
* Stopping Criteria::
* Algorithms with Derivatives::
* Algorithms without Derivatives: Algorithms without Derivatives<2>.
* Examples: Examples<30>.
* References and Further Reading: References and Further Reading<32>.


File: gsl-ref.info,  Node: Overview<4>,  Next: Caveats<3>,  Up: Multidimensional Minimization

39.1 Overview
=============

The problem of multidimensional minimization requires finding a point x
such that the scalar function,

     f(x_1, \dots, x_n)

takes a value which is lower than at any neighboring point.  For smooth
functions the gradient g = \nabla f vanishes at the minimum.  In general
there are no bracketing methods available for the minimization of
n-dimensional functions.  The algorithms proceed from an initial guess
using a search algorithm which attempts to move in a downhill direction.

Algorithms making use of the gradient of the function perform a
one-dimensional line minimisation along this direction until the lowest
point is found to a suitable tolerance.  The search direction is then
updated with local information from the function and its derivatives,
and the whole process repeated until the true n-dimensional minimum is
found.

Algorithms which do not require the gradient of the function use
different strategies.  For example, the Nelder-Mead Simplex algorithm
maintains n+1 trial parameter vectors as the vertices of a n-dimensional
simplex.  On each iteration it tries to improve the worst vertex of the
simplex by geometrical transformations.  The iterations are continued
until the overall size of the simplex has decreased sufficiently.

Both types of algorithms use a standard framework.  The user provides a
high-level driver for the algorithms, and the library provides the
individual functions necessary for each of the steps.  There are three
main phases of the iteration.  The steps are,

   * initialize minimizer state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself.  The state for the minimizers is held in a *note
gsl_multimin_fdfminimizer: af9. struct or a *note
gsl_multimin_fminimizer: afa. struct.


File: gsl-ref.info,  Node: Caveats<3>,  Next: Initializing the Multidimensional Minimizer,  Prev: Overview<4>,  Up: Multidimensional Minimization

39.2 Caveats
============

Note that the minimization algorithms can only search for one local
minimum at a time.  When there are several local minima in the search
area, the first minimum to be found will be returned; however it is
difficult to predict which of the minima this will be.  In most cases,
no error will be reported if you try to find a local minimum in an area
where there is more than one.

It is also important to note that the minimization algorithms find local
minima; there is no way to determine whether a minimum is a global
minimum of the function in question.


File: gsl-ref.info,  Node: Initializing the Multidimensional Minimizer,  Next: Providing a function to minimize,  Prev: Caveats<3>,  Up: Multidimensional Minimization

39.3 Initializing the Multidimensional Minimizer
================================================

The following function initializes a multidimensional minimizer.  The
minimizer itself depends only on the dimension of the problem and the
algorithm and can be reused for different problems.

 -- Type: gsl_multimin_fdfminimizer

     This is a workspace for minimizing functions using derivatives.

 -- Type: gsl_multimin_fminimizer

     This is a workspace for minimizing functions without derivatives.

 -- Function: *note gsl_multimin_fdfminimizer: af9.
          *gsl_multimin_fdfminimizer_alloc (const
          gsl_multimin_fdfminimizer_type *T, size_t n)
 -- Function: *note gsl_multimin_fminimizer: afa.
          *gsl_multimin_fminimizer_alloc (const
          gsl_multimin_fminimizer_type *T, size_t n)

     This function returns a pointer to a newly allocated instance of a
     minimizer of type *note T: afe. for an *note n: afe.-dimension
     function.  If there is insufficient memory to create the minimizer
     then the function returns a null pointer and the error handler is
     invoked with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_multimin_fdfminimizer_set
          (gsl_multimin_fdfminimizer *s, gsl_multimin_function_fdf *fdf,
          const gsl_vector *x, double step_size, double tol)
 -- Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer
          *s, gsl_multimin_function *f, const gsl_vector *x, const
          gsl_vector *step_size)

     The function *note gsl_multimin_fdfminimizer_set(): aff.
     initializes the minimizer *note s: b00. to minimize the function
     ‘fdf’ starting from the initial point *note x: b00.  The size of
     the first trial step is given by *note step_size: b00.  The
     accuracy of the line minimization is specified by ‘tol’.  The
     precise meaning of this parameter depends on the method used.
     Typically the line minimization is considered successful if the
     gradient of the function g is orthogonal to the current search
     direction p to a relative accuracy of ‘tol’, where p \cdot g < tol
     |p| |g|.  A ‘tol’ value of 0.1 is suitable for most purposes, since
     line minimization only needs to be carried out approximately.  Note
     that setting ‘tol’ to zero will force the use of “exact”
     line-searches, which are extremely expensive.

     The function *note gsl_multimin_fminimizer_set(): b00. initializes
     the minimizer *note s: b00. to minimize the function *note f: b00,
     starting from the initial point *note x: b00.  The size of the
     initial trial steps is given in vector *note step_size: b00.  The
     precise meaning of this parameter depends on the method used.

 -- Function: void gsl_multimin_fdfminimizer_free
          (gsl_multimin_fdfminimizer *s)
 -- Function: void gsl_multimin_fminimizer_free (gsl_multimin_fminimizer
          *s)

     This function frees all the memory associated with the minimizer
     *note s: b02.

 -- Function: const char *gsl_multimin_fdfminimizer_name (const
          gsl_multimin_fdfminimizer *s)
 -- Function: const char *gsl_multimin_fminimizer_name (const
          gsl_multimin_fminimizer *s)

     This function returns a pointer to the name of the minimizer.  For
     example:

          printf ("s is a '%s' minimizer\n", gsl_multimin_fdfminimizer_name (s));

     would print something like ‘s is a 'conjugate_pr' minimizer’.


File: gsl-ref.info,  Node: Providing a function to minimize,  Next: Iteration<4>,  Prev: Initializing the Multidimensional Minimizer,  Up: Multidimensional Minimization

39.4 Providing a function to minimize
=====================================

You must provide a parametric function of n variables for the minimizers
to operate on.  You may also need to provide a routine which calculates
the gradient of the function and a third routine which calculates both
the function value and the gradient together.  In order to allow for
general parameters the functions are defined by the following data
types:

 -- Type: gsl_multimin_function_fdf

     This data type defines a general function of n variables with
     parameters and the corresponding gradient vector of derivatives,

     ‘double (* f) (const gsl_vector * x, void * params)’

          this function should return the result f(x,params) for
          argument ‘x’ and parameters ‘params’.  If the function cannot
          be computed, an error value of *note GSL_NAN: 3c. should be
          returned.

     ‘void (* df) (const gsl_vector * x, void * params, gsl_vector * g)’

          this function should store the ‘n’-dimensional gradient

               g_i = d f(x,params) / d x_i

          in the vector ‘g’ for argument ‘x’ and parameters ‘params’,
          returning an appropriate error code if the function cannot be
          computed.

     ‘void (* fdf) (const gsl_vector * x, void * params, double * f,
     gsl_vector * g)’

          This function should set the values of the ‘f’ and ‘g’ as
          above, for arguments ‘x’ and parameters ‘params’.  This
          function provides an optimization of the separate functions
          for f(x) and g(x)—it is always faster to compute the function
          and its derivative at the same time.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’.

     ‘void * params’

          a pointer to the parameters of the function.

 -- Type: gsl_multimin_function

     This data type defines a general function of n variables with
     parameters,

     ‘double (* f) (const gsl_vector * x, void * params)’

          this function should return the result f(x,params) for
          argument ‘x’ and parameters ‘params’.  If the function cannot
          be computed, an error value of *note GSL_NAN: 3c. should be
          returned.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’.

     ‘void * params’

          a pointer to the parameters of the function.
The following example function defines a simple two-dimensional
paraboloid with five parameters,

     /* Paraboloid centered on (p[0],p[1]), with
        scale factors (p[2],p[3]) and minimum p[4] */

     double
     my_f (const gsl_vector *v, void *params)
     {
       double x, y;
       double *p = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       return p[2] * (x - p[0]) * (x - p[0]) +
                p[3] * (y - p[1]) * (y - p[1]) + p[4];
     }

     /* The gradient of f, df = (df/dx, df/dy). */
     void
     my_df (const gsl_vector *v, void *params,
            gsl_vector *df)
     {
       double x, y;
       double *p = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0]));
       gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[1]));
     }

     /* Compute both f and df together. */
     void
     my_fdf (const gsl_vector *x, void *params,
             double *f, gsl_vector *df)
     {
       *f = my_f(x, params);
       my_df(x, params, df);
     }

The function can be initialized using the following code:

     gsl_multimin_function_fdf my_func;

     /* Paraboloid center at (1,2), scale factors (10, 20),
        minimum value 30 */
     double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };

     my_func.n = 2;  /* number of function components */
     my_func.f = &my_f;
     my_func.df = &my_df;
     my_func.fdf = &my_fdf;
     my_func.params = (void *)p;


File: gsl-ref.info,  Node: Iteration<4>,  Next: Stopping Criteria,  Prev: Providing a function to minimize,  Up: Multidimensional Minimization

39.5 Iteration
==============

The following function drives the iteration of each algorithm.  The
function performs one iteration to update the state of the minimizer.
The same function works for all minimizers so that different methods can
be substituted at runtime without modifications to the code.

 -- Function: int gsl_multimin_fdfminimizer_iterate
          (gsl_multimin_fdfminimizer *s)
 -- Function: int gsl_multimin_fminimizer_iterate
          (gsl_multimin_fminimizer *s)

     These functions perform a single iteration of the minimizer *note
     s: b0b.  If the iteration encounters an unexpected problem then an
     error code will be returned.  The error code ‘GSL_ENOPROG’
     signifies that the minimizer is unable to improve on its current
     estimate, either due to numerical difficulty or because a genuine
     local minimum has been reached.

The minimizer maintains a current best estimate of the minimum at all
times.  This information can be accessed with the following auxiliary
functions,

 -- Function: *note gsl_vector: 35f. *gsl_multimin_fdfminimizer_x (const
          gsl_multimin_fdfminimizer *s)
 -- Function: *note gsl_vector: 35f. *gsl_multimin_fminimizer_x (const
          gsl_multimin_fminimizer *s)
 -- Function: double gsl_multimin_fdfminimizer_minimum (const
          gsl_multimin_fdfminimizer *s)
 -- Function: double gsl_multimin_fminimizer_minimum (const
          gsl_multimin_fminimizer *s)
 -- Function: *note gsl_vector: 35f. *gsl_multimin_fdfminimizer_gradient
          (const gsl_multimin_fdfminimizer *s)
 -- Function: *note gsl_vector: 35f. *gsl_multimin_fdfminimizer_dx
          (const gsl_multimin_fdfminimizer *s)
 -- Function: double gsl_multimin_fminimizer_size (const
          gsl_multimin_fminimizer *s)

     These functions return the current best estimate of the location of
     the minimum, the value of the function at that point, its gradient,
     the last step increment of the estimate, and minimizer specific
     characteristic size for the minimizer *note s: b12.

 -- Function: int gsl_multimin_fdfminimizer_restart
          (gsl_multimin_fdfminimizer *s)

     This function resets the minimizer *note s: b13. to use the current
     point as a new starting point.


File: gsl-ref.info,  Node: Stopping Criteria,  Next: Algorithms with Derivatives,  Prev: Iteration<4>,  Up: Multidimensional Minimization

39.6 Stopping Criteria
======================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result.

 -- Function: int gsl_multimin_test_gradient (const gsl_vector *g,
          double epsabs)

     This function tests the norm of the gradient *note g: b15. against
     the absolute tolerance *note epsabs: b15.  The gradient of a
     multidimensional function goes to zero at a minimum.  The test
     returns ‘GSL_SUCCESS’ if the following condition is achieved,

          |g| < epsabs

     and returns ‘GSL_CONTINUE’ otherwise.  A suitable choice of *note
     epsabs: b15. can be made from the desired accuracy in the function
     for small variations in x.  The relationship between these
     quantities is given by \delta{f} = g\,\delta{x}.

 -- Function: int gsl_multimin_test_size (const double size, double
          epsabs)

     This function tests the minimizer specific characteristic size (if
     applicable to the used minimizer) against absolute tolerance *note
     epsabs: b16.  The test returns ‘GSL_SUCCESS’ if the size is smaller
     than tolerance, otherwise ‘GSL_CONTINUE’ is returned.


File: gsl-ref.info,  Node: Algorithms with Derivatives,  Next: Algorithms without Derivatives<2>,  Prev: Stopping Criteria,  Up: Multidimensional Minimization

39.7 Algorithms with Derivatives
================================

There are several minimization methods available.  The best choice of
algorithm depends on the problem.  The algorithms described in this
section use the value of the function and its gradient at each
evaluation point.

 -- Type: gsl_multimin_fdfminimizer_type

     This type specifies a minimization algorithm using gradients.

      -- Variable: *note gsl_multimin_fdfminimizer_type: b18.
               *gsl_multimin_fdfminimizer_conjugate_fr

          This is the Fletcher-Reeves conjugate gradient algorithm.  The
          conjugate gradient algorithm proceeds as a succession of line
          minimizations.  The sequence of search directions is used to
          build up an approximation to the curvature of the function in
          the neighborhood of the minimum.

          An initial search direction ‘p’ is chosen using the gradient,
          and line minimization is carried out in that direction.  The
          accuracy of the line minimization is specified by the
          parameter ‘tol’.  The minimum along this line occurs when the
          function gradient ‘g’ and the search direction ‘p’ are
          orthogonal.  The line minimization terminates when p\cdot g <
          tol |p| |g|.  The search direction is updated using the
          Fletcher-Reeves formula p' = g' - \beta p where
          \beta=-|g'|^2/|g|^2, and the line minimization is then
          repeated for the new search direction.

      -- Variable: *note gsl_multimin_fdfminimizer_type: b18.
               *gsl_multimin_fdfminimizer_conjugate_pr

          This is the Polak-Ribiere conjugate gradient algorithm.  It is
          similar to the Fletcher-Reeves method, differing only in the
          choice of the coefficient \beta.  Both methods work well when
          the evaluation point is close enough to the minimum of the
          objective function that it is well approximated by a quadratic
          hypersurface.

      -- Variable: *note gsl_multimin_fdfminimizer_type: b18.
               *gsl_multimin_fdfminimizer_vector_bfgs2
      -- Variable: *note gsl_multimin_fdfminimizer_type: b18.
               *gsl_multimin_fdfminimizer_vector_bfgs

          These methods use the vector Broyden-Fletcher-Goldfarb-Shanno
          (BFGS) algorithm.  This is a quasi-Newton method which builds
          up an approximation to the second derivatives of the function
          f using the difference between successive gradient vectors.
          By combining the first and second derivatives the algorithm is
          able to take Newton-type steps towards the function minimum,
          assuming quadratic behavior in that region.

          The ‘bfgs2’ version of this minimizer is the most efficient
          version available, and is a faithful implementation of the
          line minimization scheme described in Fletcher’s `Practical
          Methods of Optimization', Algorithms 2.6.2 and 2.6.4.  It
          supersedes the original ‘bfgs’ routine and requires
          substantially fewer function and gradient evaluations.  The
          user-supplied tolerance ‘tol’ corresponds to the parameter
          \sigma used by Fletcher.  A value of 0.1 is recommended for
          typical use (larger values correspond to less accurate line
          searches).

      -- Variable: *note gsl_multimin_fdfminimizer_type: b18.
               *gsl_multimin_fdfminimizer_steepest_descent

          The steepest descent algorithm follows the downhill gradient
          of the function at each step.  When a downhill step is
          successful the step-size is increased by a factor of two.  If
          the downhill step leads to a higher function value then the
          algorithm backtracks and the step size is decreased using the
          parameter ‘tol’.  A suitable value of ‘tol’ for most
          applications is 0.1.  The steepest descent method is
          inefficient and is included only for demonstration purposes.


File: gsl-ref.info,  Node: Algorithms without Derivatives<2>,  Next: Examples<30>,  Prev: Algorithms with Derivatives,  Up: Multidimensional Minimization

39.8 Algorithms without Derivatives
===================================

The algorithms described in this section use only the value of the
function at each evaluation point.

 -- Type: gsl_multimin_fminimizer_type

     This type specifies minimization algorithms which do not use
     gradients.

      -- Variable: *note gsl_multimin_fminimizer_type: b1f.
               *gsl_multimin_fminimizer_nmsimplex2
      -- Variable: *note gsl_multimin_fminimizer_type: b1f.
               *gsl_multimin_fminimizer_nmsimplex

          These methods use the Simplex algorithm of Nelder and Mead.
          Starting from the initial vector x = p_0, the algorithm
          constructs an additional n vectors p_i using the step size
          vector s = step\_size as follows:

               p_0 = (x_0, x_1, ... , x_n)
               p_1 = (x_0 + s_0, x_1, ... , x_n)
               p_2 = (x_0, x_1 + s_1, ... , x_n)
               ... = ...
               p_n = (x_0, x_1, ... , x_n + s_n)

          These vectors form the n+1 vertices of a simplex in n
          dimensions.  On each iteration the algorithm uses simple
          geometrical transformations to update the vector corresponding
          to the highest function value.  The geometric transformations
          are reflection, reflection followed by expansion, contraction
          and multiple contraction.  Using these transformations the
          simplex moves through the space towards the minimum, where it
          contracts itself.

          After each iteration, the best vertex is returned.  Note, that
          due to the nature of the algorithm not every step improves the
          current best parameter vector.  Usually several iterations are
          required.

          The minimizer-specific characteristic size is calculated as
          the average distance from the geometrical center of the
          simplex to all its vertices.  This size can be used as a
          stopping criteria, as the simplex contracts itself near the
          minimum.  The size is returned by the function *note
          gsl_multimin_fminimizer_size(): b12.

          The *note gsl_multimin_fminimizer_nmsimplex2: b20. version of
          this minimiser is a new O(N) operations implementation of the
          earlier O(N^2) operations *note
          gsl_multimin_fminimizer_nmsimplex: b21. minimiser.  It uses
          the same underlying algorithm, but the simplex updates are
          computed more efficiently for high-dimensional problems.  In
          addition, the size of simplex is calculated as the RMS
          distance of each vertex from the center rather than the mean
          distance, allowing a linear update of this quantity on each
          step.  The memory usage is O(N^2) for both algorithms.

      -- Variable: *note gsl_multimin_fminimizer_type: b1f.
               *gsl_multimin_fminimizer_nmsimplex2rand

          This method is a variant of *note
          gsl_multimin_fminimizer_nmsimplex2: b20. which initialises the
          simplex around the starting point ‘x’ using a
          randomly-oriented set of basis vectors instead of the fixed
          coordinate axes.  The final dimensions of the simplex are
          scaled along the coordinate axes by the vector ‘step_size’.
          The randomisation uses a simple deterministic generator so
          that repeated calls to *note gsl_multimin_fminimizer_set():
          b00. for a given solver object will vary the orientation in a
          well-defined way.


File: gsl-ref.info,  Node: Examples<30>,  Next: References and Further Reading<32>,  Prev: Algorithms without Derivatives<2>,  Up: Multidimensional Minimization

39.9 Examples
=============

This example program finds the minimum of the *note paraboloid function:
b08. defined earlier.  The location of the minimum is offset from the
origin in x and y, and the function value at the minimum is non-zero.
The main program is given below, it requires the example function given
earlier in this chapter.

     int
     main (void)
     {
       size_t iter = 0;
       int status;

       const gsl_multimin_fdfminimizer_type *T;
       gsl_multimin_fdfminimizer *s;

       /* Position of the minimum (1,2), scale factors
          10,20, height 30. */
       double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };

       gsl_vector *x;
       gsl_multimin_function_fdf my_func;

       my_func.n = 2;
       my_func.f = my_f;
       my_func.df = my_df;
       my_func.fdf = my_fdf;
       my_func.params = par;

       /* Starting point, x = (5,7) */
       x = gsl_vector_alloc (2);
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);

       T = gsl_multimin_fdfminimizer_conjugate_fr;
       s = gsl_multimin_fdfminimizer_alloc (T, 2);

       gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);

       do
         {
           iter++;
           status = gsl_multimin_fdfminimizer_iterate (s);

           if (status)
             break;

           status = gsl_multimin_test_gradient (s->gradient, 1e-3);

           if (status == GSL_SUCCESS)
             printf ("Minimum found at:\n");

           printf ("%5d %.5f %.5f %10.5f\n", iter,
                   gsl_vector_get (s->x, 0),
                   gsl_vector_get (s->x, 1),
                   s->f);

         }
       while (status == GSL_CONTINUE && iter < 100);

       gsl_multimin_fdfminimizer_free (s);
       gsl_vector_free (x);

       return 0;
     }

The initial step-size is chosen as 0.01, a conservative estimate in this
case, and the line minimization parameter is set at 0.0001.  The program
terminates when the norm of the gradient has been reduced below 0.001.
The output of the program is shown below,

              x       y         f
         1 4.99629 6.99072  687.84780
         2 4.98886 6.97215  683.55456
         3 4.97400 6.93501  675.01278
         4 4.94429 6.86073  658.10798
         5 4.88487 6.71217  625.01340
         6 4.76602 6.41506  561.68440
         7 4.52833 5.82083  446.46694
         8 4.05295 4.63238  261.79422
         9 3.10219 2.25548   75.49762
        10 2.85185 1.62963   67.03704
        11 2.19088 1.76182   45.31640
        12 0.86892 2.02622   30.18555
     Minimum found at:
        13 1.00000 2.00000   30.00000

Note that the algorithm gradually increases the step size as it
successfully moves downhill, as can be seen by plotting the successive
points in Fig.  %s.

[gsl-ref-figures/multimin]
Figure: Function contours with path taken by minimization algorithm

The conjugate gradient algorithm finds the minimum on its second
direction because the function is purely quadratic.  Additional
iterations would be needed for a more complicated function.

Here is another example using the Nelder-Mead Simplex algorithm to
minimize the same example object function, as above.

     int
     main(void)
     {
       double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0};

       const gsl_multimin_fminimizer_type *T =
         gsl_multimin_fminimizer_nmsimplex2;
       gsl_multimin_fminimizer *s = NULL;
       gsl_vector *ss, *x;
       gsl_multimin_function minex_func;

       size_t iter = 0;
       int status;
       double size;

       /* Starting point */
       x = gsl_vector_alloc (2);
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);

       /* Set initial step sizes to 1 */
       ss = gsl_vector_alloc (2);
       gsl_vector_set_all (ss, 1.0);

       /* Initialize method and iterate */
       minex_func.n = 2;
       minex_func.f = my_f;
       minex_func.params = par;

       s = gsl_multimin_fminimizer_alloc (T, 2);
       gsl_multimin_fminimizer_set (s, &minex_func, x, ss);

       do
         {
           iter++;
           status = gsl_multimin_fminimizer_iterate(s);

           if (status)
             break;

           size = gsl_multimin_fminimizer_size (s);
           status = gsl_multimin_test_size (size, 1e-2);

           if (status == GSL_SUCCESS)
             {
               printf ("converged to minimum at\n");
             }

           printf ("%5d %10.3e %10.3e f() = %7.3f size = %.3f\n",
                   iter,
                   gsl_vector_get (s->x, 0),
                   gsl_vector_get (s->x, 1),
                   s->fval, size);
         }
       while (status == GSL_CONTINUE && iter < 100);

       gsl_vector_free(x);
       gsl_vector_free(ss);
       gsl_multimin_fminimizer_free (s);

       return status;
     }

The minimum search stops when the Simplex size drops to 0.01.  The
output is shown below.

         1  6.500e+00  5.000e+00 f() = 512.500 size = 1.130
         2  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
         3  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
         4  5.500e+00  1.000e+00 f() = 252.500 size = 1.409
         5  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
         6  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
         7  0.000e+00  3.000e+00 f() =  60.000 size = 1.847
         8  2.094e+00  1.875e+00 f() =  42.275 size = 1.321
         9  2.578e-01  1.906e+00 f() =  35.684 size = 1.069
        10  5.879e-01  2.445e+00 f() =  35.664 size = 0.841
        11  1.258e+00  2.025e+00 f() =  30.680 size = 0.476
        12  1.258e+00  2.025e+00 f() =  30.680 size = 0.367
        13  1.093e+00  1.849e+00 f() =  30.539 size = 0.300
        14  8.830e-01  2.004e+00 f() =  30.137 size = 0.172
        15  8.830e-01  2.004e+00 f() =  30.137 size = 0.126
        16  9.582e-01  2.060e+00 f() =  30.090 size = 0.106
        17  1.022e+00  2.004e+00 f() =  30.005 size = 0.063
        18  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
        19  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
        20  1.022e+00  2.004e+00 f() =  30.005 size = 0.027
        21  1.022e+00  2.004e+00 f() =  30.005 size = 0.022
        22  9.920e-01  1.997e+00 f() =  30.001 size = 0.016
        23  9.920e-01  1.997e+00 f() =  30.001 size = 0.013
     converged to minimum at
        24  9.920e-01  1.997e+00 f() =  30.001 size = 0.008

The simplex size first increases, while the simplex moves towards the
minimum.  After a while the size begins to decrease as the simplex
contracts around the minimum.


File: gsl-ref.info,  Node: References and Further Reading<32>,  Prev: Examples<30>,  Up: Multidimensional Minimization

39.10 References and Further Reading
====================================

The conjugate gradient and BFGS methods are described in detail in the
following book,

   * R. Fletcher, `Practical Methods of Optimization (Second Edition)'
     Wiley (1987), ISBN 0471915475.

A brief description of multidimensional minimization algorithms and more
recent references can be found in,

   * C.W. Ueberhuber, `Numerical Computation (Volume 2)', Chapter 14,
     Section 4.4 “Minimization Methods”, p.: 325–335, Springer (1997),
     ISBN 3-540-62057-5.

The simplex algorithm is described in the following paper,

   * J.A. Nelder and R. Mead, `A simplex method for function
     minimization', Computer Journal vol.: 7 (1965), 308–313.


File: gsl-ref.info,  Node: Linear Least-Squares Fitting,  Next: Nonlinear Least-Squares Fitting,  Prev: Multidimensional Minimization,  Up: Top

40 Linear Least-Squares Fitting
*******************************

This chapter describes routines for performing least squares fits to
experimental data using linear combinations of functions.  The data may
be weighted or unweighted, i.e.  with known or unknown errors.  For
weighted data the functions compute the best fit parameters and their
associated covariance matrix.  For unweighted data the covariance matrix
is estimated from the scatter of the points, giving a
variance-covariance matrix.

The functions are divided into separate versions for simple one- or
two-parameter regression and multiple-parameter fits.

* Menu:

* Overview: Overview<5>.
* Linear regression::
* Multi-parameter regression::
* Regularized regression::
* Robust linear regression::
* Large dense linear systems::
* Troubleshooting::
* Examples: Examples<31>.
* References and Further Reading: References and Further Reading<33>.


File: gsl-ref.info,  Node: Overview<5>,  Next: Linear regression,  Up: Linear Least-Squares Fitting

40.1 Overview
=============

Least-squares fits are found by minimizing \chi^2 (chi-squared), the
weighted sum of squared residuals over n experimental datapoints (x_i,
y_i) for the model Y(c,x),

     \chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2

The p parameters of the model are c = \{c_0, c_1, \dots\}.  The weight
factors w_i are given by w_i = 1/\sigma_i^2 where \sigma_i is the
experimental error on the data-point y_i.  The errors are assumed to be
Gaussian and uncorrelated.  For unweighted data the chi-squared sum is
computed without any weight factors.

The fitting routines return the best-fit parameters c and their p \times
p covariance matrix.  The covariance matrix measures the statistical
errors on the best-fit parameters resulting from the errors on the data,
\sigma_i, and is defined as

     C_{ab} = <\delta c_a \delta c_b>

where \langle \, \rangle denotes an average over the Gaussian error
distributions of the underlying datapoints.

The covariance matrix is calculated by error propagation from the data
errors \sigma_i.  The change in a fitted parameter \delta c_a caused by
a small change in the data \delta y_i is given by

     \delta c_a = \sum_i (dc_a/dy_i) \delta y_i

allowing the covariance matrix to be written in terms of the errors on
the data,

     C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>

For uncorrelated data the fluctuations of the underlying datapoints
satisfy

     <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}

giving a corresponding parameter covariance matrix of

     C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)

When computing the covariance matrix for unweighted data, i.e.  data
with unknown errors, the weight factors w_i in this sum are replaced by
the single estimate w = 1/\sigma^2, where \sigma^2 is the computed
variance of the residuals about the best-fit model, \sigma^2 = \sum (y_i
- Y(c,x_i))^2 / (n-p).  This is referred to as the `variance-covariance
matrix'.

The standard deviations of the best-fit parameters are given by the
square root of the corresponding diagonal elements of the covariance
matrix, \sigma_{c_a} = \sqrt{C_{aa}}.  The correlation coefficient of
the fit parameters c_a and c_b is given by \rho_{ab} = C_{ab} /
\sqrt{C_{aa} C_{bb}}.


File: gsl-ref.info,  Node: Linear regression,  Next: Multi-parameter regression,  Prev: Overview<5>,  Up: Linear Least-Squares Fitting

40.2 Linear regression
======================

The functions in this section are used to fit simple one or two
parameter linear regression models.  The functions are declared in the
header file ‘gsl_fit.h’.

* Menu:

* Linear regression with a constant term::
* Linear regression without a constant term::


File: gsl-ref.info,  Node: Linear regression with a constant term,  Next: Linear regression without a constant term,  Up: Linear regression

40.2.1 Linear regression with a constant term
---------------------------------------------

The functions described in this section can be used to perform
least-squares fits to a straight line model, Y(c,x) = c_0 + c_1 x.

 -- Function: int gsl_fit_linear (const double *x, const size_t xstride,
          const double *y, const size_t ystride, size_t n, double *c0,
          double *c1, double *cov00, double *cov01, double *cov11,
          double *sumsq)

     This function computes the best-fit linear regression coefficients
     (*note c0: b2c, *note c1: b2c.) of the model Y = c_0 + c_1 X for
     the dataset (*note x: b2c, *note y: b2c.), two vectors of length
     *note n: b2c. with strides *note xstride: b2c. and *note ystride:
     b2c.  The errors on *note y: b2c. are assumed unknown so the
     variance-covariance matrix for the parameters (*note c0: b2c, *note
     c1: b2c.) is estimated from the scatter of the points around the
     best-fit line and returned via the parameters (*note cov00: b2c,
     *note cov01: b2c, *note cov11: b2c.).  The sum of squares of the
     residuals from the best-fit line is returned in *note sumsq: b2c.
     Note: the correlation coefficient of the data can be computed using
     *note gsl_stats_correlation(): 7ef, it does not depend on the fit.

 -- Function: int gsl_fit_wlinear (const double *x, const size_t
          xstride, const double *w, const size_t wstride, const double
          *y, const size_t ystride, size_t n, double *c0, double *c1,
          double *cov00, double *cov01, double *cov11, double *chisq)

     This function computes the best-fit linear regression coefficients
     (*note c0: b2d, *note c1: b2d.) of the model Y = c_0 + c_1 X for
     the weighted dataset (*note x: b2d, *note y: b2d.), two vectors of
     length *note n: b2d. with strides *note xstride: b2d. and *note
     ystride: b2d.  The vector *note w: b2d, of length *note n: b2d. and
     stride *note wstride: b2d, specifies the weight of each datapoint.
     The weight is the reciprocal of the variance for each datapoint in
     *note y: b2d.

     The covariance matrix for the parameters (*note c0: b2d, *note c1:
     b2d.) is computed using the weights and returned via the parameters
     (*note cov00: b2d, *note cov01: b2d, *note cov11: b2d.).  The
     weighted sum of squares of the residuals from the best-fit line,
     \chi^2, is returned in *note chisq: b2d.

 -- Function: int gsl_fit_linear_est (double x, double c0, double c1,
          double cov00, double cov01, double cov11, double *y, double
          *y_err)

     This function uses the best-fit linear regression coefficients
     *note c0: b2e, *note c1: b2e. and their covariance *note cov00:
     b2e, *note cov01: b2e, *note cov11: b2e. to compute the fitted
     function *note y: b2e. and its standard deviation *note y_err: b2e.
     for the model Y = c_0 + c_1 X at the point *note x: b2e.


File: gsl-ref.info,  Node: Linear regression without a constant term,  Prev: Linear regression with a constant term,  Up: Linear regression

40.2.2 Linear regression without a constant term
------------------------------------------------

The functions described in this section can be used to perform
least-squares fits to a straight line model without a constant term, Y =
c_1 X.

 -- Function: int gsl_fit_mul (const double *x, const size_t xstride,
          const double *y, const size_t ystride, size_t n, double *c1,
          double *cov11, double *sumsq)

     This function computes the best-fit linear regression coefficient
     *note c1: b30. of the model Y = c_1 X for the datasets (*note x:
     b30, *note y: b30.), two vectors of length *note n: b30. with
     strides *note xstride: b30. and *note ystride: b30.  The errors on
     *note y: b30. are assumed unknown so the variance of the parameter
     *note c1: b30. is estimated from the scatter of the points around
     the best-fit line and returned via the parameter *note cov11: b30.
     The sum of squares of the residuals from the best-fit line is
     returned in *note sumsq: b30.

 -- Function: int gsl_fit_wmul (const double *x, const size_t xstride,
          const double *w, const size_t wstride, const double *y, const
          size_t ystride, size_t n, double *c1, double *cov11, double
          *sumsq)

     This function computes the best-fit linear regression coefficient
     *note c1: b31. of the model Y = c_1 X for the weighted datasets
     (*note x: b31, *note y: b31.), two vectors of length *note n: b31.
     with strides *note xstride: b31. and *note ystride: b31.  The
     vector *note w: b31, of length *note n: b31. and stride *note
     wstride: b31, specifies the weight of each datapoint.  The weight
     is the reciprocal of the variance for each datapoint in *note y:
     b31.

     The variance of the parameter *note c1: b31. is computed using the
     weights and returned via the parameter *note cov11: b31.  The
     weighted sum of squares of the residuals from the best-fit line,
     \chi^2, is returned in ‘chisq’.

 -- Function: int gsl_fit_mul_est (double x, double c1, double cov11,
          double *y, double *y_err)

     This function uses the best-fit linear regression coefficient *note
     c1: b32. and its covariance *note cov11: b32. to compute the fitted
     function *note y: b32. and its standard deviation *note y_err: b32.
     for the model Y = c_1 X at the point *note x: b32.


File: gsl-ref.info,  Node: Multi-parameter regression,  Next: Regularized regression,  Prev: Linear regression,  Up: Linear Least-Squares Fitting

40.3 Multi-parameter regression
===============================

This section describes routines which perform least squares fits to a
linear model by minimizing the cost function

     \chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2 = || y - Xc ||_W^2

where y is a vector of n observations, X is an n-by-p matrix of
predictor variables, c is a vector of the p unknown best-fit parameters
to be estimated, and ||r||_W^2 = r^T W r.  The matrix W =
\diag(w_1,w_2,...,w_n) defines the weights or uncertainties of the
observation vector.

This formulation can be used for fits to any number of functions and/or
variables by preparing the n-by-p matrix X appropriately.  For example,
to fit to a p-th order polynomial in ‘x’, use the following matrix,

     X_{ij} = x_i^j

where the index i runs over the observations and the index j runs from 0
to p-1.

To fit to a set of p sinusoidal functions with fixed frequencies
\omega_1, \omega_2, \ldots, \omega_p, use,

     X_{ij} = \sin(\omega_j x_i)

To fit to p independent variables x_1, x_2, \ldots, x_p, use,

     X_{ij} = x_j(i)

where x_j(i) is the i-th value of the predictor variable x_j.

The solution of the general linear least-squares system requires an
additional working space for intermediate results, such as the singular
value decomposition of the matrix X.

These functions are declared in the header file ‘gsl_multifit.h’.

 -- Type: gsl_multifit_linear_workspace

     This workspace contains internal variables for fitting
     multi-parameter models.

 -- Function: *note gsl_multifit_linear_workspace: b34.
          *gsl_multifit_linear_alloc (const size_t n, const size_t p)

     This function allocates a workspace for fitting a model to a
     maximum of *note n: b35. observations using a maximum of *note p:
     b35. parameters.  The user may later supply a smaller least squares
     system if desired.  The size of the workspace is O(np + p^2).

 -- Function: void gsl_multifit_linear_free
          (gsl_multifit_linear_workspace *work)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_multifit_linear_svd (const gsl_matrix *X,
          gsl_multifit_linear_workspace *work)

     This function performs a singular value decomposition of the matrix
     *note X: b37. and stores the SVD factors internally in *note work:
     b37.

 -- Function: int gsl_multifit_linear_bsvd (const gsl_matrix *X,
          gsl_multifit_linear_workspace *work)

     This function performs a singular value decomposition of the matrix
     *note X: b38. and stores the SVD factors internally in *note work:
     b38.  The matrix *note X: b38. is first balanced by applying column
     scaling factors to improve the accuracy of the singular values.

 -- Function: int gsl_multifit_linear (const gsl_matrix *X, const
          gsl_vector *y, gsl_vector *c, gsl_matrix *cov, double *chisq,
          gsl_multifit_linear_workspace *work)

     This function computes the best-fit parameters *note c: b39. of the
     model y = X c for the observations *note y: b39. and the matrix of
     predictor variables *note X: b39, using the preallocated workspace
     provided in *note work: b39.  The p-by-p variance-covariance matrix
     of the model parameters *note cov: b39. is set to \sigma^2 (X^T
     X)^{-1}, where \sigma is the standard deviation of the fit
     residuals.  The sum of squares of the residuals from the best-fit,
     \chi^2, is returned in *note chisq: b39.  If the coefficient of
     determination is desired, it can be computed from the expression
     R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of the
     observations *note y: b39. may be computed from *note
     gsl_stats_tss(): 7dc.

     The best-fit is found by singular value decomposition of the matrix
     *note X: b39. using the modified Golub-Reinsch SVD algorithm, with
     column scaling to improve the accuracy of the singular values.  Any
     components which have zero singular value (to machine precision)
     are discarded from the fit.

 -- Function: int gsl_multifit_linear_tsvd (const gsl_matrix *X, const
          gsl_vector *y, const double tol, gsl_vector *c, gsl_matrix
          *cov, double *chisq, size_t *rank,
          gsl_multifit_linear_workspace *work)

     This function computes the best-fit parameters *note c: b3a. of the
     model y = X c for the observations *note y: b3a. and the matrix of
     predictor variables *note X: b3a, using a truncated SVD expansion.
     Singular values which satisfy s_i \le tol \times s_0 are discarded
     from the fit, where s_0 is the largest singular value.  The p-by-p
     variance-covariance matrix of the model parameters *note cov: b3a.
     is set to \sigma^2 (X^T X)^{-1}, where \sigma is the standard
     deviation of the fit residuals.  The sum of squares of the
     residuals from the best-fit, \chi^2, is returned in *note chisq:
     b3a.  The effective rank (number of singular values used in
     solution) is returned in *note rank: b3a.  If the coefficient of
     determination is desired, it can be computed from the expression
     R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of the
     observations *note y: b3a. may be computed from *note
     gsl_stats_tss(): 7dc.

 -- Function: int gsl_multifit_wlinear (const gsl_matrix *X, const
          gsl_vector *w, const gsl_vector *y, gsl_vector *c, gsl_matrix
          *cov, double *chisq, gsl_multifit_linear_workspace *work)

     This function computes the best-fit parameters *note c: b3b. of the
     weighted model y = X c for the observations *note y: b3b. with
     weights *note w: b3b. and the matrix of predictor variables *note
     X: b3b, using the preallocated workspace provided in *note work:
     b3b.  The p-by-p covariance matrix of the model parameters *note
     cov: b3b. is computed as (X^T W X)^{-1}.  The weighted sum of
     squares of the residuals from the best-fit, \chi^2, is returned in
     *note chisq: b3b.  If the coefficient of determination is desired,
     it can be computed from the expression R^2 = 1 - \chi^2 / WTSS,
     where the weighted total sum of squares (WTSS) of the observations
     *note y: b3b. may be computed from *note gsl_stats_wtss(): 7f9.

 -- Function: int gsl_multifit_wlinear_tsvd (const gsl_matrix *X, const
          gsl_vector *w, const gsl_vector *y, const double tol,
          gsl_vector *c, gsl_matrix *cov, double *chisq, size_t *rank,
          gsl_multifit_linear_workspace *work)

     This function computes the best-fit parameters *note c: b3c. of the
     weighted model y = X c for the observations *note y: b3c. with
     weights *note w: b3c. and the matrix of predictor variables *note
     X: b3c, using a truncated SVD expansion.  Singular values which
     satisfy s_i \le tol \times s_0 are discarded from the fit, where
     s_0 is the largest singular value.  The p-by-p covariance matrix of
     the model parameters *note cov: b3c. is computed as (X^T W X)^{-1}.
     The weighted sum of squares of the residuals from the best-fit,
     \chi^2, is returned in *note chisq: b3c.  The effective rank of the
     system (number of singular values used in the solution) is returned
     in *note rank: b3c.  If the coefficient of determination is
     desired, it can be computed from the expression R^2 = 1 - \chi^2 /
     WTSS, where the weighted total sum of squares (WTSS) of the
     observations *note y: b3c. may be computed from *note
     gsl_stats_wtss(): 7f9.

 -- Function: int gsl_multifit_linear_est (const gsl_vector *x, const
          gsl_vector *c, const gsl_matrix *cov, double *y, double
          *y_err)

     This function uses the best-fit multilinear regression coefficients
     *note c: b3d. and their covariance matrix *note cov: b3d. to
     compute the fitted function value *note y: b3d. and its standard
     deviation *note y_err: b3d. for the model y = x.c at the point
     *note x: b3d.

 -- Function: int gsl_multifit_linear_residuals (const gsl_matrix *X,
          const gsl_vector *y, const gsl_vector *c, gsl_vector *r)

     This function computes the vector of residuals r = y - X c for the
     observations *note y: b3e, coefficients *note c: b3e. and matrix of
     predictor variables *note X: b3e.

 -- Function: size_t gsl_multifit_linear_rank (const double tol, const
          gsl_multifit_linear_workspace *work)

     This function returns the rank of the matrix X which must first
     have its singular value decomposition computed.  The rank is
     computed by counting the number of singular values \sigma_j which
     satisfy \sigma_j > tol \times \sigma_0, where \sigma_0 is the
     largest singular value.


File: gsl-ref.info,  Node: Regularized regression,  Next: Robust linear regression,  Prev: Multi-parameter regression,  Up: Linear Least-Squares Fitting

40.4 Regularized regression
===========================

Ordinary weighted least squares models seek a solution vector c which
minimizes the residual

     \chi^2 = || y - Xc ||_W^2

where y is the n-by-1 observation vector, X is the n-by-p design matrix,
c is the p-by-1 solution vector, W = \diag(w_1,...,w_n) is the data
weighting matrix, and ||r||_W^2 = r^T W r.  In cases where the least
squares matrix X is ill-conditioned, small perturbations (ie: noise) in
the observation vector could lead to widely different solution vectors
c.  One way of dealing with ill-conditioned matrices is to use a
“truncated SVD” in which small singular values, below some given
tolerance, are discarded from the solution.  The truncated SVD method is
available using the functions *note gsl_multifit_linear_tsvd(): b3a. and
*note gsl_multifit_wlinear_tsvd(): b3c.  Another way to help solve
ill-posed problems is to include a regularization term in the least
squares minimization

     \chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2

for a suitably chosen regularization parameter \lambda and matrix L.
This type of regularization is known as Tikhonov, or ridge, regression.
In some applications, L is chosen as the identity matrix, giving
preference to solution vectors c with smaller norms.  Including this
regularization term leads to the explicit “normal equations” solution

     c = ( X^T W X + \lambda^2 L^T L )^-1 X^T W y

which reduces to the ordinary least squares solution when L = 0.  In
practice, it is often advantageous to transform a regularized least
squares system into the form

     \chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2

This is known as the Tikhonov “standard form” and has the normal
equations solution

     \tilde{c} = ( \tilde{X}^T \tilde{X} + \lambda^2 I )^{-1} \tilde{X}^T \tilde{y}

For an m-by-p matrix L which is full rank and has m >= p (ie: L is
square or has more rows than columns), we can calculate the “thin” QR
decomposition of L, and note that ||L c|| = ||R c|| since the Q factor
will not change the norm.  Since R is p-by-p, we can then use the
transformation

     X~ = sqrt(W) X R^-1
     y~ = sqrt(W) y
     c~ = R c

to achieve the standard form.  For a rectangular matrix L with m < p, a
more sophisticated approach is needed (see Hansen 1998, chapter 2.3).
In practice, the normal equations solution above is not desirable due to
numerical instabilities, and so the system is solved using the singular
value decomposition of the matrix \tilde{X}.  The matrix L is often
chosen as the identity matrix, or as a first or second finite difference
operator, to ensure a smoothly varying coefficient vector c, or as a
diagonal matrix to selectively damp each model parameter differently.
If L \ne I, the user must first convert the least squares problem to
standard form using *note gsl_multifit_linear_stdform1(): b42. or *note
gsl_multifit_linear_stdform2(): b43, solve the system, and then
backtransform the solution vector to recover the solution of the
original problem (see *note gsl_multifit_linear_genform1(): b44. and
*note gsl_multifit_linear_genform2(): b45.).

In many regularization problems, care must be taken when choosing the
regularization parameter \lambda.  Since both the residual norm ||y - X
c|| and solution norm ||L c|| are being minimized, the parameter \lambda
represents a tradeoff between minimizing either the residuals or the
solution vector.  A common tool for visualizing the comprimise between
the minimization of these two quantities is known as the L-curve.  The
L-curve is a log-log plot of the residual norm ||y - X c|| on the
horizontal axis and the solution norm ||L c|| on the vertical axis.
This curve nearly always as an L shaped appearance, with a distinct
corner separating the horizontal and vertical sections of the curve.
The regularization parameter corresponding to this corner is often
chosen as the optimal value.  GSL provides routines to calculate the
L-curve for all relevant regularization parameters as well as locating
the corner.

Another method of choosing the regularization parameter is known as
Generalized Cross Validation (GCV). This method is based on the idea
that if an arbitrary element y_i is left out of the right hand side, the
resulting regularized solution should predict this element accurately.
This leads to choosing the parameter \lambda which minimizes the GCV
function

     G(\lambda) = (||y - X c_{\lambda}||^2) / Tr(I_n - X X^I)^2

where X_{\lambda}^I is the matrix which relates the solution c_{\lambda}
to the right hand side y, ie: c_{\lambda} = X_{\lambda}^I y.  GSL
provides routines to compute the GCV curve and its minimum.

For most applications, the steps required to solve a regularized least
squares problem are as follows:

  1. Construct the least squares system (X, y, W, L)

  2. Transform the system to standard form (\tilde{X}, \tilde{y}).  This
     step can be skipped if L = I_p and W = I_n.

  3. Calculate the SVD of \tilde{X}.

  4. Determine an appropriate regularization parameter \lambda (using
     for example L-curve or GCV analysis).

  5. Solve the standard form system using the chosen \lambda and the SVD
     of \tilde{X}.

  6. Backtransform the standard form solution \tilde{c} to recover the
     original solution vector c.

 -- Function: int gsl_multifit_linear_stdform1 (const gsl_vector *L,
          const gsl_matrix *X, const gsl_vector *y, gsl_matrix *Xs,
          gsl_vector *ys, gsl_multifit_linear_workspace *work)
 -- Function: int gsl_multifit_linear_wstdform1 (const gsl_vector *L,
          const gsl_matrix *X, const gsl_vector *w, const gsl_vector *y,
          gsl_matrix *Xs, gsl_vector *ys, gsl_multifit_linear_workspace
          *work)

     These functions define a regularization matrix L =
     \diag(l_0,l_1,...,l_{p-1}).  The diagonal matrix element l_i is
     provided by the i-th element of the input vector *note L: b46.  The
     n-by-p least squares matrix *note X: b46. and vector *note y: b46.
     of length n are then converted to standard form as described above
     and the parameters (\tilde{X}, \tilde{y}) are stored in *note Xs:
     b46. and *note ys: b46. on output.  *note Xs: b46. and *note ys:
     b46. have the same dimensions as *note X: b46. and *note y: b46.
     Optional data weights may be supplied in the vector *note w: b46.
     of length n.  In order to apply this transformation, L^{-1} must
     exist and so none of the l_i may be zero.  After the standard form
     system has been solved, use *note gsl_multifit_linear_genform1():
     b44. to recover the original solution vector.  It is allowed to
     have *note X: b46. = *note Xs: b46. and *note y: b46. = *note ys:
     b46. for an in-place transform.  In order to perform a weighted
     regularized fit with L = I, the user may call *note
     gsl_multifit_linear_applyW(): b47. to convert to standard form.

 -- Function: int gsl_multifit_linear_L_decomp (gsl_matrix *L,
          gsl_vector *tau)

     This function factors the m-by-p regularization matrix *note L:
     b48. into a form needed for the later transformation to standard
     form.  *note L: b48. may have any number of rows m.  If m \ge p the
     QR decomposition of *note L: b48. is computed and stored in *note
     L: b48. on output.  If m < p, the QR decomposition of L^T is
     computed and stored in *note L: b48. on output.  On output, the
     Householder scalars are stored in the vector *note tau: b48. of
     size MIN(m,p).  These outputs will be used by *note
     gsl_multifit_linear_wstdform2(): b49. to complete the
     transformation to standard form.

 -- Function: int gsl_multifit_linear_stdform2 (const gsl_matrix *LQR,
          const gsl_vector *Ltau, const gsl_matrix *X, const gsl_vector
          *y, gsl_matrix *Xs, gsl_vector *ys, gsl_matrix *M,
          gsl_multifit_linear_workspace *work)
 -- Function: int gsl_multifit_linear_wstdform2 (const gsl_matrix *LQR,
          const gsl_vector *Ltau, const gsl_matrix *X, const gsl_vector
          *w, const gsl_vector *y, gsl_matrix *Xs, gsl_vector *ys,
          gsl_matrix *M, gsl_multifit_linear_workspace *work)

     These functions convert the least squares system (*note X: b49,
     *note y: b49, ‘W’, L) to standard form (\tilde{X}, \tilde{y}) which
     are stored in *note Xs: b49. and *note ys: b49. respectively.  The
     m-by-p regularization matrix ‘L’ is specified by the inputs *note
     LQR: b49. and *note Ltau: b49, which are outputs from *note
     gsl_multifit_linear_L_decomp(): b48.  The dimensions of the
     standard form parameters (\tilde{X}, \tilde{y}) depend on whether m
     is larger or less than p.  For m \ge p, *note Xs: b49. is n-by-p,
     *note ys: b49. is n-by-1, and *note M: b49. is not used.  For m <
     p, *note Xs: b49. is (n - p + m)-by-m, *note ys: b49. is (n - p +
     m)-by-1, and *note M: b49. is additional n-by-p workspace, which is
     required to recover the original solution vector after the system
     has been solved (see *note gsl_multifit_linear_genform2(): b45.).
     Optional data weights may be supplied in the vector *note w: b49.
     of length n, where W = \diag(w).

 -- Function: int gsl_multifit_linear_solve (const double lambda, const
          gsl_matrix *Xs, const gsl_vector *ys, gsl_vector *cs, double
          *rnorm, double *snorm, gsl_multifit_linear_workspace *work)

     This function computes the regularized best-fit parameters
     \tilde{c} which minimize the cost function \chi^2 = || \tilde{y} -
     \tilde{X} \tilde{c} ||^2 + \lambda^2 || \tilde{c} ||^2 which is in
     standard form.  The least squares system must therefore be
     converted to standard form prior to calling this function.  The
     observation vector \tilde{y} is provided in *note ys: b4a. and the
     matrix of predictor variables \tilde{X} in *note Xs: b4a.  The
     solution vector \tilde{c} is returned in *note cs: b4a, which has
     length min(m,p).  The SVD of *note Xs: b4a. must be computed prior
     to calling this function, using *note gsl_multifit_linear_svd():
     b37.  The regularization parameter \lambda is provided in *note
     lambda: b4a.  The residual norm || \tilde{y} - \tilde{X} \tilde{c}
     || = ||y - X c||_W is returned in *note rnorm: b4a.  The solution
     norm || \tilde{c} || = ||L c|| is returned in *note snorm: b4a.

 -- Function: int gsl_multifit_linear_genform1 (const gsl_vector *L,
          const gsl_vector *cs, gsl_vector *c,
          gsl_multifit_linear_workspace *work)

     After a regularized system has been solved with L =
     \diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
     standard form solution vector *note cs: b44. to recover the
     solution vector of the original problem *note c: b44.  The diagonal
     matrix elements l_i are provided in the vector *note L: b44.  It is
     allowed to have *note c: b44. = *note cs: b44. for an in-place
     transform.

 -- Function: int gsl_multifit_linear_genform2 (const gsl_matrix *LQR,
          const gsl_vector *Ltau, const gsl_matrix *X, const gsl_vector
          *y, const gsl_vector *cs, const gsl_matrix *M, gsl_vector *c,
          gsl_multifit_linear_workspace *work)
 -- Function: int gsl_multifit_linear_wgenform2 (const gsl_matrix *LQR,
          const gsl_vector *Ltau, const gsl_matrix *X, const gsl_vector
          *w, const gsl_vector *y, const gsl_vector *cs, const
          gsl_matrix *M, gsl_vector *c, gsl_multifit_linear_workspace
          *work)

     After a regularized system has been solved with a general
     rectangular matrix L, specified by (*note LQR: b4b, *note Ltau:
     b4b.), this function backtransforms the standard form solution
     *note cs: b4b. to recover the solution vector of the original
     problem, which is stored in *note c: b4b, of length p.  The
     original least squares matrix and observation vector are provided
     in *note X: b4b. and *note y: b4b. respectively.  *note M: b4b. is
     the matrix computed by *note gsl_multifit_linear_stdform2(): b43.
     For weighted fits, the weight vector *note w: b4b. must also be
     supplied.

 -- Function: int gsl_multifit_linear_applyW (const gsl_matrix *X, const
          gsl_vector *w, const gsl_vector *y, gsl_matrix *WX, gsl_vector
          *Wy)

     For weighted least squares systems with L = I, this function may be
     used to convert the system to standard form by applying the weight
     matrix W = \diag(w) to the least squares matrix *note X: b47. and
     observation vector *note y: b47.  On output, *note WX: b47. is
     equal to W^{1/2} X and *note Wy: b47. is equal to W^{1/2} y.  It is
     allowed for *note WX: b47. = *note X: b47. and *note Wy: b47. =
     *note y: b47. for an in-place transform.

 -- Function: int gsl_multifit_linear_lcurve (const gsl_vector *y,
          gsl_vector *reg_param, gsl_vector *rho, gsl_vector *eta,
          gsl_multifit_linear_workspace *work)

     This function computes the L-curve for a least squares system using
     the right hand side vector *note y: b4c. and the SVD decomposition
     of the least squares matrix ‘X’, which must be provided to *note
     gsl_multifit_linear_svd(): b37. prior to calling this function.
     The output vectors *note reg_param: b4c, *note rho: b4c, and *note
     eta: b4c. must all be the same size, and will contain the
     regularization parameters \lambda_i, residual norms ||y - X c_i||,
     and solution norms || L c_i || which compose the L-curve, where c_i
     is the regularized solution vector corresponding to \lambda_i.  The
     user may determine the number of points on the L-curve by adjusting
     the size of these input arrays.  The regularization parameters
     \lambda_i are estimated from the singular values of ‘X’, and chosen
     to represent the most relevant portion of the L-curve.

 -- Function: int gsl_multifit_linear_lcurvature (const gsl_vector *y,
          const gsl_vector *reg_param, const gsl_vector *rho, const
          gsl_vector *eta, gsl_vector *kappa,
          gsl_multifit_linear_workspace *work)

     This function computes the curvature of the L-curve as a function
     of the regularization parameter \lambda, using the right hand side
     vector *note y: b4d, the vector of regularization parameters, *note
     reg_param: b4d, vector of residual norms, *note rho: b4d, and
     vector of solution norms, *note eta: b4d.  The arrays *note
     reg_param: b4d, *note rho: b4d, and *note eta: b4d. can be computed
     by *note gsl_multifit_linear_lcurve(): b4c.  The curvature is
     defined as

          \kappa(\lambda) = \frac{\hat{\rho}' \hat{\eta}'' - \hat{\rho}'' \hat{\eta}'}{\left( (\hat{\rho}')^2 + (\hat{\eta}')^2 \right)^{\frac{3}{2}}}

     where \hat{\rho}(\lambda) = \log{||y - X c_{\lambda}||} and
     \hat{\eta}(\lambda) = \log{|| L c_{\lambda} ||}.  The curvature
     values are stored in *note kappa: b4d. on output.

 -- Function: int gsl_multifit_linear_lcorner (const gsl_vector *rho,
          const gsl_vector *eta, size_t *idx)

     This function attempts to locate the corner of the L-curve (||y - X
     c||, ||L c||) defined by the *note rho: b4e. and *note eta: b4e.
     input arrays respectively.  The corner is defined as the point of
     maximum curvature of the L-curve in log-log scale.  The *note rho:
     b4e. and *note eta: b4e. arrays can be outputs of *note
     gsl_multifit_linear_lcurve(): b4c.  The algorithm used simply fits
     a circle to 3 consecutive points on the L-curve and uses the
     circle’s radius to determine the curvature at the middle point.
     Therefore, the input array sizes must be \ge 3.  With more points
     provided for the L-curve, a better estimate of the curvature can be
     obtained.  The array index corresponding to maximum curvature (ie:
     the corner) is returned in *note idx: b4e.  If the input arrays
     contain colinear points, this function could fail and return *note
     GSL_EINVAL: 2b.

 -- Function: int gsl_multifit_linear_lcorner2 (const gsl_vector
          *reg_param, const gsl_vector *eta, size_t *idx)

     This function attempts to locate the corner of an alternate L-curve
     (\lambda^2, ||L c||^2) studied by Rezghi and Hosseini, 2009.  This
     alternate L-curve can provide better estimates of the
     regularization parameter for smooth solution vectors.  The
     regularization parameters \lambda and solution norms ||L c|| are
     provided in the *note reg_param: b4f. and *note eta: b4f. input
     arrays respectively.  The corner is defined as the point of maximum
     curvature of this alternate L-curve in linear scale.  The *note
     reg_param: b4f. and *note eta: b4f. arrays can be outputs of *note
     gsl_multifit_linear_lcurve(): b4c.  The algorithm used simply fits
     a circle to 3 consecutive points on the L-curve and uses the
     circle’s radius to determine the curvature at the middle point.
     Therefore, the input array sizes must be \ge 3.  With more points
     provided for the L-curve, a better estimate of the curvature can be
     obtained.  The array index corresponding to maximum curvature (ie:
     the corner) is returned in *note idx: b4f.  If the input arrays
     contain colinear points, this function could fail and return *note
     GSL_EINVAL: 2b.

 -- Function: int gsl_multifit_linear_gcv_init (const gsl_vector *y,
          gsl_vector *reg_param, gsl_vector *UTy, double *delta0,
          gsl_multifit_linear_workspace *work)

     This function performs some initialization in preparation for
     computing the GCV curve and its minimum.  The right hand side
     vector is provided in *note y: b50.  On output, *note reg_param:
     b50. is set to a vector of regularization parameters in decreasing
     order and may be of any size.  The vector *note UTy: b50. of size p
     is set to U^T y.  The parameter *note delta0: b50. is needed for
     subsequent steps of the GCV calculation.

 -- Function: int gsl_multifit_linear_gcv_curve (const gsl_vector
          *reg_param, const gsl_vector *UTy, const double delta0,
          gsl_vector *G, gsl_multifit_linear_workspace *work)

     This funtion calculates the GCV curve G(\lambda) and stores it in
     *note G: b51. on output, which must be the same size as *note
     reg_param: b51.  The inputs *note reg_param: b51, *note UTy: b51.
     and *note delta0: b51. are computed in *note
     gsl_multifit_linear_gcv_init(): b50.

 -- Function: int gsl_multifit_linear_gcv_min (const gsl_vector
          *reg_param, const gsl_vector *UTy, const gsl_vector *G, const
          double delta0, double *lambda, gsl_multifit_linear_workspace
          *work)

     This function computes the value of the regularization parameter
     which minimizes the GCV curve G(\lambda) and stores it in *note
     lambda: b52.  The input *note G: b52. is calculated by *note
     gsl_multifit_linear_gcv_curve(): b51. and the inputs *note
     reg_param: b52, *note UTy: b52. and *note delta0: b52. are computed
     by *note gsl_multifit_linear_gcv_init(): b50.

 -- Function: double gsl_multifit_linear_gcv_calc (const double lambda,
          const gsl_vector *UTy, const double delta0,
          gsl_multifit_linear_workspace *work)

     This function returns the value of the GCV curve G(\lambda)
     corresponding to the input *note lambda: b53.

 -- Function: int gsl_multifit_linear_gcv (const gsl_vector *y,
          gsl_vector *reg_param, gsl_vector *G, double *lambda, double
          *G_lambda, gsl_multifit_linear_workspace *work)

     This function combines the steps ‘gcv_init’, ‘gcv_curve’, and
     ‘gcv_min’ defined above into a single function.  The input *note y:
     b54. is the right hand side vector.  On output, *note reg_param:
     b54. and *note G: b54, which must be the same size, are set to
     vectors of \lambda and G(\lambda) values respectively.  The output
     *note lambda: b54. is set to the optimal value of \lambda which
     minimizes the GCV curve.  The minimum value of the GCV curve is
     returned in *note G_lambda: b54.

 -- Function: int gsl_multifit_linear_Lk (const size_t p, const size_t
          k, gsl_matrix *L)

     This function computes the discrete approximation to the derivative
     operator L_k of order *note k: b55. on a regular grid of *note p:
     b55. points and stores it in *note L: b55.  The dimensions of *note
     L: b55. are (p-k)-by-p.

 -- Function: int gsl_multifit_linear_Lsobolev (const size_t p, const
          size_t kmax, const gsl_vector *alpha, gsl_matrix *L,
          gsl_multifit_linear_workspace *work)

     This function computes the regularization matrix *note L: b56.
     corresponding to the weighted Sobolov norm ||L c||^2 = \sum_k
     \alpha_k^2 ||L_k c||^2 where L_k approximates the derivative
     operator of order k.  This regularization norm can be useful in
     applications where it is necessary to smooth several derivatives of
     the solution.  *note p: b56. is the number of model parameters,
     *note kmax: b56. is the highest derivative to include in the
     summation above, and *note alpha: b56. is the vector of weights of
     size *note kmax: b56. + 1, where ‘alpha[k]’ = \alpha_k is the
     weight assigned to the derivative of order k.  The output matrix
     *note L: b56. is size *note p: b56.-by-*note p: b56. and upper
     triangular.

 -- Function: double gsl_multifit_linear_rcond (const
          gsl_multifit_linear_workspace *work)

     This function returns the reciprocal condition number of the least
     squares matrix X, defined as the ratio of the smallest and largest
     singular values, rcond = \sigma_{min}/\sigma_{max}.  The routine
     *note gsl_multifit_linear_svd(): b37. must first be called to
     compute the SVD of X.


File: gsl-ref.info,  Node: Robust linear regression,  Next: Large dense linear systems,  Prev: Regularized regression,  Up: Linear Least-Squares Fitting

40.5 Robust linear regression
=============================

Ordinary least squares (OLS) models are often heavily influenced by the
presence of outliers.  Outliers are data points which do not follow the
general trend of the other observations, although there is strictly no
precise definition of an outlier.  Robust linear regression refers to
regression algorithms which are robust to outliers.  The most common
type of robust regression is M-estimation.  The general M-estimator
minimizes the objective function

     \sum_i \rho(e_i) = \sum_i \rho (y_i - Y(c, x_i))

where e_i = y_i - Y(c, x_i) is the residual of the ith data point, and
\rho(e_i) is a function which should have the following properties:

   * \rho(e) \ge 0

   * \rho(0) = 0

   * \rho(-e) = \rho(e)

   * \rho(e_1) > \rho(e_2) for |e_1| > |e_2|

The special case of ordinary least squares is given by \rho(e_i) =
e_i^2.  Letting \psi = \rho' be the derivative of \rho, differentiating
the objective function with respect to the coefficients c and setting
the partial derivatives to zero produces the system of equations

     \sum_i \psi(e_i) X_i = 0

where X_i is a vector containing row i of the design matrix X. Next, we
define a weight function w(e) = \psi(e)/e, and let w_i = w(e_i):

     \sum_i w_i e_i X_i = 0

This system of equations is equivalent to solving a weighted ordinary
least squares problem, minimizing \chi^2 = \sum_i w_i e_i^2.  The
weights however, depend on the residuals e_i, which depend on the
coefficients c, which depend on the weights.  Therefore, an iterative
solution is used, called Iteratively Reweighted Least Squares (IRLS).

  1. Compute initial estimates of the coefficients c^{(0)} using
     ordinary least squares

  2. For iteration k, form the residuals e_i^{(k)} = (y_i - X_i
     c^{(k-1)})/(t \sigma^{(k)} \sqrt{1 - h_i}), where t is a tuning
     constant depending on the choice of \psi, and h_i are the
     statistical leverages (diagonal elements of the matrix X (X^T
     X)^{-1} X^T). Including t and h_i in the residual calculation has
     been shown to improve the convergence of the method.  The residual
     standard deviation is approximated as \sigma^{(k)} = MAD / 0.6745,
     where MAD is the Median-Absolute-Deviation of the n-p largest
     residuals from the previous iteration.

  3. Compute new weights w_i^{(k)} = \psi(e_i^{(k)})/e_i^{(k)}.

  4. Compute new coefficients c^{(k)} by solving the weighted least
     squares problem with weights w_i^{(k)}.

  5. Steps 2 through 4 are iterated until the coefficients converge or
     until some maximum iteration limit is reached.  Coefficients are
     tested for convergence using the critera:

          |c_i^(k) - c_i^(k-1)| <= \epsilon * max(|c_i^(k)|, |c_i^(k-1)|)

     for all 0 \le i < p where \epsilon is a small tolerance factor.

The key to this method lies in selecting the function \psi(e_i) to
assign smaller weights to large residuals, and larger weights to smaller
residuals.  As the iteration proceeds, outliers are assigned smaller and
smaller weights, eventually having very little or no effect on the
fitted model.

 -- Type: gsl_multifit_robust_workspace

     This workspace is used for robust least squares fitting.

 -- Function: *note gsl_multifit_robust_workspace: b59.
          *gsl_multifit_robust_alloc (const gsl_multifit_robust_type *T,
          const size_t n, const size_t p)

     This function allocates a workspace for fitting a model to *note n:
     b5a. observations using *note p: b5a. parameters.  The size of the
     workspace is O(np + p^2).  The type *note T: b5a. specifies the
     function \psi and can be selected from the following choices.

      -- Type: gsl_multifit_robust_type

           -- Variable: *note gsl_multifit_robust_type: b5b.
                    *gsl_multifit_robust_default

               This specifies the *note gsl_multifit_robust_bisquare:
               b5d. type (see below) and is a good general purpose
               choice for robust regression.

           -- Variable: *note gsl_multifit_robust_type: b5b.
                    *gsl_multifit_robust_bisquare

               This is Tukey’s biweight (bisquare) function and is a
               good general purpose choice for robust regression.  The
               weight function is given by

                    w(e) = { (1 - e^2)^2, |e| <= 1
                           {     0,       |e| > 1

               and the default tuning constant is t = 4.685.

           -- Variable: *note gsl_multifit_robust_type: b5b.
                    *gsl_multifit_robust_cauchy

               This is Cauchy’s function, also known as the Lorentzian
               function.  This function does not guarantee a unique
               solution, meaning different choices of the coefficient
               vector ‘c’ could minimize the objective function.
               Therefore this option should be used with care.  The
               weight function is given by

                    w(e) = 1 / (1 + e^2)

               and the default tuning constant is t = 2.385.

           -- Variable: *note gsl_multifit_robust_type: b5b.
                    *gsl_multifit_robust_fair

               This is the fair \rho function, which guarantees a unique
               solution and has continuous derivatives to three orders.
               The weight function is given by

                    w(e) = 1 / (1 + |e|)

               and the default tuning constant is t = 1.400.

           -- Variable: *note gsl_multifit_robust_type: b5b.
                    *gsl_multifit_robust_huber

               This specifies Huber’s \rho function, which is a parabola
               in the vicinity of zero and increases linearly for a
               given threshold |e| > t.  This function is also
               considered an excellent general purpose robust estimator,
               however, occasional difficulties can be encountered due
               to the discontinuous first derivative of the \psi
               function.  The weight function is given by

                    w(e) = 1/max(1,|e|)

               and the default tuning constant is t = 1.345.

           -- Variable: *note gsl_multifit_robust_type: b5b.
                    *gsl_multifit_robust_ols

               This specifies the ordinary least squares solution, which
               can be useful for quickly checking the difference between
               the various robust and OLS solutions.  The weight
               function is given by

                    w(e) = 1

               and the default tuning constant is t = 1.

           -- Variable: *note gsl_multifit_robust_type: b5b.
                    *gsl_multifit_robust_welsch

               This specifies the Welsch function which can perform well
               in cases where the residuals have an exponential
               distribution.  The weight function is given by

                    w(e) = \exp{(-e^2)}

               and the default tuning constant is t = 2.985.

 -- Function: void gsl_multifit_robust_free
          (gsl_multifit_robust_workspace *w)

     This function frees the memory associated with the workspace *note
     w: b63.

 -- Function: const char *gsl_multifit_robust_name (const
          gsl_multifit_robust_workspace *w)

     This function returns the name of the robust type ‘T’ specified to
     *note gsl_multifit_robust_alloc(): b5a.

 -- Function: int gsl_multifit_robust_tune (const double tune,
          gsl_multifit_robust_workspace *w)

     This function sets the tuning constant t used to adjust the
     residuals at each iteration to *note tune: b65.  Decreasing the
     tuning constant increases the downweight assigned to large
     residuals, while increasing the tuning constant decreases the
     downweight assigned to large residuals.

 -- Function: int gsl_multifit_robust_maxiter (const size_t maxiter,
          gsl_multifit_robust_workspace *w)

     This function sets the maximum number of iterations in the
     iteratively reweighted least squares algorithm to *note maxiter:
     b66.  By default, this value is set to 100 by *note
     gsl_multifit_robust_alloc(): b5a.

 -- Function: int gsl_multifit_robust_weights (const gsl_vector *r,
          gsl_vector *wts, gsl_multifit_robust_workspace *w)

     This function assigns weights to the vector *note wts: b67. using
     the residual vector *note r: b67. and previously specified
     weighting function.  The output weights are given by wts_i = w(r_i
     / (t \sigma)), where the weighting functions w are detailed in
     *note gsl_multifit_robust_alloc(): b5a.  \sigma is an estimate of
     the residual standard deviation based on the
     Median-Absolute-Deviation and t is the tuning constant.  This
     function is useful if the user wishes to implement their own robust
     regression rather than using the supplied *note
     gsl_multifit_robust(): b68. routine below.

 -- Function: int gsl_multifit_robust (const gsl_matrix *X, const
          gsl_vector *y, gsl_vector *c, gsl_matrix *cov,
          gsl_multifit_robust_workspace *w)

     This function computes the best-fit parameters *note c: b68. of the
     model y = X c for the observations *note y: b68. and the matrix of
     predictor variables *note X: b68, attemping to reduce the influence
     of outliers using the algorithm outlined above.  The p-by-p
     variance-covariance matrix of the model parameters *note cov: b68.
     is estimated as \sigma^2 (X^T X)^{-1}, where \sigma is an
     approximation of the residual standard deviation using the theory
     of robust regression.  Special care must be taken when estimating
     \sigma and other statistics such as R^2, and so these are computed
     internally and are available by calling the function *note
     gsl_multifit_robust_statistics(): b69.

     If the coefficients do not converge within the maximum iteration
     limit, the function returns ‘GSL_EMAXITER’.  In this case, the
     current estimates of the coefficients and covariance matrix are
     returned in *note c: b68. and *note cov: b68. and the internal fit
     statistics are computed with these estimates.

 -- Function: int gsl_multifit_robust_est (const gsl_vector *x, const
          gsl_vector *c, const gsl_matrix *cov, double *y, double
          *y_err)

     This function uses the best-fit robust regression coefficients
     *note c: b6a. and their covariance matrix *note cov: b6a. to
     compute the fitted function value *note y: b6a. and its standard
     deviation *note y_err: b6a. for the model y = x \cdot c at the
     point *note x: b6a.

 -- Function: int gsl_multifit_robust_residuals (const gsl_matrix *X,
          const gsl_vector *y, const gsl_vector *c, gsl_vector *r,
          gsl_multifit_robust_workspace *w)

     This function computes the vector of studentized residuals r_i =
     {y_i - (X c)_i \over \sigma \sqrt{1 - h_i}} for the observations
     *note y: b6b, coefficients *note c: b6b. and matrix of predictor
     variables *note X: b6b.  The routine *note gsl_multifit_robust():
     b68. must first be called to compute the statisical leverages h_i
     of the matrix *note X: b6b. and residual standard deviation
     estimate \sigma.

 -- Function: *note gsl_multifit_robust_stats: b6c.
          gsl_multifit_robust_statistics (const
          gsl_multifit_robust_workspace *w)

     This function returns a structure containing relevant statistics
     from a robust regression.  The function *note
     gsl_multifit_robust(): b68. must be called first to perform the
     regression and calculate these statistics.  The returned *note
     gsl_multifit_robust_stats: b6c. structure contains the following
     fields.

      -- Type: gsl_multifit_robust_stats

          ‘double sigma_ols’

               This contains the standard deviation of the residuals as
               computed from ordinary least squares (OLS).

          ‘double sigma_mad’

               This contains an estimate of the standard deviation of
               the final residuals using the Median-Absolute-Deviation
               statistic

          ‘double sigma_rob’

               This contains an estimate of the standard deviation of
               the final residuals from the theory of robust regression
               (see Street et al, 1988).

          ‘double sigma’

               This contains an estimate of the standard deviation of
               the final residuals by attemping to reconcile ‘sigma_rob’
               and ‘sigma_ols’ in a reasonable way.

          ‘double Rsq’

               This contains the R^2 coefficient of determination
               statistic using the estimate ‘sigma’.

          ‘double adj_Rsq’

               This contains the adjusted R^2 coefficient of
               determination statistic using the estimate ‘sigma’.

          ‘double rmse’

               This contains the root mean squared error of the final
               residuals

          ‘double sse’

               This contains the residual sum of squares taking into
               account the robust covariance matrix.

          ‘size_t dof’

               This contains the number of degrees of freedom n - p

          ‘size_t numit’

               Upon successful convergence, this contains the number of
               iterations performed

          ‘gsl_vector * weights’

               This contains the final weight vector of length ‘n’

          ‘gsl_vector * r’

               This contains the final residual vector of length ‘n’, r
               = y - X c


File: gsl-ref.info,  Node: Large dense linear systems,  Next: Troubleshooting,  Prev: Robust linear regression,  Up: Linear Least-Squares Fitting

40.6 Large dense linear systems
===============================

This module is concerned with solving large dense least squares systems
X c = y where the n-by-p matrix X has n >> p (ie: many more rows than
columns).  This type of matrix is called a “tall skinny” matrix, and for
some applications, it may not be possible to fit the entire matrix in
memory at once to use the standard SVD approach.  Therefore, the
algorithms in this module are designed to allow the user to construct
smaller blocks of the matrix X and accumulate those blocks into the
larger system one at a time.  The algorithms in this module never need
to store the entire matrix X in memory.  The large linear least squares
routines support data weights and Tikhonov regularization, and are
designed to minimize the residual

     \chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2

where y is the n-by-1 observation vector, X is the n-by-p design matrix,
c is the p-by-1 solution vector, W = \diag(w_1,...,w_n) is the data
weighting matrix, L is an m-by-p regularization matrix, \lambda is a
regularization parameter, and ||r||_W^2 = r^T W r.  In the discussion
which follows, we will assume that the system has been converted into
Tikhonov standard form,

     \chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2

and we will drop the tilde characters from the various parameters.  For
a discussion of the transformation to standard form, see *note
Regularized regression: b41.

The basic idea is to partition the matrix X and observation vector y as

     [ X_1 ] c = [ y_1 ]
     [ X_2 ]     [ y_2 ]
     [ X_3 ]     [ y_3 ]
     [ ... ]     [ ... ]
     [ X_k ]     [ y_k ]

into k blocks, where each block (X_i,y_i) may have any number of rows,
but each X_i has p columns.  The sections below describe the methods
available for solving this partitioned system.  The functions are
declared in the header file ‘gsl_multilarge.h’.

* Menu:

* Normal Equations Approach::
* Tall Skinny QR (TSQR) Approach: Tall Skinny QR TSQR Approach.
* Large Dense Linear Systems Solution Steps::
* Large Dense Linear Least Squares Routines::


File: gsl-ref.info,  Node: Normal Equations Approach,  Next: Tall Skinny QR TSQR Approach,  Up: Large dense linear systems

40.6.1 Normal Equations Approach
--------------------------------

The normal equations approach to the large linear least squares problem
described above is popular due to its speed and simplicity.  Since the
normal equations solution to the problem is given by

     c = ( X^T X + \lambda^2 I )^-1 X^T y

only the p-by-p matrix X^T X and p-by-1 vector X^T y need to be stored.
Using the partition scheme described above, these are given by

     X^T X = \sum_i X_i^T X_i
     X^T y = \sum_i X_i^T y_i

Since the matrix X^T X is symmetric, only half of it needs to be
calculated.  Once all of the blocks (X_i,y_i) have been accumulated into
the final X^T X and X^T y, the system can be solved with a Cholesky
factorization of the X^T X matrix.  The X^T X matrix is first
transformed via a diagonal scaling transformation to attempt to reduce
its condition number as much as possible to recover a more accurate
solution vector.  The normal equations approach is the fastest method
for solving the large least squares problem, and is accurate for
well-conditioned matrices X. However, for ill-conditioned matrices, as
is often the case for large systems, this method can suffer from
numerical instabilities (see Trefethen and Bau, 1997).  The number of
operations for this method is O(np^2 + {1 \over 3}p^3).


File: gsl-ref.info,  Node: Tall Skinny QR TSQR Approach,  Next: Large Dense Linear Systems Solution Steps,  Prev: Normal Equations Approach,  Up: Large dense linear systems

40.6.2 Tall Skinny QR (TSQR) Approach
-------------------------------------

An algorithm which has better numerical stability for ill-conditioned
problems is known as the Tall Skinny QR (TSQR) method.  This method is
based on computing the thin QR decomposition of the least squares matrix
X = Q R, where Q is an n-by-p matrix with orthogonal columns, and R is a
p-by-p upper triangular matrix.  Once these factors are calculated, the
residual becomes

     \chi^2 = || Q^T y - R c ||^2 + \lambda^2 || c ||^2

which can be written as the matrix equation

     [ R ; \lambda I ] c = [ Q^T b ; 0 ]

The matrix on the left hand side is now a much smaller 2p-by-p matrix
which can be solved with a standard SVD approach.  The Q matrix is just
as large as the original matrix X, however it does not need to be
explicitly constructed.  The TSQR algorithm computes only the p-by-p
matrix R and the p-by-1 vector Q^T y, and updates these quantities as
new blocks are added to the system.  Each time a new block of rows
(X_i,y_i) is added, the algorithm performs a QR decomposition of the
matrix

     [ R_(i-1) ; X_i ]

where R_{i-1} is the upper triangular R factor for the matrix

     [ X_1 ; ... ; X_(i-1) ]

This QR decomposition is done efficiently taking into account the sparse
structure of R_{i-1}.  See Demmel et al, 2008 for more details on how
this is accomplished.  The number of operations for this method is
O(2np^2 - {2 \over 3}p^3).


File: gsl-ref.info,  Node: Large Dense Linear Systems Solution Steps,  Next: Large Dense Linear Least Squares Routines,  Prev: Tall Skinny QR TSQR Approach,  Up: Large dense linear systems

40.6.3 Large Dense Linear Systems Solution Steps
------------------------------------------------

The typical steps required to solve large regularized linear least
squares problems are as follows:

  1. Choose the regularization matrix L.

  2. Construct a block of rows of the least squares matrix, right hand
     side vector, and weight vector (X_i, y_i, w_i).

  3. Transform the block to standard form (\tilde{X_i}, \tilde{y_i}).
     This step can be skipped if L = I and W = I.

  4. Accumulate the standard form block (\tilde{X_i}, \tilde{y_i}) into
     the system.

  5. Repeat steps 2-4 until the entire matrix and right hand side vector
     have been accumulated.

  6. Determine an appropriate regularization parameter \lambda (using
     for example L-curve analysis).

  7. Solve the standard form system using the chosen \lambda.

  8. Backtransform the standard form solution \tilde{c} to recover the
     original solution vector c.


File: gsl-ref.info,  Node: Large Dense Linear Least Squares Routines,  Prev: Large Dense Linear Systems Solution Steps,  Up: Large dense linear systems

40.6.4 Large Dense Linear Least Squares Routines
------------------------------------------------

 -- Type: gsl_multilarge_linear_workspace

     This workspace contains parameters for solving large linear least
     squares problems.

 -- Function: *note gsl_multilarge_linear_workspace: b72.
          *gsl_multilarge_linear_alloc (const gsl_multilarge_linear_type
          *T, const size_t p)

     This function allocates a workspace for solving large linear least
     squares systems.  The least squares matrix X has *note p: b73.
     columns, but may have any number of rows.

      -- Type: gsl_multilarge_linear_type

          The parameter *note T: b73. specifies the method to be used
          for solving the large least squares system and may be selected
          from the following choices

           -- Variable: *note gsl_multilarge_linear_type: b74.
                    *gsl_multilarge_linear_normal

               This specifies the normal equations approach for solving
               the least squares system.  This method is suitable in
               cases where performance is critical and it is known that
               the least squares matrix X is well conditioned.  The size
               of this workspace is O(p^2).

           -- Variable: *note gsl_multilarge_linear_type: b74.
                    *gsl_multilarge_linear_tsqr

               This specifies the sequential Tall Skinny QR (TSQR)
               approach for solving the least squares system.  This
               method is a good general purpose choice for large
               systems, but requires about twice as many operations as
               the normal equations method for n >> p.  The size of this
               workspace is O(p^2).

 -- Function: void gsl_multilarge_linear_free
          (gsl_multilarge_linear_workspace *w)

     This function frees the memory associated with the workspace *note
     w: b77.

 -- Function: const char *gsl_multilarge_linear_name
          (gsl_multilarge_linear_workspace *w)

     This function returns a string pointer to the name of the
     multilarge solver.

 -- Function: int gsl_multilarge_linear_reset
          (gsl_multilarge_linear_workspace *w)

     This function resets the workspace *note w: b79. so it can begin to
     accumulate a new least squares system.

 -- Function: int gsl_multilarge_linear_stdform1 (const gsl_vector *L,
          const gsl_matrix *X, const gsl_vector *y, gsl_matrix *Xs,
          gsl_vector *ys, gsl_multilarge_linear_workspace *work)
 -- Function: int gsl_multilarge_linear_wstdform1 (const gsl_vector *L,
          const gsl_matrix *X, const gsl_vector *w, const gsl_vector *y,
          gsl_matrix *Xs, gsl_vector *ys,
          gsl_multilarge_linear_workspace *work)

     These functions define a regularization matrix L =
     \diag(l_0,l_1,...,l_{p-1}).  The diagonal matrix element l_i is
     provided by the i-th element of the input vector *note L: b7b.  The
     block (*note X: b7b, *note y: b7b.) is converted to standard form
     and the parameters (\tilde{X}, \tilde{y}) are stored in *note Xs:
     b7b. and *note ys: b7b. on output.  *note Xs: b7b. and *note ys:
     b7b. have the same dimensions as *note X: b7b. and *note y: b7b.
     Optional data weights may be supplied in the vector *note w: b7b.
     In order to apply this transformation, L^{-1} must exist and so
     none of the l_i may be zero.  After the standard form system has
     been solved, use *note gsl_multilarge_linear_genform1(): b7c. to
     recover the original solution vector.  It is allowed to have *note
     X: b7b. = *note Xs: b7b. and *note y: b7b. = *note ys: b7b. for an
     in-place transform.

 -- Function: int gsl_multilarge_linear_L_decomp (gsl_matrix *L,
          gsl_vector *tau)

     This function calculates the QR decomposition of the m-by-p
     regularization matrix *note L: b7d.  *note L: b7d. must have m \ge
     p.  On output, the Householder scalars are stored in the vector
     *note tau: b7d. of size p.  These outputs will be used by *note
     gsl_multilarge_linear_wstdform2(): b7e. to complete the
     transformation to standard form.

 -- Function: int gsl_multilarge_linear_stdform2 (const gsl_matrix *LQR,
          const gsl_vector *Ltau, const gsl_matrix *X, const gsl_vector
          *y, gsl_matrix *Xs, gsl_vector *ys,
          gsl_multilarge_linear_workspace *work)
 -- Function: int gsl_multilarge_linear_wstdform2 (const gsl_matrix
          *LQR, const gsl_vector *Ltau, const gsl_matrix *X, const
          gsl_vector *w, const gsl_vector *y, gsl_matrix *Xs, gsl_vector
          *ys, gsl_multilarge_linear_workspace *work)

     These functions convert a block of rows (*note X: b7e, *note y:
     b7e, *note w: b7e.) to standard form (\tilde{X}, \tilde{y}) which
     are stored in *note Xs: b7e. and *note ys: b7e. respectively.
     *note X: b7e, *note y: b7e, and *note w: b7e. must all have the
     same number of rows.  The m-by-p regularization matrix ‘L’ is
     specified by the inputs *note LQR: b7e. and *note Ltau: b7e, which
     are outputs from *note gsl_multilarge_linear_L_decomp(): b7d.
     *note Xs: b7e. and *note ys: b7e. have the same dimensions as *note
     X: b7e. and *note y: b7e.  After the standard form system has been
     solved, use *note gsl_multilarge_linear_genform2(): b80. to recover
     the original solution vector.  Optional data weights may be
     supplied in the vector *note w: b7e, where W = \diag(w).

 -- Function: int gsl_multilarge_linear_accumulate (gsl_matrix *X,
          gsl_vector *y, gsl_multilarge_linear_workspace *w)

     This function accumulates the standard form block (X,y) into the
     current least squares system.  *note X: b81. and *note y: b81. have
     the same number of rows, which can be arbitrary.  *note X: b81.
     must have p columns.  For the TSQR method, *note X: b81. and *note
     y: b81. are destroyed on output.  For the normal equations method,
     they are both unchanged.

 -- Function: int gsl_multilarge_linear_solve (const double lambda,
          gsl_vector *c, double *rnorm, double *snorm,
          gsl_multilarge_linear_workspace *w)

     After all blocks (X_i,y_i) have been accumulated into the large
     least squares system, this function will compute the solution
     vector which is stored in *note c: b82. on output.  The
     regularization parameter \lambda is provided in *note lambda: b82.
     On output, *note rnorm: b82. contains the residual norm ||y - X
     c||_W and *note snorm: b82. contains the solution norm ||L c||.

 -- Function: int gsl_multilarge_linear_genform1 (const gsl_vector *L,
          const gsl_vector *cs, gsl_vector *c,
          gsl_multilarge_linear_workspace *work)

     After a regularized system has been solved with L =
     \diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
     standard form solution vector *note cs: b7c. to recover the
     solution vector of the original problem *note c: b7c.  The diagonal
     matrix elements l_i are provided in the vector *note L: b7c.  It is
     allowed to have *note c: b7c. = *note cs: b7c. for an in-place
     transform.

 -- Function: int gsl_multilarge_linear_genform2 (const gsl_matrix *LQR,
          const gsl_vector *Ltau, const gsl_vector *cs, gsl_vector *c,
          gsl_multilarge_linear_workspace *work)

     After a regularized system has been solved with a regularization
     matrix L, specified by (*note LQR: b80, *note Ltau: b80.), this
     function backtransforms the standard form solution *note cs: b80.
     to recover the solution vector of the original problem, which is
     stored in *note c: b80, of length p.

 -- Function: int gsl_multilarge_linear_lcurve (gsl_vector *reg_param,
          gsl_vector *rho, gsl_vector *eta,
          gsl_multilarge_linear_workspace *work)

     This function computes the L-curve for a large least squares system
     after it has been fully accumulated into the workspace *note work:
     b83.  The output vectors *note reg_param: b83, *note rho: b83, and
     *note eta: b83. must all be the same size, and will contain the
     regularization parameters \lambda_i, residual norms ||y - X c_i||,
     and solution norms || L c_i || which compose the L-curve, where c_i
     is the regularized solution vector corresponding to \lambda_i.  The
     user may determine the number of points on the L-curve by adjusting
     the size of these input arrays.  For the TSQR method, the
     regularization parameters \lambda_i are estimated from the singular
     values of the triangular R factor.  For the normal equations
     method, they are estimated from the eigenvalues of the X^T X
     matrix.

 -- Function: const *note gsl_matrix: 3a2.
          *gsl_multilarge_linear_matrix_ptr (const
          gsl_multilarge_linear_workspace *work)

     For the normal equations method, this function returns a pointer to
     the X^T X matrix.  For the TSQR method, this function returns a
     pointer to the upper triangular R matrix.

 -- Function: const *note gsl_vector: 35f.
          *gsl_multilarge_linear_rhs_ptr (const
          gsl_multilarge_linear_workspace *work)

     For the normal equations method, this function returns a pointer to
     the X^T y right hand side vector.  For the TSQR method, this
     function returns a pointer to the Q^T y right hand side vector.

 -- Function: int gsl_multilarge_linear_rcond (double *rcond,
          gsl_multilarge_linear_workspace *work)

     This function computes the reciprocal condition number, stored in
     *note rcond: b86, of the least squares matrix after it has been
     accumulated into the workspace *note work: b86.  For the TSQR
     algorithm, this is accomplished by calculating the SVD of the R
     factor, which has the same singular values as the matrix X. For the
     normal equations method, this is done by computing the eigenvalues
     of X^T X, which could be inaccurate for ill-conditioned matrices X.


File: gsl-ref.info,  Node: Troubleshooting,  Next: Examples<31>,  Prev: Large dense linear systems,  Up: Linear Least-Squares Fitting

40.7 Troubleshooting
====================

When using models based on polynomials, care should be taken when
constructing the design matrix X. If the x values are large, then the
matrix X could be ill-conditioned since its columns are powers of x,
leading to unstable least-squares solutions.  In this case it can often
help to center and scale the x values using the mean and standard
deviation:

     x' = (x - mu)/sigma

and then construct the X matrix using the transformed values x'.


File: gsl-ref.info,  Node: Examples<31>,  Next: References and Further Reading<33>,  Prev: Troubleshooting,  Up: Linear Least-Squares Fitting

40.8 Examples
=============

The example programs in this section demonstrate the various linear
regression methods.

* Menu:

* Simple Linear Regression Example::
* Multi-parameter Linear Regression Example::
* Regularized Linear Regression Example 1::
* Regularized Linear Regression Example 2::
* Robust Linear Regression Example::
* Large Dense Linear Regression Example::


File: gsl-ref.info,  Node: Simple Linear Regression Example,  Next: Multi-parameter Linear Regression Example,  Up: Examples<31>

40.8.1 Simple Linear Regression Example
---------------------------------------

The following program computes a least squares straight-line fit to a
simple dataset, and outputs the best-fit line and its associated one
standard-deviation error bars.

     #include <stdio.h>
     #include <gsl/gsl_fit.h>

     int
     main (void)
     {
       int i, n = 4;
       double x[4] = { 1970, 1980, 1990, 2000 };
       double y[4] = {   12,   11,   14,   13 };
       double w[4] = {  0.1,  0.2,  0.3,  0.4 };

       double c0, c1, cov00, cov01, cov11, chisq;

       gsl_fit_wlinear (x, 1, w, 1, y, 1, n,
                        &c0, &c1, &cov00, &cov01, &cov11,
                        &chisq);

       printf ("# best fit: Y = %g + %g X\n", c0, c1);
       printf ("# covariance matrix:\n");
       printf ("# [ %g, %g\n#   %g, %g]\n",
               cov00, cov01, cov01, cov11);
       printf ("# chisq = %g\n", chisq);

       for (i = 0; i < n; i++)
         printf ("data: %g %g %g\n",
                        x[i], y[i], 1/sqrt(w[i]));

       printf ("\n");

       for (i = -30; i < 130; i++)
         {
           double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
           double yf, yf_err;

           gsl_fit_linear_est (xf,
                               c0, c1,
                               cov00, cov01, cov11,
                               &yf, &yf_err);

           printf ("fit: %g %g\n", xf, yf);
           printf ("hi : %g %g\n", xf, yf + yf_err);
           printf ("lo : %g %g\n", xf, yf - yf_err);
         }
       return 0;
     }

The following commands extract the data from the output of the program
and display it using the GNU plotutils “graph” utility:

     $ ./demo > tmp
     $ more tmp
     # best fit: Y = -106.6 + 0.06 X
     # covariance matrix:
     # [ 39602, -19.9
     #   -19.9, 0.01]
     # chisq = 0.8

     $ for n in data fit hi lo ;
        do
          grep "^$n" tmp | cut -d: -f2 > $n ;
        done
     $ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
          -S 0 -I a -m 1 fit -m 2 hi -m 2 lo

The result is shown in Fig.  %s.

[gsl-ref-figures/fit-wlinear]
Figure: Straight line fit with 1-\sigma error bars


File: gsl-ref.info,  Node: Multi-parameter Linear Regression Example,  Next: Regularized Linear Regression Example 1,  Prev: Simple Linear Regression Example,  Up: Examples<31>

40.8.2 Multi-parameter Linear Regression Example
------------------------------------------------

The following program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2
to a weighted dataset using the generalised linear fitting function
*note gsl_multifit_wlinear(): b3b.  The model matrix X for a quadratic
fit is given by,

     X = [ 1   , x_0  , x_0^2 ;
           1   , x_1  , x_1^2 ;
           1   , x_2  , x_2^2 ;
           ... , ...  , ...   ]

where the column of ones corresponds to the constant term c_0.  The two
remaining columns corresponds to the terms c_1 x and c_2 x^2.

The program reads ‘n’ lines of data in the format (‘x’, ‘y’, ‘err’)
where ‘err’ is the error (standard deviation) in the value ‘y’.

     #include <stdio.h>
     #include <gsl/gsl_multifit.h>

     int
     main (int argc, char **argv)
     {
       int i, n;
       double xi, yi, ei, chisq;
       gsl_matrix *X, *cov;
       gsl_vector *y, *w, *c;

       if (argc != 2)
         {
           fprintf (stderr,"usage: fit n < data\n");
           exit (-1);
         }

       n = atoi (argv[1]);

       X = gsl_matrix_alloc (n, 3);
       y = gsl_vector_alloc (n);
       w = gsl_vector_alloc (n);

       c = gsl_vector_alloc (3);
       cov = gsl_matrix_alloc (3, 3);

       for (i = 0; i < n; i++)
         {
           int count = fscanf (stdin, "%lg %lg %lg",
                               &xi, &yi, &ei);

           if (count != 3)
             {
               fprintf (stderr, "error reading file\n");
               exit (-1);
             }

           printf ("%g %g +/- %g\n", xi, yi, ei);

           gsl_matrix_set (X, i, 0, 1.0);
           gsl_matrix_set (X, i, 1, xi);
           gsl_matrix_set (X, i, 2, xi*xi);

           gsl_vector_set (y, i, yi);
           gsl_vector_set (w, i, 1.0/(ei*ei));
         }

       {
         gsl_multifit_linear_workspace * work
           = gsl_multifit_linear_alloc (n, 3);
         gsl_multifit_wlinear (X, w, y, c, cov,
                               &chisq, work);
         gsl_multifit_linear_free (work);
       }

     #define C(i) (gsl_vector_get(c,(i)))
     #define COV(i,j) (gsl_matrix_get(cov,(i),(j)))

       {
         printf ("# best fit: Y = %g + %g X + %g X^2\n",
                 C(0), C(1), C(2));

         printf ("# covariance matrix:\n");
         printf ("[ %+.5e, %+.5e, %+.5e  \n",
                    COV(0,0), COV(0,1), COV(0,2));
         printf ("  %+.5e, %+.5e, %+.5e  \n",
                    COV(1,0), COV(1,1), COV(1,2));
         printf ("  %+.5e, %+.5e, %+.5e ]\n",
                    COV(2,0), COV(2,1), COV(2,2));
         printf ("# chisq = %g\n", chisq);
       }

       gsl_matrix_free (X);
       gsl_vector_free (y);
       gsl_vector_free (w);
       gsl_vector_free (c);
       gsl_matrix_free (cov);

       return 0;
     }

A suitable set of data for fitting can be generated using the following
program.  It outputs a set of points with gaussian errors from the curve
y = e^x in the region 0 < x < 2.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_randist.h>

     int
     main (void)
     {
       double x;
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (x = 0.1; x < 2; x+= 0.1)
         {
           double y0 = exp (x);
           double sigma = 0.1 * y0;
           double dy = gsl_ran_gaussian (r, sigma);

           printf ("%g %g %g\n", x, y0 + dy, sigma);
         }

       gsl_rng_free(r);

       return 0;
     }

The data can be prepared by running the resulting executable program:

     $ GSL_RNG_TYPE=mt19937_1999 ./generate > exp.dat
     $ more exp.dat
     0.1 0.97935 0.110517
     0.2 1.3359 0.12214
     0.3 1.52573 0.134986
     0.4 1.60318 0.149182
     0.5 1.81731 0.164872
     0.6 1.92475 0.182212
     ....

To fit the data use the previous program, with the number of data points
given as the first argument.  In this case there are 19 data points:

     $ ./fit 19 < exp.dat
     0.1 0.97935 +/- 0.110517
     0.2 1.3359 +/- 0.12214
     ...
     # best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
     # covariance matrix:
     [ +1.25612e-02, -3.64387e-02, +1.94389e-02
       -3.64387e-02, +1.42339e-01, -8.48761e-02
       +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
     # chisq = 23.0987

The parameters of the quadratic fit match the coefficients of the
expansion of e^x, taking into account the errors on the parameters and
the O(x^3) difference between the exponential and quadratic functions
for the larger values of x.  The errors on the parameters are given by
the square-root of the corresponding diagonal elements of the covariance
matrix.  The chi-squared per degree of freedom is 1.4, indicating a
reasonable fit to the data.

Fig.  %s shows the resulting fit.

[gsl-ref-figures/fit-wlinear2]
Figure: Weighted fit example with error bars


File: gsl-ref.info,  Node: Regularized Linear Regression Example 1,  Next: Regularized Linear Regression Example 2,  Prev: Multi-parameter Linear Regression Example,  Up: Examples<31>

40.8.3 Regularized Linear Regression Example 1
----------------------------------------------

The next program demonstrates the difference between ordinary and
regularized least squares when the design matrix is near-singular.  In
this program, we generate two random normally distributed variables u
and v, with v = u + noise so that u and v are nearly colinear.  We then
set a third dependent variable y = u + v + noise and solve for the
coefficients c_1,c_2 of the model Y(c_1,c_2) = c_1 u + c_2 v.  Since u
\approx v, the design matrix X is nearly singular, leading to unstable
ordinary least squares solutions.

Here is the program output:

     matrix condition number = 1.025113e+04

     === Unregularized fit ===
     best fit: y = -43.6588 u + 45.6636 v
     residual norm = 31.6248
     solution norm = 63.1764
     chisq/dof = 1.00213

     === Regularized fit (L-curve) ===
     optimal lambda: 4.51103
     best fit: y = 1.00113 u + 1.0032 v
     residual norm = 31.6547
     solution norm = 1.41728
     chisq/dof = 1.04499

     === Regularized fit (GCV) ===
     optimal lambda: 0.0232029
     best fit: y = -19.8367 u + 21.8417 v
     residual norm = 31.6332
     solution norm = 29.5051
     chisq/dof = 1.00314

We see that the ordinary least squares solution is completely wrong,
while the L-curve regularized method with the optimal \lambda = 4.51103
finds the correct solution c_1 \approx c_2 \approx 1.  The GCV
regularized method finds a regularization parameter \lambda = 0.0232029
which is too small to give an accurate solution, although it performs
better than OLS. The L-curve and its computed corner, as well as the GCV
curve and its minimum are plotted in Fig.  %s.

[gsl-ref-figures/regularized]
Figure: L-curve and GCV curve for example program.

The program is given below.

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_multifit.h>

     int
     main()
     {
       const size_t n = 1000; /* number of observations */
       const size_t p = 2;    /* number of model parameters */
       size_t i;
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_matrix *X = gsl_matrix_alloc(n, p);
       gsl_vector *y = gsl_vector_alloc(n);

       for (i = 0; i < n; ++i)
         {
           /* generate first random variable u */
           double ui = 5.0 * gsl_ran_gaussian(r, 1.0);

           /* set v = u + noise */
           double vi = ui + gsl_ran_gaussian(r, 0.001);

           /* set y = u + v + noise */
           double yi = ui + vi + gsl_ran_gaussian(r, 1.0);

           /* since u =~ v, the matrix X is ill-conditioned */
           gsl_matrix_set(X, i, 0, ui);
           gsl_matrix_set(X, i, 1, vi);

           /* rhs vector */
           gsl_vector_set(y, i, yi);
         }

       {
         const size_t npoints = 200;                   /* number of points on L-curve and GCV curve */
         gsl_multifit_linear_workspace *w =
           gsl_multifit_linear_alloc(n, p);
         gsl_vector *c = gsl_vector_alloc(p);          /* OLS solution */
         gsl_vector *c_lcurve = gsl_vector_alloc(p);   /* regularized solution (L-curve) */
         gsl_vector *c_gcv = gsl_vector_alloc(p);      /* regularized solution (GCV) */
         gsl_vector *reg_param = gsl_vector_alloc(npoints);
         gsl_vector *rho = gsl_vector_alloc(npoints);  /* residual norms */
         gsl_vector *eta = gsl_vector_alloc(npoints);  /* solution norms */
         gsl_vector *G = gsl_vector_alloc(npoints);    /* GCV function values */
         double lambda_l;                              /* optimal regularization parameter (L-curve) */
         double lambda_gcv;                            /* optimal regularization parameter (GCV) */
         double G_gcv;                                 /* G(lambda_gcv) */
         size_t reg_idx;                               /* index of optimal lambda */
         double rcond;                                 /* reciprocal condition number of X */
         double chisq, rnorm, snorm;

         /* compute SVD of X */
         gsl_multifit_linear_svd(X, w);

         rcond = gsl_multifit_linear_rcond(w);
         fprintf(stderr, "matrix condition number = %e\n\n", 1.0 / rcond);

         /* unregularized (standard) least squares fit, lambda = 0 */
         gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0);

         fprintf(stderr, "=== Unregularized fit ===\n");
         fprintf(stderr, "best fit: y = %g u + %g v\n",
           gsl_vector_get(c, 0), gsl_vector_get(c, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate L-curve and find its corner */
         gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
         gsl_multifit_linear_lcorner(rho, eta, &reg_idx);

         /* store optimal regularization parameter */
         lambda_l = gsl_vector_get(reg_param, reg_idx);

         /* regularize with lambda_l */
         gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_l);
         fprintf(stderr, "best fit: y = %g u + %g v\n",
                 gsl_vector_get(c_lcurve, 0), gsl_vector_get(c_lcurve, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate GCV curve and find its minimum */
         gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);

         /* regularize with lambda_gcv */
         gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
         fprintf(stderr, "best fit: y = %g u + %g v\n",
                 gsl_vector_get(c_gcv, 0), gsl_vector_get(c_gcv, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* output L-curve and GCV curve */
         for (i = 0; i < npoints; ++i)
           {
             printf("%e %e %e %e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i),
                    gsl_vector_get(G, i));
           }

         /* output L-curve corner point */
         printf("\n\n%f %f\n",
                gsl_vector_get(rho, reg_idx),
                gsl_vector_get(eta, reg_idx));

         /* output GCV curve corner minimum */
         printf("\n\n%e %e\n",
                lambda_gcv,
                G_gcv);

         gsl_multifit_linear_free(w);
         gsl_vector_free(c);
         gsl_vector_free(c_lcurve);
         gsl_vector_free(reg_param);
         gsl_vector_free(rho);
         gsl_vector_free(eta);
         gsl_vector_free(G);
       }

       gsl_rng_free(r);
       gsl_matrix_free(X);
       gsl_vector_free(y);

       return 0;
     }


File: gsl-ref.info,  Node: Regularized Linear Regression Example 2,  Next: Robust Linear Regression Example,  Prev: Regularized Linear Regression Example 1,  Up: Examples<31>

40.8.4 Regularized Linear Regression Example 2
----------------------------------------------

The following example program minimizes the cost function

     ||y - X c||^2 + \lambda^2 ||x||^2

where X is the 10-by-8 Hilbert matrix whose entries are given by

     X_{ij} = 1 / (i + j - 1)

and the right hand side vector is given by y =
[1,-1,1,-1,1,-1,1,-1,1,-1]^T. Solutions are computed for \lambda = 0
(unregularized) as well as for optimal parameters \lambda chosen by
analyzing the L-curve and GCV curve.

Here is the program output:

     matrix condition number = 3.565872e+09

     === Unregularized fit ===
     residual norm = 2.15376
     solution norm = 2.92217e+09
     chisq/dof = 2.31934

     === Regularized fit (L-curve) ===
     optimal lambda: 7.11407e-07
     residual norm = 2.60386
     solution norm = 424507
     chisq/dof = 3.43565

     === Regularized fit (GCV) ===
     optimal lambda: 1.72278
     residual norm = 3.1375
     solution norm = 0.139357
     chisq/dof = 4.95076

Here we see the unregularized solution results in a large solution norm
due to the ill-conditioned matrix.  The L-curve solution finds a small
value of \lambda = 7.11e-7 which still results in a badly conditioned
system and a large solution norm.  The GCV method finds a parameter
\lambda = 1.72 which results in a well-conditioned system and small
solution norm.

The L-curve and its computed corner, as well as the GCV curve and its
minimum are plotted in Fig.  %s.

[gsl-ref-figures/regularized2]
Figure: L-curve and GCV curve for example program.

The program is given below.

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_blas.h>

     static int
     hilbert_matrix(gsl_matrix * m)
     {
       const size_t N = m->size1;
       const size_t M = m->size2;
       size_t i, j;

       for (i = 0; i < N; i++)
         {
           for (j = 0; j < M; j++)
             {
               gsl_matrix_set(m, i, j, 1.0/(i+j+1.0));
             }
         }

       return GSL_SUCCESS;
     }

     int
     main()
     {
       const size_t n = 10; /* number of observations */
       const size_t p = 8;  /* number of model parameters */
       size_t i;
       gsl_matrix *X = gsl_matrix_alloc(n, p);
       gsl_vector *y = gsl_vector_alloc(n);

       /* construct Hilbert matrix and rhs vector */
       hilbert_matrix(X);

       {
         double val = 1.0;
         for (i = 0; i < n; ++i)
           {
             gsl_vector_set(y, i, val);
             val *= -1.0;
           }
       }

       {
         const size_t npoints = 200;                   /* number of points on L-curve and GCV curve */
         gsl_multifit_linear_workspace *w =
           gsl_multifit_linear_alloc(n, p);
         gsl_vector *c = gsl_vector_alloc(p);          /* OLS solution */
         gsl_vector *c_lcurve = gsl_vector_alloc(p);   /* regularized solution (L-curve) */
         gsl_vector *c_gcv = gsl_vector_alloc(p);      /* regularized solution (GCV) */
         gsl_vector *reg_param = gsl_vector_alloc(npoints);
         gsl_vector *rho = gsl_vector_alloc(npoints);  /* residual norms */
         gsl_vector *eta = gsl_vector_alloc(npoints);  /* solution norms */
         gsl_vector *G = gsl_vector_alloc(npoints);    /* GCV function values */
         double lambda_l;                              /* optimal regularization parameter (L-curve) */
         double lambda_gcv;                            /* optimal regularization parameter (GCV) */
         double G_gcv;                                 /* G(lambda_gcv) */
         size_t reg_idx;                               /* index of optimal lambda */
         double rcond;                                 /* reciprocal condition number of X */
         double chisq, rnorm, snorm;

         /* compute SVD of X */
         gsl_multifit_linear_svd(X, w);

         rcond = gsl_multifit_linear_rcond(w);
         fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);

         /* unregularized (standard) least squares fit, lambda = 0 */
         gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0);

         fprintf(stderr, "\n=== Unregularized fit ===\n");
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate L-curve and find its corner */
         gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
         gsl_multifit_linear_lcorner(rho, eta, &reg_idx);

         /* store optimal regularization parameter */
         lambda_l = gsl_vector_get(reg_param, reg_idx);

         /* regularize with lambda_l */
         gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_l);
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate GCV curve and find its minimum */
         gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);

         /* regularize with lambda_gcv */
         gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* output L-curve and GCV curve */
         for (i = 0; i < npoints; ++i)
           {
             printf("%e %e %e %e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i),
                    gsl_vector_get(G, i));
           }

         /* output L-curve corner point */
         printf("\n\n%f %f\n",
                gsl_vector_get(rho, reg_idx),
                gsl_vector_get(eta, reg_idx));

         /* output GCV curve corner minimum */
         printf("\n\n%e %e\n",
                lambda_gcv,
                G_gcv);

         gsl_multifit_linear_free(w);
         gsl_vector_free(c);
         gsl_vector_free(c_lcurve);
         gsl_vector_free(reg_param);
         gsl_vector_free(rho);
         gsl_vector_free(eta);
         gsl_vector_free(G);
       }

       gsl_matrix_free(X);
       gsl_vector_free(y);

       return 0;
     }


File: gsl-ref.info,  Node: Robust Linear Regression Example,  Next: Large Dense Linear Regression Example,  Prev: Regularized Linear Regression Example 2,  Up: Examples<31>

40.8.5 Robust Linear Regression Example
---------------------------------------

The next program demonstrates the advantage of robust least squares on a
dataset with outliers.  The program generates linear (x,y) data pairs on
the line y = 1.45 x + 3.88, adds some random noise, and inserts 3
outliers into the dataset.  Both the robust and ordinary least squares
(OLS) coefficients are computed for comparison.

     #include <stdio.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_randist.h>

     int
     dofit(const gsl_multifit_robust_type *T,
           const gsl_matrix *X, const gsl_vector *y,
           gsl_vector *c, gsl_matrix *cov)
     {
       int s;
       gsl_multifit_robust_workspace * work
         = gsl_multifit_robust_alloc (T, X->size1, X->size2);

       s = gsl_multifit_robust (X, y, c, cov, work);
       gsl_multifit_robust_free (work);

       return s;
     }

     int
     main (int argc, char **argv)
     {
       size_t i;
       size_t n;
       const size_t p = 2; /* linear fit */
       gsl_matrix *X, *cov;
       gsl_vector *x, *y, *c, *c_ols;
       const double a = 1.45; /* slope */
       const double b = 3.88; /* intercept */
       gsl_rng *r;

       if (argc != 2)
         {
           fprintf (stderr,"usage: robfit n\n");
           exit (-1);
         }

       n = atoi (argv[1]);

       X = gsl_matrix_alloc (n, p);
       x = gsl_vector_alloc (n);
       y = gsl_vector_alloc (n);

       c = gsl_vector_alloc (p);
       c_ols = gsl_vector_alloc (p);
       cov = gsl_matrix_alloc (p, p);

       r = gsl_rng_alloc(gsl_rng_default);

       /* generate linear dataset */
       for (i = 0; i < n - 3; i++)
         {
           double dx = 10.0 / (n - 1.0);
           double ei = gsl_rng_uniform(r);
           double xi = -5.0 + i * dx;
           double yi = a * xi + b;

           gsl_vector_set (x, i, xi);
           gsl_vector_set (y, i, yi + ei);
         }

       /* add a few outliers */
       gsl_vector_set(x, n - 3, 4.7);
       gsl_vector_set(y, n - 3, -8.3);

       gsl_vector_set(x, n - 2, 3.5);
       gsl_vector_set(y, n - 2, -6.7);

       gsl_vector_set(x, n - 1, 4.1);
       gsl_vector_set(y, n - 1, -6.0);

       /* construct design matrix X for linear fit */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);

           gsl_matrix_set (X, i, 0, 1.0);
           gsl_matrix_set (X, i, 1, xi);
         }

       /* perform robust and OLS fit */
       dofit(gsl_multifit_robust_ols, X, y, c_ols, cov);
       dofit(gsl_multifit_robust_bisquare, X, y, c, cov);

       /* output data and model */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           gsl_vector_view v = gsl_matrix_row(X, i);
           double y_ols, y_rob, y_err;

           gsl_multifit_robust_est(&v.vector, c, cov, &y_rob, &y_err);
           gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err);

           printf("%g %g %g %g\n", xi, yi, y_rob, y_ols);
         }

     #define C(i) (gsl_vector_get(c,(i)))
     #define COV(i,j) (gsl_matrix_get(cov,(i),(j)))

       {
         printf ("# best fit: Y = %g + %g X\n",
                 C(0), C(1));

         printf ("# covariance matrix:\n");
         printf ("# [ %+.5e, %+.5e\n",
                    COV(0,0), COV(0,1));
         printf ("#   %+.5e, %+.5e\n",
                    COV(1,0), COV(1,1));
       }

       gsl_matrix_free (X);
       gsl_vector_free (x);
       gsl_vector_free (y);
       gsl_vector_free (c);
       gsl_vector_free (c_ols);
       gsl_matrix_free (cov);
       gsl_rng_free(r);

       return 0;
     }

The output from the program is shown in Fig.  %s.

[gsl-ref-figures/robust]
Figure: Linear fit to dataset with outliers.


File: gsl-ref.info,  Node: Large Dense Linear Regression Example,  Prev: Robust Linear Regression Example,  Up: Examples<31>

40.8.6 Large Dense Linear Regression Example
--------------------------------------------

The following program demostrates the large dense linear least squares
solvers.  This example is adapted from Trefethen and Bau, and fits the
function f(t) = \exp{(\sin^3{(10t)}}) on the interval [0,1] with a
degree 15 polynomial.  The program generates n = 50000 equally spaced
points t_i on this interval, calculates the function value and adds
random noise to determine the observation value y_i.  The entries of the
least squares matrix are X_{ij} = t_i^j, representing a polynomial fit.
The matrix is highly ill-conditioned, with a condition number of about
2.4 \cdot 10^{11}.  The program accumulates the matrix into the least
squares system in 5 blocks, each with 10000 rows.  This way the full
matrix X is never stored in memory.  We solve the system with both the
normal equations and TSQR methods.  The results are shown in Fig.  %s.
In the top left plot, the TSQR solution provides a reasonable agreement
to the exact solution, while the normal equations method fails
completely since the Cholesky factorization fails due to the
ill-conditioning of the matrix.  In the bottom left plot, we show the
L-curve calculated from TSQR, which exhibits multiple corners.  In the
top right panel, we plot a regularized solution using \lambda = 10^{-5}.
The TSQR and normal solutions now agree, however they are unable to
provide a good fit due to the damping.  This indicates that for some
ill-conditioned problems, regularizing the normal equations does not
improve the solution.  This is further illustrated in the bottom right
panel, where we plot the L-curve calculated from the normal equations.
The curve agrees with the TSQR curve for larger damping parameters, but
for small \lambda, the normal equations approach cannot provide accurate
solution vectors leading to numerical inaccuracies in the left portion
of the curve.

[gsl-ref-figures/multilarge]
Figure: Top left: unregularized solutions; top right: regularized
solutions; bottom left: L-curve for TSQR method; bottom right: L-curve
from normal equations method.

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_multilarge.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_errno.h>

     /* function to be fitted */
     double
     func(const double t)
     {
       double x = sin(10.0 * t);
       return exp(x*x*x);
     }

     /* construct a row of the least squares matrix */
     int
     build_row(const double t, gsl_vector *row)
     {
       const size_t p = row->size;
       double Xj = 1.0;
       size_t j;

       for (j = 0; j < p; ++j)
         {
           gsl_vector_set(row, j, Xj);
           Xj *= t;
         }

       return 0;
     }

     int
     solve_system(const int print_data, const gsl_multilarge_linear_type * T,
                  const double lambda, const size_t n, const size_t p,
                  gsl_vector * c)
     {
       const size_t nblock = 5;         /* number of blocks to accumulate */
       const size_t nrows = n / nblock; /* number of rows per block */
       gsl_multilarge_linear_workspace * w =
         gsl_multilarge_linear_alloc(T, p);
       gsl_matrix *X = gsl_matrix_alloc(nrows, p);
       gsl_vector *y = gsl_vector_alloc(nrows);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       const size_t nlcurve = 200;
       gsl_vector *reg_param = gsl_vector_alloc(nlcurve);
       gsl_vector *rho = gsl_vector_calloc(nlcurve);
       gsl_vector *eta = gsl_vector_calloc(nlcurve);
       size_t rowidx = 0;
       double rnorm, snorm, rcond;
       double t = 0.0;
       double dt = 1.0 / (n - 1.0);

       while (rowidx < n)
         {
           size_t nleft = n - rowidx;         /* number of rows left to accumulate */
           size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */
           gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p);
           gsl_vector_view yv = gsl_vector_subvector(y, 0, nr);
           size_t i;

           /* build (X,y) block with 'nr' rows */
           for (i = 0; i < nr; ++i)
             {
               gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i);
               double fi = func(t);
               double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */
               double yi = fi + ei;

               /* construct this row of LS matrix */
               build_row(t, &row.vector);

               /* set right hand side value with added noise */
               gsl_vector_set(&yv.vector, i, yi);

               if (print_data && (i % 100 == 0))
                 printf("%f %f\n", t, yi);

               t += dt;
             }

           /* accumulate (X,y) block into LS system */
           gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w);

           rowidx += nr;
         }

       if (print_data)
         printf("\n\n");

       /* compute L-curve */
       gsl_multilarge_linear_lcurve(reg_param, rho, eta, w);

       /* solve large LS system and store solution in c */
       gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w);

       /* compute reciprocal condition number */
       gsl_multilarge_linear_rcond(&rcond, w);

       fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w));
       fprintf(stderr, "condition number = %e\n", 1.0 / rcond);
       fprintf(stderr, "residual norm    = %e\n", rnorm);
       fprintf(stderr, "solution norm    = %e\n", snorm);

       /* output L-curve */
       {
         size_t i;
         for (i = 0; i < nlcurve; ++i)
           {
             printf("%.12e %.12e %.12e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i));
           }
         printf("\n\n");
       }

       gsl_matrix_free(X);
       gsl_vector_free(y);
       gsl_multilarge_linear_free(w);
       gsl_rng_free(r);
       gsl_vector_free(reg_param);
       gsl_vector_free(rho);
       gsl_vector_free(eta);

       return 0;
     }

     int
     main(int argc, char *argv[])
     {
       const size_t n = 50000;   /* number of observations */
       const size_t p = 16;      /* polynomial order + 1 */
       double lambda = 0.0;      /* regularization parameter */
       gsl_vector *c_tsqr = gsl_vector_calloc(p);
       gsl_vector *c_normal = gsl_vector_calloc(p);

       if (argc > 1)
         lambda = atof(argv[1]);

       /* turn off error handler so normal equations method won't abort */
       gsl_set_error_handler_off();

       /* solve system with TSQR method */
       solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);

       /* solve system with Normal equations method */
       solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal);

       /* output solutions */
       {
         gsl_vector *v = gsl_vector_alloc(p);
         double t;

         for (t = 0.0; t <= 1.0; t += 0.01)
           {
             double f_exact = func(t);
             double f_tsqr, f_normal;

             build_row(t, v);
             gsl_blas_ddot(v, c_tsqr, &f_tsqr);
             gsl_blas_ddot(v, c_normal, &f_normal);

             printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal);
           }

         gsl_vector_free(v);
       }

       gsl_vector_free(c_tsqr);
       gsl_vector_free(c_normal);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<33>,  Prev: Examples<31>,  Up: Linear Least-Squares Fitting

40.9 References and Further Reading
===================================

A summary of formulas and techniques for least squares fitting can be
found in the “Statistics” chapter of the Annual Review of Particle
Physics prepared by the Particle Data Group,

   * `Review of Particle Properties', R.M. Barnett et al., Physical
     Review D54, 1 (1996) ‘http://pdg.lbl.gov’

The Review of Particle Physics is available online at the website given
above.

The tests used to prepare these routines are based on the NIST
Statistical Reference Datasets.  The datasets and their documentation
are available from NIST at the following website,

‘http://www.nist.gov/itl/div898/strd/index.html’

More information on Tikhonov regularization can be found in

   * Hansen, P. C. (1998), Rank-Deficient and Discrete Ill-Posed
     Problems: Numerical Aspects of Linear Inversion.  SIAM Monogr.  on
     Mathematical Modeling and Computation, Society for Industrial and
     Applied Mathematics

   * M. Rezghi and S. M. Hosseini (2009), A new variant of L-curve for
     Tikhonov regularization, Journal of Computational and Applied
     Mathematics, Volume 231, Issue 2, pages 914-924.

The GSL implementation of robust linear regression closely follows the
publications

   * DuMouchel, W. and F. O’Brien (1989), “Integrating a robust option
     into a multiple regression computing environment,” Computer Science
     and Statistics: Proceedings of the 21st Symposium on the Interface,
     American Statistical Association

   * Street, J.O., R.J. Carroll, and D. Ruppert (1988), “A note on
     computing robust regression estimates via iteratively reweighted
     least squares,” The American Statistician, v.  42, pp.  152-154.

More information about the normal equations and TSQR approach for
solving large linear least squares systems can be found in the
publications

   * Trefethen, L. N. and Bau, D. (1997), “Numerical Linear Algebra”,
     SIAM.

   * Demmel, J., Grigori, L., Hoemmen, M. F., and Langou, J.
     “Communication-optimal parallel and sequential QR and LU
     factorizations”, UCB Technical Report No.  UCB/EECS-2008-89, 2008.


File: gsl-ref.info,  Node: Nonlinear Least-Squares Fitting,  Next: Basis Splines,  Prev: Linear Least-Squares Fitting,  Up: Top

41 Nonlinear Least-Squares Fitting
**********************************

This chapter describes functions for multidimensional nonlinear
least-squares fitting.  There are generally two classes of algorithms
for solving nonlinear least squares problems, which fall under line
search methods and trust region methods.  GSL currently implements only
trust region methods and provides the user with full access to
intermediate steps of the iteration.  The user also has the ability to
tune a number of parameters which affect low-level aspects of the
algorithm which can help to accelerate convergence for the specific
problem at hand.  GSL provides two separate interfaces for nonlinear
least squares fitting.  The first is designed for small to moderate
sized problems, and the second is designed for very large problems,
which may or may not have significant sparse structure.

The header file ‘gsl_multifit_nlinear.h’ contains prototypes for the
multidimensional nonlinear fitting functions and related declarations
relating to the small to moderate sized systems.

The header file ‘gsl_multilarge_nlinear.h’ contains prototypes for the
multidimensional nonlinear fitting functions and related declarations
relating to large systems.

* Menu:

* Overview: Overview<6>.
* Solving the Trust Region Subproblem (TRS): Solving the Trust Region Subproblem TRS.
* Weighted Nonlinear Least-Squares::
* Tunable Parameters::
* Initializing the Solver: Initializing the Solver<3>.
* Providing the Function to be Minimized::
* Iteration: Iteration<5>.
* Testing for Convergence::
* High Level Driver::
* Covariance matrix of best fit parameters::
* Troubleshooting: Troubleshooting<2>.
* Examples: Examples<32>.
* References and Further Reading: References and Further Reading<34>.


File: gsl-ref.info,  Node: Overview<6>,  Next: Solving the Trust Region Subproblem TRS,  Up: Nonlinear Least-Squares Fitting

41.1 Overview
=============

The problem of multidimensional nonlinear least-squares fitting requires
the minimization of the squared residuals of n functions, f_i, in p
parameters, x_i,

     \Phi(x) = (1/2) || f(x) ||^2
             = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2

In trust region methods, the objective (or cost) function \Phi(x) is
approximated by a model function m_k(\delta) in the vicinity of some
point x_k.  The model function is often simply a second order Taylor
series expansion around the point x_k, ie:

     \Phi(x_k + \delta) ~=~ m_k(\delta) = \Phi(x_k) + g_k^T \delta + 1/2 \delta^T B_k \delta

where g_k = \nabla \Phi(x_k) = J^T f is the gradient vector at the point
x_k, B_k = \nabla^2 \Phi(x_k) is the Hessian matrix at x_k, or some
approximation to it, and J is the n-by-p Jacobian matrix

     J_{ij} = d f_i / d x_j

In order to find the next step \delta, we minimize the model function
m_k(\delta), but search for solutions only within a region where we
trust that m_k(\delta) is a good approximation to the objective function
\Phi(x_k + \delta).  In other words, we seek a solution of the trust
region subproblem (TRS)

     \min_(\delta \in R^p) m_k(\delta), s.t. || D_k \delta || <= \Delta_k

where \Delta_k > 0 is the trust region radius and D_k is a scaling
matrix.  If D_k = I, then the trust region is a ball of radius \Delta_k
centered at x_k.  In some applications, the parameter vector x may have
widely different scales.  For example, one parameter might be a
temperature on the order of 10^3 K, while another might be a length on
the order of 10^{-6} m.  In such cases, a spherical trust region may not
be the best choice, since if \Phi changes rapidly along directions with
one scale, and more slowly along directions with a different scale, the
model function m_k may be a poor approximation to \Phi along the rapidly
changing directions.  In such problems, it may be best to use an
elliptical trust region, by setting D_k to a diagonal matrix whose
entries are designed so that the scaled step D_k \delta has entries of
approximately the same order of magnitude.

The trust region subproblem above normally amounts to solving a linear
least squares system (or multiple systems) for the step \delta.  Once
\delta is computed, it is checked whether or not it reduces the
objective function \Phi(x).  A useful statistic for this is to look at
the ratio

     \rho_k = ( \Phi(x_k) - \Phi(x_k + \delta_k) / ( m_k(0) - m_k(\delta_k) )

where the numerator is the actual reduction of the objective function
due to the step \delta_k, and the denominator is the predicted reduction
due to the model m_k.  If \rho_k is negative, it means that the step
\delta_k increased the objective function and so it is rejected.  If
\rho_k is positive, then we have found a step which reduced the
objective function and it is accepted.  Furthermore, if \rho_k is close
to 1, then this indicates that the model function is a good
approximation to the objective function in the trust region, and so on
the next iteration the trust region is enlarged in order to take more
ambitious steps.  When a step is rejected, the trust region is made
smaller and the TRS is solved again.  An outline for the general trust
region method used by GSL can now be given.

`Trust Region Algorithm'

  1. Initialize: given x_0, construct m_0(\delta), D_0 and \Delta_0 > 0

  2. For k = 0, 1, 2, …

       a. If converged, then stop

       b. Solve TRS for trial step \delta_k

       c. Evaluate trial step by computing \rho_k

          1). if step is accepted, set x_{k+1} = x_k + \delta_k and increase radius,

               \Delta_{k+1} = \alpha \Delta_k

          2). if step is rejected, set x_{k+1} = x_k and decrease radius,

               \Delta_{k+1} = {\Delta_k \over \beta}; goto 2(b)

       d. Construct m_{k+1}(\delta) and D_{k+1}

GSL offers the user a number of different algorithms for solving the
trust region subproblem in 2(b), as well as different choices of scaling
matrices D_k and different methods of updating the trust region radius
\Delta_k.  Therefore, while reasonable default methods are provided, the
user has a lot of control to fine-tune the various steps of the
algorithm for their specific problem.


File: gsl-ref.info,  Node: Solving the Trust Region Subproblem TRS,  Next: Weighted Nonlinear Least-Squares,  Prev: Overview<6>,  Up: Nonlinear Least-Squares Fitting

41.2 Solving the Trust Region Subproblem (TRS)
==============================================

Below we describe the methods available for solving the trust region
subproblem.  The methods available provide either exact or approximate
solutions to the trust region subproblem.  In all algorithms below, the
Hessian matrix B_k is approximated as B_k \approx J_k^T J_k, where J_k =
J(x_k).  In all methods, the solution of the TRS involves solving a
linear least squares system involving the Jacobian matrix.  For small to
moderate sized problems (‘gsl_multifit_nlinear’ interface), this is
accomplished by factoring the full Jacobian matrix, which is provided by
the user, with the Cholesky, QR, or SVD decompositions.  For large
systems (‘gsl_multilarge_nlinear’ interface), the user has two choices.
One is to solve the system iteratively, without needing to store the
full Jacobian matrix in memory.  With this method, the user must provide
a routine to calculate the matrix-vector products J u or J^T u for a
given vector u.  This iterative method is particularly useful for
systems where the Jacobian has sparse structure, since forming
matrix-vector products can be done cheaply.  The second option for large
systems involves forming the normal equations matrix J^T J and then
factoring it using a Cholesky decomposition.  The normal equations
matrix is p-by-p, typically much smaller than the full n-by-p Jacobian,
and can usually be stored in memory even if the full Jacobian matrix
cannot.  This option is useful for large, dense systems, or if the
iterative method has difficulty converging.

* Menu:

* Levenberg-Marquardt::
* Levenberg-Marquardt with Geodesic Acceleration::
* Dogleg::
* Double Dogleg::
* Two Dimensional Subspace::
* Steihaug-Toint Conjugate Gradient::


File: gsl-ref.info,  Node: Levenberg-Marquardt,  Next: Levenberg-Marquardt with Geodesic Acceleration,  Up: Solving the Trust Region Subproblem TRS

41.2.1 Levenberg-Marquardt
--------------------------

There is a theorem which states that if \delta_k is a solution to the
trust region subproblem given above, then there exists \mu_k \ge 0 such
that

     ( B_k + \mu_k D_k^T D_k ) \delta_k = -g_k

with \mu_k (\Delta_k - ||D_k \delta_k||) = 0.  This forms the basis of
the Levenberg-Marquardt algorithm, which controls the trust region size
by adjusting the parameter \mu_k rather than the radius \Delta_k
directly.  For each radius \Delta_k, there is a unique parameter \mu_k
which solves the TRS, and they have an inverse relationship, so that
large values of \mu_k correspond to smaller trust regions, while small
values of \mu_k correspond to larger trust regions.

With the approximation B_k \approx J_k^T J_k, on each iteration, in
order to calculate the step \delta_k, the following linear least squares
problem is solved:

     [J_k; sqrt(mu_k) D_k] \delta_k = - [f_k; 0]

If the step \delta_k is accepted, then \mu_k is decreased on the next
iteration in order to take a larger step, otherwise it is increased to
take a smaller step.  The Levenberg-Marquardt algorithm provides an
exact solution of the trust region subproblem, but typically has a
higher computational cost per iteration than the approximate methods
discussed below, since it may need to solve the least squares system
above several times for different values of \mu_k.


File: gsl-ref.info,  Node: Levenberg-Marquardt with Geodesic Acceleration,  Next: Dogleg,  Prev: Levenberg-Marquardt,  Up: Solving the Trust Region Subproblem TRS

41.2.2 Levenberg-Marquardt with Geodesic Acceleration
-----------------------------------------------------

This method applies a so-called geodesic acceleration correction to the
standard Levenberg-Marquardt step \delta_k (Transtrum et al, 2011).  By
interpreting \delta_k as a first order step along a geodesic in the
model parameter space (ie: a velocity \delta_k = v_k), the geodesic
acceleration a_k is a second order correction along the geodesic which
is determined by solving the linear least squares system

     [J_k; sqrt(mu_k) D_k] a_k = - [f_vv(x_k); 0]

where f_{vv} is the second directional derivative of the residual vector
in the velocity direction v, f_{vv}(x) = D_v^2 f = \sum_{\alpha\beta}
v_{\alpha} v_{\beta} \partial_{\alpha} \partial_{\beta} f(x), where
\alpha and \beta are summed over the p parameters.  The new total step
is then \delta_k' = v_k + {1 \over 2}a_k.  The second order correction
a_k can be calculated with a modest additional cost, and has been shown
to dramatically reduce the number of iterations (and expensive Jacobian
evaluations) required to reach convergence on a variety of different
problems.  In order to utilize the geodesic acceleration, the user must
supply a function which provides the second directional derivative
vector f_{vv}(x), or alternatively the library can use a finite
difference method to estimate this vector with one additional function
evaluation of f(x + h v) where h is a tunable step size (see the ‘h_fvv’
parameter description).


File: gsl-ref.info,  Node: Dogleg,  Next: Double Dogleg,  Prev: Levenberg-Marquardt with Geodesic Acceleration,  Up: Solving the Trust Region Subproblem TRS

41.2.3 Dogleg
-------------

This is Powell’s dogleg method, which finds an approximate solution to
the trust region subproblem, by restricting its search to a piecewise
linear “dogleg” path, composed of the origin, the Cauchy point which
represents the model minimizer along the steepest descent direction, and
the Gauss-Newton point, which is the overall minimizer of the
unconstrained model.  The Gauss-Newton step is calculated by solving

     J_k \delta_{gn} = -f_k

which is the main computational task for each iteration, but only needs
to be performed once per iteration.  If the Gauss-Newton point is inside
the trust region, it is selected as the step.  If it is outside, the
method then calculates the Cauchy point, which is located along the
gradient direction.  If the Cauchy point is also outside the trust
region, the method assumes that it is still far from the minimum and so
proceeds along the gradient direction, truncating the step at the trust
region boundary.  If the Cauchy point is inside the trust region, with
the Gauss-Newton point outside, the method uses a dogleg step, which is
a linear combination of the gradient direction and the Gauss-Newton
direction, stopping at the trust region boundary.


File: gsl-ref.info,  Node: Double Dogleg,  Next: Two Dimensional Subspace,  Prev: Dogleg,  Up: Solving the Trust Region Subproblem TRS

41.2.4 Double Dogleg
--------------------

This method is an improvement over the classical dogleg algorithm, which
attempts to include information about the Gauss-Newton step while the
iteration is still far from the minimum.  When the Cauchy point is
inside the trust region and the Gauss-Newton point is outside, the
method computes a scaled Gauss-Newton point and then takes a dogleg step
between the Cauchy point and the scaled Gauss-Newton point.  The scaling
is calculated to ensure that the reduction in the model m_k is about the
same as the reduction provided by the Cauchy point.


File: gsl-ref.info,  Node: Two Dimensional Subspace,  Next: Steihaug-Toint Conjugate Gradient,  Prev: Double Dogleg,  Up: Solving the Trust Region Subproblem TRS

41.2.5 Two Dimensional Subspace
-------------------------------

The dogleg methods restrict the search for the TRS solution to a 1D
curve defined by the Cauchy and Gauss-Newton points.  An improvement to
this is to search for a solution using the full two dimensional subspace
spanned by the Cauchy and Gauss-Newton directions.  The dogleg path is
of course inside this subspace, and so this method solves the TRS at
least as accurately as the dogleg methods.  Since this method searches a
larger subspace for a solution, it can converge more quickly than dogleg
on some problems.  Because the subspace is only two dimensional, this
method is very efficient and the main computation per iteration is to
determine the Gauss-Newton point.


File: gsl-ref.info,  Node: Steihaug-Toint Conjugate Gradient,  Prev: Two Dimensional Subspace,  Up: Solving the Trust Region Subproblem TRS

41.2.6 Steihaug-Toint Conjugate Gradient
----------------------------------------

One difficulty of the dogleg methods is calculating the Gauss-Newton
step when the Jacobian matrix is singular.  The Steihaug-Toint method
also computes a generalized dogleg step, but avoids solving for the
Gauss-Newton step directly, instead using an iterative conjugate
gradient algorithm.  This method performs well at points where the
Jacobian is singular, and is also suitable for large-scale problems
where factoring the Jacobian matrix could be prohibitively expensive.


File: gsl-ref.info,  Node: Weighted Nonlinear Least-Squares,  Next: Tunable Parameters,  Prev: Solving the Trust Region Subproblem TRS,  Up: Nonlinear Least-Squares Fitting

41.3 Weighted Nonlinear Least-Squares
=====================================

Weighted nonlinear least-squares fitting minimizes the function

     \Phi(x) = (1/2) || f(x) ||_W^2
             = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2

where W = \diag(w_1,w_2,...,w_n) is the weighting matrix, and ||f||_W^2
= f^T W f.  The weights w_i are commonly defined as w_i = 1/\sigma_i^2,
where \sigma_i is the error in the i-th measurement.  A simple change of
variables \tilde{f} = W^{1 \over 2} f yields \Phi(x) = {1 \over 2}
||\tilde{f}||^2, which is in the same form as the unweighted case.  The
user can either perform this transform directly on their function
residuals and Jacobian, or use the *note gsl_multifit_nlinear_winit():
ba1. interface which automatically performs the correct scaling.  To
manually perform this transformation, the residuals and Jacobian should
be modified according to

     f~_i = f_i / \sigma_i
     J~_ij = 1 / \sigma_i df_i/dx_j

For large systems, the user must perform their own weighting.


File: gsl-ref.info,  Node: Tunable Parameters,  Next: Initializing the Solver<3>,  Prev: Weighted Nonlinear Least-Squares,  Up: Nonlinear Least-Squares Fitting

41.4 Tunable Parameters
=======================

The user can tune nearly all aspects of the iteration at allocation
time.  For the ‘gsl_multifit_nlinear’ interface, the user may modify the
*note gsl_multifit_nlinear_parameters: ba4. structure, which is defined
as follows:

 -- Type: gsl_multifit_nlinear_parameters

          typedef struct
          {
            const gsl_multifit_nlinear_trs *trs;        /* trust region subproblem method */
            const gsl_multifit_nlinear_scale *scale;    /* scaling method */
            const gsl_multifit_nlinear_solver *solver;  /* solver method */
            gsl_multifit_nlinear_fdtype fdtype;         /* finite difference method */
            double factor_up;                           /* factor for increasing trust radius */
            double factor_down;                         /* factor for decreasing trust radius */
            double avmax;                               /* max allowed |a|/|v| */
            double h_df;                                /* step size for finite difference Jacobian */
            double h_fvv;                               /* step size for finite difference fvv */
          } gsl_multifit_nlinear_parameters;

For the ‘gsl_multilarge_nlinear’ interface, the user may modify the
*note gsl_multilarge_nlinear_parameters: ba5. structure, which is
defined as follows:

 -- Type: gsl_multilarge_nlinear_parameters

          typedef struct
          {
            const gsl_multilarge_nlinear_trs *trs;       /* trust region subproblem method */
            const gsl_multilarge_nlinear_scale *scale;   /* scaling method */
            const gsl_multilarge_nlinear_solver *solver; /* solver method */
            gsl_multilarge_nlinear_fdtype fdtype;        /* finite difference method */
            double factor_up;                            /* factor for increasing trust radius */
            double factor_down;                          /* factor for decreasing trust radius */
            double avmax;                                /* max allowed |a|/|v| */
            double h_df;                                 /* step size for finite difference Jacobian */
            double h_fvv;                                /* step size for finite difference fvv */
            size_t max_iter;                             /* maximum iterations for trs method */
            double tol;                                  /* tolerance for solving trs */
          } gsl_multilarge_nlinear_parameters;

Each of these parameters is discussed in further detail below.

 -- Type: gsl_multifit_nlinear_trs
 -- Type: gsl_multilarge_nlinear_trs

     The parameter ‘trs’ determines the method used to solve the trust
     region subproblem, and may be selected from the following choices,

      -- Variable: *note gsl_multifit_nlinear_trs: ba6.
               *gsl_multifit_nlinear_trs_lm
      -- Variable: *note gsl_multilarge_nlinear_trs: ba7.
               *gsl_multilarge_nlinear_trs_lm

          This selects the Levenberg-Marquardt algorithm.

      -- Variable: *note gsl_multifit_nlinear_trs: ba6.
               *gsl_multifit_nlinear_trs_lmaccel
      -- Variable: *note gsl_multilarge_nlinear_trs: ba7.
               *gsl_multilarge_nlinear_trs_lmaccel

          This selects the Levenberg-Marquardt algorithm with geodesic
          acceleration.

      -- Variable: *note gsl_multifit_nlinear_trs: ba6.
               *gsl_multifit_nlinear_trs_dogleg
      -- Variable: *note gsl_multilarge_nlinear_trs: ba7.
               *gsl_multilarge_nlinear_trs_dogleg

          This selects the dogleg algorithm.

      -- Variable: *note gsl_multifit_nlinear_trs: ba6.
               *gsl_multifit_nlinear_trs_ddogleg
      -- Variable: *note gsl_multilarge_nlinear_trs: ba7.
               *gsl_multilarge_nlinear_trs_ddogleg

          This selects the double dogleg algorithm.

      -- Variable: *note gsl_multifit_nlinear_trs: ba6.
               *gsl_multifit_nlinear_trs_subspace2D
      -- Variable: *note gsl_multilarge_nlinear_trs: ba7.
               *gsl_multilarge_nlinear_trs_subspace2D

          This selects the 2D subspace algorithm.

      -- Variable: *note gsl_multilarge_nlinear_trs: ba7.
               *gsl_multilarge_nlinear_trs_cgst

          This selects the Steihaug-Toint conjugate gradient algorithm.
          This method is available only for large systems.

 -- Type: gsl_multifit_nlinear_scale
 -- Type: gsl_multilarge_nlinear_scale

     The parameter ‘scale’ determines the diagonal scaling matrix D and
     may be selected from the following choices,

      -- Variable: *note gsl_multifit_nlinear_scale: bb3.
               *gsl_multifit_nlinear_scale_more
      -- Variable: *note gsl_multilarge_nlinear_scale: bb4.
               *gsl_multilarge_nlinear_scale_more

          This damping strategy was suggested by Moré, and corresponds
          to D^T D = \max(\diag(J^T J)), in other words the maximum
          elements of \diag(J^T J) encountered thus far in the
          iteration.  This choice of D makes the problem
          scale-invariant, so that if the model parameters x_i are each
          scaled by an arbitrary constant, \tilde{x}_i = a_i x_i, then
          the sequence of iterates produced by the algorithm would be
          unchanged.  This method can work very well in cases where the
          model parameters have widely different scales (ie: if some
          parameters are measured in nanometers, while others are
          measured in degrees Kelvin).  This strategy has been proven
          effective on a large class of problems and so it is the
          library default, but it may not be the best choice for all
          problems.

      -- Variable: *note gsl_multifit_nlinear_scale: bb3.
               *gsl_multifit_nlinear_scale_levenberg
      -- Variable: *note gsl_multilarge_nlinear_scale: bb4.
               *gsl_multilarge_nlinear_scale_levenberg

          This damping strategy was originally suggested by Levenberg,
          and corresponds to D^T D = I. This method has also proven
          effective on a large class of problems, but is not
          scale-invariant.  However, some authors (e.g.  Transtrum and
          Sethna 2012) argue that this choice is better for problems
          which are susceptible to parameter evaporation (ie: parameters
          go to infinity)

      -- Variable: *note gsl_multifit_nlinear_scale: bb3.
               *gsl_multifit_nlinear_scale_marquardt
      -- Variable: *note gsl_multilarge_nlinear_scale: bb4.
               *gsl_multilarge_nlinear_scale_marquardt

          This damping strategy was suggested by Marquardt, and
          corresponds to D^T D = \diag(J^T J). This method is
          scale-invariant, but it is generally considered inferior to
          both the Levenberg and Moré strategies, though may work well
          on certain classes of problems.

 -- Type: gsl_multifit_nlinear_solver
 -- Type: gsl_multilarge_nlinear_solver

     Solving the trust region subproblem on each iteration almost always
     requires the solution of the following linear least squares system

          [J; sqrt(mu) D] \delta = - [f; 0]

     The ‘solver’ parameter determines how the system is solved and can
     be selected from the following choices:

      -- Variable: *note gsl_multifit_nlinear_solver: bbb.
               *gsl_multifit_nlinear_solver_qr

          This method solves the system using a rank revealing QR
          decomposition of the Jacobian J. This method will produce
          reliable solutions in cases where the Jacobian is rank
          deficient or near-singular but does require about twice as
          many operations as the Cholesky method discussed below.

      -- Variable: *note gsl_multifit_nlinear_solver: bbb.
               *gsl_multifit_nlinear_solver_cholesky
      -- Variable: *note gsl_multilarge_nlinear_solver: bbc.
               *gsl_multilarge_nlinear_solver_cholesky

          This method solves the alternate normal equations problem

               ( J^T J + \mu D^T D ) \delta = -J^T f

          by using a Cholesky decomposition of the matrix J^T J + \mu
          D^T D. This method is faster than the QR approach, however it
          is susceptible to numerical instabilities if the Jacobian
          matrix is rank deficient or near-singular.  In these cases, an
          attempt is made to reduce the condition number of the matrix
          using Jacobi preconditioning, but for highly ill-conditioned
          problems the QR approach is better.  If it is known that the
          Jacobian matrix is well conditioned, this method is accurate
          and will perform faster than the QR approach.

      -- Variable: *note gsl_multifit_nlinear_solver: bbb.
               *gsl_multifit_nlinear_solver_mcholesky
      -- Variable: *note gsl_multilarge_nlinear_solver: bbc.
               *gsl_multilarge_nlinear_solver_mcholesky

          This method solves the alternate normal equations problem

               ( J^T J + \mu D^T D ) \delta = -J^T f

          by using a modified Cholesky decomposition of the matrix J^T J
          + \mu D^T D. This is more suitable for the dogleg methods
          where the parameter \mu = 0, and the matrix J^T J may be
          ill-conditioned or indefinite causing the standard Cholesky
          decomposition to fail.  This method is based on Level 2 BLAS
          and is thus slower than the standard Cholesky decomposition,
          which is based on Level 3 BLAS.

      -- Variable: *note gsl_multifit_nlinear_solver: bbb.
               *gsl_multifit_nlinear_solver_svd

          This method solves the system using a singular value
          decomposition of the Jacobian J. This method will produce the
          most reliable solutions for ill-conditioned Jacobians but is
          also the slowest solver method.

 -- Type: gsl_multifit_nlinear_fdtype

     The parameter ‘fdtype’ specifies whether to use forward or centered
     differences when approximating the Jacobian.  This is only used
     when an analytic Jacobian is not provided to the solver.  This
     parameter may be set to one of the following choices.

      -- Macro: GSL_MULTIFIT_NLINEAR_FWDIFF

          This specifies a forward finite difference to approximate the
          Jacobian matrix.  The Jacobian matrix will be calculated as

               J_ij = 1 / \Delta_j ( f_i(x + \Delta_j e_j) - f_i(x) )

          where \Delta_j = h |x_j| and e_j is the standard j-th
          Cartesian unit basis vector so that x + \Delta_j e_j
          represents a small (forward) perturbation of the j-th
          parameter by an amount \Delta_j.  The perturbation \Delta_j is
          proportional to the current value |x_j| which helps to
          calculate an accurate Jacobian when the various parameters
          have different scale sizes.  The value of h is specified by
          the ‘h_df’ parameter.  The accuracy of this method is O(h),
          and evaluating this matrix requires an additional p function
          evaluations.

      -- Macro: GSL_MULTIFIT_NLINEAR_CTRDIFF

          This specifies a centered finite difference to approximate the
          Jacobian matrix.  The Jacobian matrix will be calculated as

               J_ij = 1 / \Delta_j ( f_i(x + 1/2 \Delta_j e_j) - f_i(x - 1/2 \Delta_j e_j) )

          See above for a description of \Delta_j.  The accuracy of this
          method is O(h^2), but evaluating this matrix requires an
          additional 2p function evaluations.

‘double factor_up’

When a step is accepted, the trust region radius will be increased by
this factor.  The default value is 3.

‘double factor_down’

When a step is rejected, the trust region radius will be decreased by
this factor.  The default value is 2.

‘double avmax’

When using geodesic acceleration to solve a nonlinear least squares
problem, an important parameter to monitor is the ratio of the
acceleration term to the velocity term,

     |a| / |v|

If this ratio is small, it means the acceleration correction is
contributing very little to the step.  This could be because the problem
is not “nonlinear” enough to benefit from the acceleration.  If the
ratio is large (> 1) it means that the acceleration is larger than the
velocity, which shouldn’t happen since the step represents a truncated
series and so the second order term a should be smaller than the first
order term v to guarantee convergence.  Therefore any steps with a ratio
larger than the parameter ‘avmax’ are rejected.  ‘avmax’ is set to 0.75
by default.  For problems which experience difficulty converging, this
threshold could be lowered.

‘double h_df’

This parameter specifies the step size for approximating the Jacobian
matrix with finite differences.  It is set to \sqrt{\epsilon} by
default, where \epsilon is ‘GSL_DBL_EPSILON’.

‘double h_fvv’

When using geodesic acceleration, the user must either supply a function
to calculate f_{vv}(x) or the library can estimate this second
directional derivative using a finite difference method.  When using
finite differences, the library must calculate f(x + h v) where h
represents a small step in the velocity direction.  The parameter
‘h_fvv’ defines this step size and is set to 0.02 by default.


File: gsl-ref.info,  Node: Initializing the Solver<3>,  Next: Providing the Function to be Minimized,  Prev: Tunable Parameters,  Up: Nonlinear Least-Squares Fitting

41.5 Initializing the Solver
============================

 -- Type: gsl_multifit_nlinear_type

     This structure specifies the type of algorithm which will be used
     to solve a nonlinear least squares problem.  It may be selected
     from the following choices,

      -- Variable: *note gsl_multifit_nlinear_type: bc7.
               *gsl_multifit_nlinear_trust

          This specifies a trust region method.  It is currently the
          only implemented nonlinear least squares method.

 -- Function: gsl_multifit_nlinear_workspace *gsl_multifit_nlinear_alloc
          (const gsl_multifit_nlinear_type *T, const
          gsl_multifit_nlinear_parameters *params, const size_t n, const
          size_t p)
 -- Function: gsl_multilarge_nlinear_workspace
          *gsl_multilarge_nlinear_alloc (const
          gsl_multilarge_nlinear_type *T, const
          gsl_multilarge_nlinear_parameters *params, const size_t n,
          const size_t p)

     These functions return a pointer to a newly allocated instance of a
     derivative solver of type *note T: bca. for *note n: bca.
     observations and *note p: bca. parameters.  The *note params: bca.
     input specifies a tunable set of parameters which will affect
     important details in each iteration of the trust region subproblem
     algorithm.  It is recommended to start with the suggested default
     parameters (see *note gsl_multifit_nlinear_default_parameters():
     bcb. and *note gsl_multilarge_nlinear_default_parameters(): bcc.)
     and then tune the parameters once the code is working correctly.
     See *note Tunable Parameters: ba2.  for descriptions of the various
     parameters.  For example, the following code creates an instance of
     a Levenberg-Marquardt solver for 100 data points and 3 parameters,
     using suggested defaults:

          const gsl_multifit_nlinear_type * T = gsl_multifit_nlinear_trust;
          gsl_multifit_nlinear_parameters params = gsl_multifit_nlinear_default_parameters();
          gsl_multifit_nlinear_workspace * w = gsl_multifit_nlinear_alloc (T, &params, 100, 3);

     The number of observations *note n: bca. must be greater than or
     equal to parameters *note p: bca.

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: *note gsl_multifit_nlinear_parameters: ba4.
          gsl_multifit_nlinear_default_parameters (void)
 -- Function: *note gsl_multilarge_nlinear_parameters: ba5.
          gsl_multilarge_nlinear_default_parameters (void)

     These functions return a set of recommended default parameters for
     use in solving nonlinear least squares problems.  The user can tune
     each parameter to improve the performance on their particular
     problem, see *note Tunable Parameters: ba2.

 -- Function: int gsl_multifit_nlinear_init (const gsl_vector *x,
          gsl_multifit_nlinear_fdf *fdf, gsl_multifit_nlinear_workspace
          *w)
 -- Function: int gsl_multifit_nlinear_winit (const gsl_vector *x, const
          gsl_vector *wts, gsl_multifit_nlinear_fdf *fdf,
          gsl_multifit_nlinear_workspace *w)
 -- Function: int gsl_multilarge_nlinear_init (const gsl_vector *x,
          gsl_multilarge_nlinear_fdf *fdf,
          gsl_multilarge_nlinear_workspace *w)

     These functions initialize, or reinitialize, an existing workspace
     *note w: bce. to use the system *note fdf: bce. and the initial
     guess *note x: bce.  See *note Providing the Function to be
     Minimized: bcf. for a description of the *note fdf: bce. structure.

     Optionally, a weight vector ‘wts’ can be given to perform a
     weighted nonlinear regression.  Here, the weighting matrix is W =
     \diag(w_1,w_2,...,w_n).

 -- Function: void gsl_multifit_nlinear_free
          (gsl_multifit_nlinear_workspace *w)
 -- Function: void gsl_multilarge_nlinear_free
          (gsl_multilarge_nlinear_workspace *w)

     These functions free all the memory associated with the workspace
     *note w: bd1.

 -- Function: const char *gsl_multifit_nlinear_name (const
          gsl_multifit_nlinear_workspace *w)
 -- Function: const char *gsl_multilarge_nlinear_name (const
          gsl_multilarge_nlinear_workspace *w)

     These functions return a pointer to the name of the solver.  For
     example:

          printf ("w is a '%s' solver\n", gsl_multifit_nlinear_name (w));

     would print something like ‘w is a 'trust-region' solver’.

 -- Function: const char *gsl_multifit_nlinear_trs_name (const
          gsl_multifit_nlinear_workspace *w)
 -- Function: const char *gsl_multilarge_nlinear_trs_name (const
          gsl_multilarge_nlinear_workspace *w)

     These functions return a pointer to the name of the trust region
     subproblem method.  For example:

          printf ("w is a '%s' solver\n", gsl_multifit_nlinear_trs_name (w));

     would print something like ‘w is a 'levenberg-marquardt' solver’.


File: gsl-ref.info,  Node: Providing the Function to be Minimized,  Next: Iteration<5>,  Prev: Initializing the Solver<3>,  Up: Nonlinear Least-Squares Fitting

41.6 Providing the Function to be Minimized
===========================================

The user must provide n functions of p variables for the minimization
algorithm to operate on.  In order to allow for arbitrary parameters the
functions are defined by the following data types:

 -- Type: gsl_multifit_nlinear_fdf

     This data type defines a general system of functions with arbitrary
     parameters, the corresponding Jacobian matrix of derivatives, and
     optionally the second directional derivative of the functions for
     geodesic acceleration.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          This function should store the n components of the vector f(x)
          in ‘f’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the function cannot be
          computed.

     ‘int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)’

          This function should store the ‘n’-by-‘p’ matrix result

               J_ij = d f_i(x) / d x_j

          in ‘J’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the matrix cannot be
          computed.  If an analytic Jacobian is unavailable, or too
          expensive to compute, this function pointer may be set to
          ‘NULL’, in which case the Jacobian will be internally computed
          using finite difference approximations of the function ‘f’.

     ‘int (* fvv) (const gsl_vector * x, const gsl_vector * v, void *
     params, gsl_vector * fvv)’

          When geodesic acceleration is enabled, this function should
          store the n components of the vector f_{vv}(x) =
          \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over
          \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x),
          representing second directional derivatives of the function to
          be minimized, into the output ‘fvv’.  The parameter vector is
          provided in ‘x’ and the velocity vector is provided in ‘v’,
          both of which have p components.  The arbitrary parameters are
          given in ‘params’.  If analytic expressions for f_{vv}(x) are
          unavailable or too difficult to compute, this function pointer
          may be set to ‘NULL’, in which case f_{vv}(x) will be computed
          internally using a finite difference approximation.

     ‘size_t n’

          the number of functions, i.e.  the number of components of the
          vector ‘f’.

     ‘size_t p’

          the number of independent variables, i.e.  the number of
          components of the vector ‘x’.

     ‘void * params’

          a pointer to the arbitrary parameters of the function.

     ‘size_t nevalf’

          This does not need to be set by the user.  It counts the
          number of function evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevaldf’

          This does not need to be set by the user.  It counts the
          number of Jacobian evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevalfvv’

          This does not need to be set by the user.  It counts the
          number of f_{vv}(x) evaluations and is initialized by the
          ‘_init’ function.

 -- Type: gsl_multilarge_nlinear_fdf

     This data type defines a general system of functions with arbitrary
     parameters, a function to compute J u or J^T u for a given vector
     u, the normal equations matrix J^T J, and optionally the second
     directional derivative of the functions for geodesic acceleration.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          This function should store the n components of the vector f(x)
          in ‘f’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the function cannot be
          computed.

     ‘int (* df) (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x, const
     gsl_vector * u, void * params, gsl_vector * v, gsl_matrix * JTJ)’

          If ‘TransJ’ is equal to ‘CblasNoTrans’, then this function
          should compute the matrix-vector product J u and store the
          result in ‘v’.  If ‘TransJ’ is equal to ‘CblasTrans’, then
          this function should compute the matrix-vector product J^T u
          and store the result in ‘v’.  Additionally, the normal
          equations matrix J^T J should be stored in the lower half of
          ‘JTJ’.  The input matrix ‘JTJ’ could be set to ‘NULL’, for
          example by iterative methods which do not require this matrix,
          so the user should check for this prior to constructing the
          matrix.  The input ‘params’ contains the arbitrary parameters.

     ‘int (* fvv) (const gsl_vector * x, const gsl_vector * v, void *
     params, gsl_vector * fvv)’

          When geodesic acceleration is enabled, this function should
          store the n components of the vector f_{vv}(x) =
          \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over
          \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x),
          representing second directional derivatives of the function to
          be minimized, into the output ‘fvv’.  The parameter vector is
          provided in ‘x’ and the velocity vector is provided in ‘v’,
          both of which have p components.  The arbitrary parameters are
          given in ‘params’.  If analytic expressions for f_{vv}(x) are
          unavailable or too difficult to compute, this function pointer
          may be set to ‘NULL’, in which case f_{vv}(x) will be computed
          internally using a finite difference approximation.

     ‘size_t n’

          the number of functions, i.e.  the number of components of the
          vector ‘f’.

     ‘size_t p’

          the number of independent variables, i.e.  the number of
          components of the vector ‘x’.

     ‘void * params’

          a pointer to the arbitrary parameters of the function.

     ‘size_t nevalf’

          This does not need to be set by the user.  It counts the
          number of function evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevaldfu’

          This does not need to be set by the user.  It counts the
          number of Jacobian matrix-vector evaluations (J u or J^T u)
          and is initialized by the ‘_init’ function.

     ‘size_t nevaldf2’

          This does not need to be set by the user.  It counts the
          number of J^T J evaluations and is initialized by the ‘_init’
          function.

     ‘size_t nevalfvv’

          This does not need to be set by the user.  It counts the
          number of f_{vv}(x) evaluations and is initialized by the
          ‘_init’ function.

Note that when fitting a non-linear model against experimental data, the
data is passed to the functions above using the ‘params’ argument and
the trial best-fit parameters through the ‘x’ argument.


File: gsl-ref.info,  Node: Iteration<5>,  Next: Testing for Convergence,  Prev: Providing the Function to be Minimized,  Up: Nonlinear Least-Squares Fitting

41.7 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration of the trust region method and updates
the state of the solver.

 -- Function: int gsl_multifit_nlinear_iterate
          (gsl_multifit_nlinear_workspace *w)
 -- Function: int gsl_multilarge_nlinear_iterate
          (gsl_multilarge_nlinear_workspace *w)

     These functions perform a single iteration of the solver *note w:
     bdb.  If the iteration encounters an unexpected problem then an
     error code will be returned.  The solver workspace maintains a
     current estimate of the best-fit parameters at all times.

The solver workspace ‘w’ contains the following entries, which can be
used to track the progress of the solution:

‘gsl_vector * x’

     The current position, length p.

‘gsl_vector * f’

     The function residual vector at the current position f(x), length
     n.

‘gsl_matrix * J’

     The Jacobian matrix at the current position J(x), size n-by-p (only
     for ‘gsl_multifit_nlinear’ interface).

‘gsl_vector * dx’

     The difference between the current position and the previous
     position, i.e.  the last step \delta, taken as a vector, length p.

These quantities can be accessed with the following functions,

 -- Function: *note gsl_vector: 35f. *gsl_multifit_nlinear_position
          (const gsl_multifit_nlinear_workspace *w)
 -- Function: *note gsl_vector: 35f. *gsl_multilarge_nlinear_position
          (const gsl_multilarge_nlinear_workspace *w)

     These functions return the current position x (i.e.  best-fit
     parameters) of the solver *note w: bdd.

 -- Function: *note gsl_vector: 35f. *gsl_multifit_nlinear_residual
          (const gsl_multifit_nlinear_workspace *w)
 -- Function: *note gsl_vector: 35f. *gsl_multilarge_nlinear_residual
          (const gsl_multilarge_nlinear_workspace *w)

     These functions return the current residual vector f(x) of the
     solver *note w: bdf.  For weighted systems, the residual vector
     includes the weighting factor \sqrt{W}.

 -- Function: *note gsl_matrix: 3a2. *gsl_multifit_nlinear_jac (const
          gsl_multifit_nlinear_workspace *w)

     This function returns a pointer to the n-by-p Jacobian matrix for
     the current iteration of the solver *note w: be0.  This function is
     available only for the ‘gsl_multifit_nlinear’ interface.

 -- Function: size_t gsl_multifit_nlinear_niter (const
          gsl_multifit_nlinear_workspace *w)
 -- Function: size_t gsl_multilarge_nlinear_niter (const
          gsl_multilarge_nlinear_workspace *w)

     These functions return the number of iterations performed so far.
     The iteration counter is updated on each call to the ‘_iterate’
     functions above, and reset to 0 in the ‘_init’ functions.

 -- Function: int gsl_multifit_nlinear_rcond (double *rcond, const
          gsl_multifit_nlinear_workspace *w)
 -- Function: int gsl_multilarge_nlinear_rcond (double *rcond, const
          gsl_multilarge_nlinear_workspace *w)

     This function estimates the reciprocal condition number of the
     Jacobian matrix at the current position x and stores it in *note
     rcond: be4.  The computed value is only an estimate to give the
     user a guideline as to the conditioning of their particular
     problem.  Its calculation is based on which factorization method is
     used (Cholesky, QR, or SVD).

        * For the Cholesky solver, the matrix J^T J is factored at each
          iteration.  Therefore this function will estimate the 1-norm
          condition number rcond^2 = 1/(||J^T J||_1 \cdot ||(J^T
          J)^{-1}||_1)

        * For the QR solver, J is factored as J = Q R at each iteration.
          For simplicity, this function calculates the 1-norm
          conditioning of only the R factor, rcond = 1 / (||R||_1 \cdot
          ||R^{-1}||_1).  This can be computed efficiently since R is
          upper triangular.

        * For the SVD solver, in order to efficiently solve the trust
          region subproblem, the matrix which is factored is J D^{-1},
          instead of J itself.  The resulting singular values are used
          to provide the 2-norm reciprocal condition number, as rcond =
          \sigma_{min} / \sigma_{max}.  Note that when using Moré
          scaling, D \ne I and the resulting *note rcond: be4. estimate
          may be significantly different from the true *note rcond: be4.
          of J itself.

 -- Function: double gsl_multifit_nlinear_avratio (const
          gsl_multifit_nlinear_workspace *w)
 -- Function: double gsl_multilarge_nlinear_avratio (const
          gsl_multilarge_nlinear_workspace *w)

     This function returns the current ratio |a| / |v| of the
     acceleration correction term to the velocity step term.  The
     acceleration term is computed only by the
     ‘gsl_multifit_nlinear_trs_lmaccel’ and
     ‘gsl_multilarge_nlinear_trs_lmaccel’ methods, so this ratio will be
     zero for other TRS methods.


File: gsl-ref.info,  Node: Testing for Convergence,  Next: High Level Driver,  Prev: Iteration<5>,  Up: Nonlinear Least-Squares Fitting

41.8 Testing for Convergence
============================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the current estimate of the best-fit
parameters in several standard ways.

 -- Function: int gsl_multifit_nlinear_test (const double xtol, const
          double gtol, const double ftol, int *info, const
          gsl_multifit_nlinear_workspace *w)
 -- Function: int gsl_multilarge_nlinear_test (const double xtol, const
          double gtol, const double ftol, int *info, const
          gsl_multilarge_nlinear_workspace *w)

     These functions test for convergence of the minimization method
     using the following criteria:

        * Testing for a small step size relative to the current
          parameter vector

               |\delta_i| \le xtol (|x_i| + xtol)

          for each 0 <= i < p.  Each element of the step vector \delta
          is tested individually in case the different parameters have
          widely different scales.  Adding *note xtol: be9. to |x_i|
          helps the test avoid breaking down in situations where the
          true solution value x_i = 0.  If this test succeeds, *note
          info: be9. is set to 1 and the function returns ‘GSL_SUCCESS’.

          A general guideline for selecting the step tolerance is to
          choose xtol = 10^{-d} where d is the number of accurate
          decimal digits desired in the solution x.  See Dennis and
          Schnabel for more information.

        * Testing for a small gradient (g = \nabla \Phi(x) = J^T f)
          indicating a local function minimum:

               ||g||_inf <= gtol

          This expression tests whether the ratio (\nabla \Phi)_i x_i /
          \Phi is small.  Testing this scaled gradient is a better than
          \nabla \Phi alone since it is a dimensionless quantity and so
          independent of the scale of the problem.  The ‘max’ arguments
          help ensure the test doesn’t break down in regions where x_i
          or \Phi(x) are close to 0.  If this test succeeds, *note info:
          be9. is set to 2 and the function returns ‘GSL_SUCCESS’.

          A general guideline for choosing the gradient tolerance is to
          set ‘gtol = GSL_DBL_EPSILON^(1/3)’.  See Dennis and Schnabel
          for more information.

     If none of the tests succeed, *note info: be9. is set to 0 and the
     function returns ‘GSL_CONTINUE’, indicating further iterations are
     required.


File: gsl-ref.info,  Node: High Level Driver,  Next: Covariance matrix of best fit parameters,  Prev: Testing for Convergence,  Up: Nonlinear Least-Squares Fitting

41.9 High Level Driver
======================

These routines provide a high level wrapper that combines the iteration
and convergence testing for easy use.

 -- Function: int gsl_multifit_nlinear_driver (const size_t maxiter,
          const double xtol, const double gtol, const double ftol, void
          (*callback)(const size_t iter, void *params, const
          gsl_multifit_linear_workspace *w), void *callback_params, int
          *info, gsl_multifit_nlinear_workspace *w)
 -- Function: int gsl_multilarge_nlinear_driver (const size_t maxiter,
          const double xtol, const double gtol, const double ftol, void
          (*callback)(const size_t iter, void *params, const
          gsl_multilarge_linear_workspace *w), void *callback_params,
          int *info, gsl_multilarge_nlinear_workspace *w)

     These functions iterate the nonlinear least squares solver *note w:
     bec. for a maximum of *note maxiter: bec. iterations.  After each
     iteration, the system is tested for convergence with the error
     tolerances *note xtol: bec, *note gtol: bec. and *note ftol: bec.
     Additionally, the user may supply a callback function *note
     callback: bec. which is called after each iteration, so that the
     user may save or print relevant quantities for each iteration.  The
     parameter *note callback_params: bec. is passed to the *note
     callback: bec. function.  The parameters *note callback: bec. and
     *note callback_params: bec. may be set to ‘NULL’ to disable this
     feature.  Upon successful convergence, the function returns
     ‘GSL_SUCCESS’ and sets *note info: bec. to the reason for
     convergence (see *note gsl_multifit_nlinear_test(): be8.).  If the
     function has not converged after *note maxiter: bec. iterations,
     ‘GSL_EMAXITER’ is returned.  In rare cases, during an iteration the
     algorithm may be unable to find a new acceptable step \delta to
     take.  In this case, ‘GSL_ENOPROG’ is returned indicating no
     further progress can be made.  If your problem is having difficulty
     converging, see *note Troubleshooting: bed. for further guidance.


File: gsl-ref.info,  Node: Covariance matrix of best fit parameters,  Next: Troubleshooting<2>,  Prev: High Level Driver,  Up: Nonlinear Least-Squares Fitting

41.10 Covariance matrix of best fit parameters
==============================================

 -- Function: int gsl_multifit_nlinear_covar (const gsl_matrix *J, const
          double epsrel, gsl_matrix *covar)
 -- Function: int gsl_multilarge_nlinear_covar (gsl_matrix *covar,
          gsl_multilarge_nlinear_workspace *w)

     This function computes the covariance matrix of best-fit parameters
     using the Jacobian matrix ‘J’ and stores it in *note covar: bf0.
     The parameter ‘epsrel’ is used to remove linear-dependent columns
     when ‘J’ is rank deficient.

     The covariance matrix is given by,

          C = (J^T J)^{-1}

     or in the weighted case,

          C = (J^T W J)^{-1}

     and is computed using the factored form of the Jacobian (Cholesky,
     QR, or SVD). Any columns of R which satisfy

          |R_{kk}| \leq epsrel |R_{11}|

     are considered linearly-dependent and are excluded from the
     covariance matrix (the corresponding rows and columns of the
     covariance matrix are set to zero).

     If the minimisation uses the weighted least-squares function f_i =
     (Y(x, t_i) - y_i) / \sigma_i then the covariance matrix above gives
     the statistical error on the best-fit parameters resulting from the
     Gaussian errors \sigma_i on the underlying data y_i.  This can be
     verified from the relation \delta f = J \delta c and the fact that
     the fluctuations in f from the data y_i are normalised by \sigma_i
     and so satisfy

          <\delta f \delta f^T> = I

     For an unweighted least-squares function f_i = (Y(x, t_i) - y_i)
     the covariance matrix above should be multiplied by the variance of
     the residuals about the best-fit \sigma^2 = \sum (y_i - Y(x,t_i))^2
     / (n-p) to give the variance-covariance matrix \sigma^2 C. This
     estimates the statistical error on the best-fit parameters from the
     scatter of the underlying data.

     For more information about covariance matrices see *note Linear
     Least-Squares Overview: b29.


File: gsl-ref.info,  Node: Troubleshooting<2>,  Next: Examples<32>,  Prev: Covariance matrix of best fit parameters,  Up: Nonlinear Least-Squares Fitting

41.11 Troubleshooting
=====================

When developing a code to solve a nonlinear least squares problem, here
are a few considerations to keep in mind.

  1. The most common difficulty is the accurate implementation of the
     Jacobian matrix.  If the analytic Jacobian is not properly provided
     to the solver, this can hinder and many times prevent convergence
     of the method.  When developing a new nonlinear least squares code,
     it often helps to compare the program output with the internally
     computed finite difference Jacobian and the user supplied analytic
     Jacobian.  If there is a large difference in coefficients, it is
     likely the analytic Jacobian is incorrectly implemented.

  2. If your code is having difficulty converging, the next thing to
     check is the starting point provided to the solver.  The methods of
     this chapter are local methods, meaning if you provide a starting
     point far away from the true minimum, the method may converge to a
     local minimum or not converge at all.  Sometimes it is possible to
     solve a linearized approximation to the nonlinear problem, and use
     the linear solution as the starting point to the nonlinear problem.

  3. If the various parameters of the coefficient vector x vary widely
     in magnitude, then the problem is said to be badly scaled.  The
     methods of this chapter do attempt to automatically rescale the
     elements of x to have roughly the same order of magnitude, but in
     extreme cases this could still cause problems for convergence.  In
     these cases it is recommended for the user to scale their parameter
     vector x so that each parameter spans roughly the same range, say
     [-1,1].  The solution vector can be backscaled to recover the
     original units of the problem.


File: gsl-ref.info,  Node: Examples<32>,  Next: References and Further Reading<34>,  Prev: Troubleshooting<2>,  Up: Nonlinear Least-Squares Fitting

41.12 Examples
==============

The following example programs demonstrate the nonlinear least squares
fitting capabilities.

* Menu:

* Exponential Fitting Example::
* Geodesic Acceleration Example 1::
* Geodesic Acceleration Example 2::
* Comparing TRS Methods Example::
* Large Nonlinear Least Squares Example::


File: gsl-ref.info,  Node: Exponential Fitting Example,  Next: Geodesic Acceleration Example 1,  Up: Examples<32>

41.12.1 Exponential Fitting Example
-----------------------------------

The following example program fits a weighted exponential model with
background to experimental data, Y = A \exp(-\lambda t) + b.  The first
part of the program sets up the functions ‘expb_f()’ and ‘expb_df()’ to
calculate the model and its Jacobian.  The appropriate fitting function
is given by,

     f_i = (A \exp(-\lambda t_i) + b) - y_i

where we have chosen t_i = i T / (N - 1), where N is the number of data
points fitted, so that t_i \in [0, T]. The Jacobian matrix J is the
derivative of these functions with respect to the three parameters (A,
\lambda, b).  It is given by,

     J_{ij} = d f_i / d x_j

where x_0 = A, x_1 = \lambda and x_2 = b.  The i-th row of the Jacobian
is therefore

     J(i,:) = [ \exp(-\lambda t_i) ; -t_i A \exp(-\lambda t_i) ; 1 ]

The main part of the program sets up a Levenberg-Marquardt solver and
some simulated random data.  The data uses the known parameters
(5.0,1.5,1.0) combined with Gaussian noise (standard deviation = 0.1)
with a maximum time T = 3 and N = 100 timesteps.  The initial guess for
the parameters is chosen as (1.0, 1.0, 0.0).  The iteration terminates
when the relative change in x is smaller than 10^{-8}, or when the
magnitude of the gradient falls below 10^{-8}.  Here are the results of
running the program:

     iter  0: A = 1.0000, lambda = 1.0000, b = 0.0000, cond(J) =      inf, |f(x)| = 88.4448
     iter  1: A = 4.5109, lambda = 2.5258, b = 1.0704, cond(J) =  26.2686, |f(x)| = 24.0646
     iter  2: A = 4.8565, lambda = 1.7442, b = 1.1669, cond(J) =  23.7470, |f(x)| = 11.9797
     iter  3: A = 4.9356, lambda = 1.5713, b = 1.0767, cond(J) =  17.5849, |f(x)| = 10.7355
     iter  4: A = 4.8678, lambda = 1.4838, b = 1.0252, cond(J) =  16.3428, |f(x)| = 10.5000
     iter  5: A = 4.8118, lambda = 1.4481, b = 1.0076, cond(J) =  15.7925, |f(x)| = 10.4786
     iter  6: A = 4.7983, lambda = 1.4404, b = 1.0041, cond(J) =  15.5840, |f(x)| = 10.4778
     iter  7: A = 4.7967, lambda = 1.4395, b = 1.0037, cond(J) =  15.5396, |f(x)| = 10.4778
     iter  8: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5344, |f(x)| = 10.4778
     iter  9: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5339, |f(x)| = 10.4778
     iter 10: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5339, |f(x)| = 10.4778
     iter 11: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5339, |f(x)| = 10.4778
     summary from method 'trust-region/levenberg-marquardt'
     number of iterations: 11
     function evaluations: 16
     Jacobian evaluations: 12
     reason for stopping: small gradient
     initial |f(x)| = 88.444756
     final   |f(x)| = 10.477801
     chisq/dof = 1.1318
     A      = 4.79653 +/- 0.18704
     lambda = 1.43937 +/- 0.07390
     b      = 1.00368 +/- 0.03473
     status = success

The approximate values of the parameters are found correctly, and the
chi-squared value indicates a good fit (the chi-squared per degree of
freedom is approximately 1).  In this case the errors on the parameters
can be estimated from the square roots of the diagonal elements of the
covariance matrix.  If the chi-squared value shows a poor fit (i.e.
\chi^2/(n-p) \gg 1 then the error estimates obtained from the covariance
matrix will be too small.  In the example program the error estimates
are multiplied by \sqrt{\chi^2/(n-p)} in this case, a common way of
increasing the errors for a poor fit.  Note that a poor fit will result
from the use of an inappropriate model, and the scaled error estimates
may then be outside the range of validity for Gaussian errors.

Additionally, we see that the condition number of J(x) stays reasonably
small throughout the iteration.  This indicates we could safely switch
to the Cholesky solver for speed improvement, although this particular
system is too small to really benefit.

Fig.  %s shows the fitted curve with the original data.

[gsl-ref-figures/fit-exp]
Figure: Exponential fitted curve with data

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     #define N      100    /* number of data points to fit */
     #define TMAX   (3.0)  /* time variable in [0,TMAX] */

     struct data {
       size_t n;
       double * t;
       double * y;
     };

     int
     expb_f (const gsl_vector * x, void *data,
             gsl_vector * f)
     {
       size_t n = ((struct data *)data)->n;
       double *t = ((struct data *)data)->t;
       double *y = ((struct data *)data)->y;

       double A = gsl_vector_get (x, 0);
       double lambda = gsl_vector_get (x, 1);
       double b = gsl_vector_get (x, 2);

       size_t i;

       for (i = 0; i < n; i++)
         {
           /* Model Yi = A * exp(-lambda * t_i) + b */
           double Yi = A * exp (-lambda * t[i]) + b;
           gsl_vector_set (f, i, Yi - y[i]);
         }

       return GSL_SUCCESS;
     }

     int
     expb_df (const gsl_vector * x, void *data,
              gsl_matrix * J)
     {
       size_t n = ((struct data *)data)->n;
       double *t = ((struct data *)data)->t;

       double A = gsl_vector_get (x, 0);
       double lambda = gsl_vector_get (x, 1);

       size_t i;

       for (i = 0; i < n; i++)
         {
           /* Jacobian matrix J(i,j) = dfi / dxj, */
           /* where fi = (Yi - yi)/sigma[i],      */
           /*       Yi = A * exp(-lambda * t_i) + b  */
           /* and the xj are the parameters (A,lambda,b) */
           double e = exp(-lambda * t[i]);
           gsl_matrix_set (J, i, 0, e);
           gsl_matrix_set (J, i, 1, -t[i] * A * e);
           gsl_matrix_set (J, i, 2, 1.0);
         }

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector *f = gsl_multifit_nlinear_residual(w);
       gsl_vector *x = gsl_multifit_nlinear_position(w);
       double rcond;

       /* compute reciprocal condition number of J(x) */
       gsl_multifit_nlinear_rcond(&rcond, w);

       fprintf(stderr, "iter %2zu: A = %.4f, lambda = %.4f, b = %.4f, cond(J) = %8.4f, |f(x)| = %.4f\n",
               iter,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1),
               gsl_vector_get(x, 2),
               1.0 / rcond,
               gsl_blas_dnrm2(f));
     }

     int
     main (void)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       gsl_multifit_nlinear_workspace *w;
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       const size_t n = N;
       const size_t p = 3;

       gsl_vector *f;
       gsl_matrix *J;
       gsl_matrix *covar = gsl_matrix_alloc (p, p);
       double t[N], y[N], weights[N];
       struct data d = { n, t, y };
       double x_init[3] = { 1.0, 1.0, 0.0 }; /* starting values */
       gsl_vector_view x = gsl_vector_view_array (x_init, p);
       gsl_vector_view wts = gsl_vector_view_array(weights, n);
       gsl_rng * r;
       double chisq, chisq0;
       int status, info;
       size_t i;

       const double xtol = 1e-8;
       const double gtol = 1e-8;
       const double ftol = 0.0;

       gsl_rng_env_setup();
       r = gsl_rng_alloc(gsl_rng_default);

       /* define the function to be minimized */
       fdf.f = expb_f;
       fdf.df = expb_df;   /* set to NULL for finite-difference Jacobian */
       fdf.fvv = NULL;     /* not using geodesic acceleration */
       fdf.n = n;
       fdf.p = p;
       fdf.params = &d;

       /* this is the data to be fitted */
       for (i = 0; i < n; i++)
         {
           double ti = i * TMAX / (n - 1.0);
           double yi = 1.0 + 5 * exp (-1.5 * ti);
           double si = 0.1 * yi;
           double dy = gsl_ran_gaussian(r, si);

           t[i] = ti;
           y[i] = yi + dy;
           weights[i] = 1.0 / (si * si);
           printf ("data: %g %g %g\n", ti, y[i], si);
         };

       /* allocate workspace with default parameters */
       w = gsl_multifit_nlinear_alloc (T, &fdf_params, n, p);

       /* initialize solver with starting point and weights */
       gsl_multifit_nlinear_winit (&x.vector, &wts.vector, &fdf, w);

       /* compute initial cost function */
       f = gsl_multifit_nlinear_residual(w);
       gsl_blas_ddot(f, f, &chisq0);

       /* solve the system with a maximum of 100 iterations */
       status = gsl_multifit_nlinear_driver(100, xtol, gtol, ftol,
                                            callback, NULL, &info, w);

       /* compute covariance of best fit parameters */
       J = gsl_multifit_nlinear_jac(w);
       gsl_multifit_nlinear_covar (J, 0.0, covar);

       /* compute final cost */
       gsl_blas_ddot(f, f, &chisq);

     #define FIT(i) gsl_vector_get(w->x, i)
     #define ERR(i) sqrt(gsl_matrix_get(covar,i,i))

       fprintf(stderr, "summary from method '%s/%s'\n",
               gsl_multifit_nlinear_name(w),
               gsl_multifit_nlinear_trs_name(w));
       fprintf(stderr, "number of iterations: %zu\n",
               gsl_multifit_nlinear_niter(w));
       fprintf(stderr, "function evaluations: %zu\n", fdf.nevalf);
       fprintf(stderr, "Jacobian evaluations: %zu\n", fdf.nevaldf);
       fprintf(stderr, "reason for stopping: %s\n",
               (info == 1) ? "small step size" : "small gradient");
       fprintf(stderr, "initial |f(x)| = %f\n", sqrt(chisq0));
       fprintf(stderr, "final   |f(x)| = %f\n", sqrt(chisq));

       {
         double dof = n - p;
         double c = GSL_MAX_DBL(1, sqrt(chisq / dof));

         fprintf(stderr, "chisq/dof = %g\n", chisq / dof);

         fprintf (stderr, "A      = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
         fprintf (stderr, "lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
         fprintf (stderr, "b      = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
       }

       fprintf (stderr, "status = %s\n", gsl_strerror (status));

       gsl_multifit_nlinear_free (w);
       gsl_matrix_free (covar);
       gsl_rng_free (r);

       return 0;
     }


File: gsl-ref.info,  Node: Geodesic Acceleration Example 1,  Next: Geodesic Acceleration Example 2,  Prev: Exponential Fitting Example,  Up: Examples<32>

41.12.2 Geodesic Acceleration Example 1
---------------------------------------

The following example program minimizes a modified Rosenbrock function,
which is characterized by a narrow canyon with steep walls.  The
starting point is selected high on the canyon wall, so the solver must
first find the canyon bottom and then navigate to the minimum.  The
problem is solved both with and without using geodesic acceleration for
comparison.  The cost function is given by

     Phi(x) = 1/2 (f1^2 + f2^2)
     f1 = 100 ( x2 - x1^2 )
     f2 = 1 - x1

The Jacobian matrix is

     J = [ -200*x1 100  ]
         [   -1     0   ]

In order to use geodesic acceleration, the user must provide the second
directional derivative of each residual in the velocity direction, D_v^2
f_i = \sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha}
\partial_{\beta} f_i.  The velocity vector v is provided by the solver.
For this example, these derivatives are

     fvv = [ -200 v1^2 ]
           [     0     ]

The solution of this minimization problem is

     x* = [ 1 ; 1 ]
     Phi(x*) = 0

The program output is shown below:

     === Solving system without acceleration ===
     NITER         = 53
     NFEV          = 56
     NJEV          = 54
     NAEV          = 0
     initial cost  = 2.250225000000e+04
     final cost    = 6.674986031430e-18
     final x       = (9.999999974165e-01, 9.999999948328e-01)
     final cond(J) = 6.000096055094e+02
     === Solving system with acceleration ===
     NITER         = 15
     NFEV          = 17
     NJEV          = 16
     NAEV          = 16
     initial cost  = 2.250225000000e+04
     final cost    = 7.518932873279e-19
     final x       = (9.999999991329e-01, 9.999999982657e-01)
     final cond(J) = 6.000097233278e+02

[gsl-ref-figures/nlfit2]
Figure: Paths taken by solver for Rosenbrock function

We can see that enabling geodesic acceleration requires less than a
third of the number of Jacobian evaluations in order to locate the
minimum.  The path taken by both methods is shown in Fig.  %s.  The
contours show the cost function \Phi(x_1,x_2).  We see that both methods
quickly find the canyon bottom, but the geodesic acceleration method
navigates along the bottom to the solution with significantly fewer
iterations.

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);

       gsl_vector_set(f, 0, 100.0 * (x2 - x1*x1));
       gsl_vector_set(f, 1, 1.0 - x1);

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       double x1 = gsl_vector_get(x, 0);

       gsl_matrix_set(J, 0, 0, -200.0*x1);
       gsl_matrix_set(J, 0, 1, 100.0);
       gsl_matrix_set(J, 1, 0, -1.0);
       gsl_matrix_set(J, 1, 1, 0.0);

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       double v1 = gsl_vector_get(v, 0);

       gsl_vector_set(fvv, 0, -200.0 * v1 * v1);
       gsl_vector_set(fvv, 1, 0.0);

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector * x = gsl_multifit_nlinear_position(w);

       /* print out current location */
       printf("%f %f\n",
              gsl_vector_get(x, 0),
              gsl_vector_get(x, 1));
     }

     void
     solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * x = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       /* initialize solver */
       gsl_multifit_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       /* print summary */

       fprintf(stderr, "NITER         = %zu\n", gsl_multifit_nlinear_niter(work));
       fprintf(stderr, "NFEV          = %zu\n", fdf->nevalf);
       fprintf(stderr, "NJEV          = %zu\n", fdf->nevaldf);
       fprintf(stderr, "NAEV          = %zu\n", fdf->nevalfvv);
       fprintf(stderr, "initial cost  = %.12e\n", chisq0);
       fprintf(stderr, "final cost    = %.12e\n", chisq);
       fprintf(stderr, "final x       = (%.12e, %.12e)\n",
               gsl_vector_get(x, 0), gsl_vector_get(x, 1));
       fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);

       printf("\n\n");

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 2;
       const size_t p = 2;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();

       /* print map of Phi(x1, x2) */
       {
         double x1, x2, chisq;
         double *f1 = gsl_vector_ptr(f, 0);
         double *f2 = gsl_vector_ptr(f, 1);

         for (x1 = -1.2; x1 < 1.3; x1 += 0.1)
           {
             for (x2 = -0.5; x2 < 2.1; x2 += 0.1)
               {
                 gsl_vector_set(x, 0, x1);
                 gsl_vector_set(x, 1, x2);
                 func_f(x, NULL, f);

                 chisq = (*f1) * (*f1) + (*f2) * (*f2);
                 printf("%f %f %f\n", x1, x2, chisq);
               }
             printf("\n");
           }
         printf("\n\n");
       }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = NULL;

       /* starting point */
       gsl_vector_set(x, 0, -0.5);
       gsl_vector_set(x, 1, 1.75);

       fprintf(stderr, "=== Solving system without acceleration ===\n");
       fdf_params.trs = gsl_multifit_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fprintf(stderr, "=== Solving system with acceleration ===\n");
       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);

       return 0;
     }


File: gsl-ref.info,  Node: Geodesic Acceleration Example 2,  Next: Comparing TRS Methods Example,  Prev: Geodesic Acceleration Example 1,  Up: Examples<32>

41.12.3 Geodesic Acceleration Example 2
---------------------------------------

The following example fits a set of data to a Gaussian model using the
Levenberg-Marquardt method with geodesic acceleration.  The cost
function is

     Phi(x) = 1/2 \sum_i f_i^2
     f_i = y_i - Y(a,b,c;t_i)

where y_i is the measured data point at time t_i, and the model is
specified by

     Y(a,b,c;t) = a exp(-1/2 ((t-b)/c)^2)

The parameters a,b,c represent the amplitude, mean, and width of the
Gaussian respectively.  The program below generates the y_i data on
[0,1] using the values a = 5, b = 0.4, c = 0.15 and adding random noise.
The i-th row of the Jacobian is

     J(i,:) = ( -e_i  -(a/c)*z_i*e_i  -(a/c)*z_i^2*e_i )

where

     z_i = (t_i - b) / c
     e_i = \exp(-1/2 z_i^2)

In order to use geodesic acceleration, we need the second directional
derivative of the residuals in the velocity direction, D_v^2 f_i =
\sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha}
\partial_{\beta} f_i, where v is provided by the solver.  To compute
this, it is helpful to make a table of all second derivatives of the
residuals f_i with respect to each combination of model parameters.
This table is

The lower half of the table is omitted since it is symmetric.  Then, the
second directional derivative of f_i is

     D_v^2 f_i = v_a^2 (d/da)^2 f_i + 2 v_a v_b (d/da) (d/db) f_i + 2 v_a v_c (d/da) (d/dc) f_i + v_b^2 (d/db)^2 f_i + 2 v_b v_c (d/db) (d/dc) f_i + v_c^2 (d/dc)^2 f_i

The factors of 2 come from the symmetry of the mixed second partial
derivatives.  The iteration is started using the initial guess a = 1, b
= 0, c = 1.  The program output is shown below:

     iter  0: a = 1.0000, b = 0.0000, c = 1.0000, |a|/|v| = 0.0000 cond(J) =      inf, |f(x)| = 35.4785
     iter  1: a = 1.5708, b = 0.5321, c = 0.5219, |a|/|v| = 0.3093 cond(J) =  29.0443, |f(x)| = 31.1042
     iter  2: a = 1.7387, b = 0.4040, c = 0.4568, |a|/|v| = 0.1199 cond(J) =   3.5256, |f(x)| = 28.7217
     iter  3: a = 2.2340, b = 0.3829, c = 0.3053, |a|/|v| = 0.3308 cond(J) =   4.5121, |f(x)| = 23.8074
     iter  4: a = 3.2275, b = 0.3952, c = 0.2243, |a|/|v| = 0.2784 cond(J) =   8.6499, |f(x)| = 15.6003
     iter  5: a = 4.3347, b = 0.3974, c = 0.1752, |a|/|v| = 0.2029 cond(J) =  15.1732, |f(x)| = 7.5908
     iter  6: a = 4.9352, b = 0.3992, c = 0.1536, |a|/|v| = 0.1001 cond(J) =  26.6621, |f(x)| = 4.8402
     iter  7: a = 5.0716, b = 0.3994, c = 0.1498, |a|/|v| = 0.0166 cond(J) =  34.6922, |f(x)| = 4.7103
     iter  8: a = 5.0828, b = 0.3994, c = 0.1495, |a|/|v| = 0.0012 cond(J) =  36.5422, |f(x)| = 4.7095
     iter  9: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6929, |f(x)| = 4.7095
     iter 10: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6975, |f(x)| = 4.7095
     iter 11: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6976, |f(x)| = 4.7095
     NITER         = 11
     NFEV          = 18
     NJEV          = 12
     NAEV          = 17
     initial cost  = 1.258724737288e+03
     final cost    = 2.217977560180e+01
     final x       = (5.083101559156e+00, 3.994484109594e-01, 1.494898e-01)
     final cond(J) = 3.669757713403e+01

We see the method converges after 11 iterations.  For comparison the
standard Levenberg-Marquardt method requires 26 iterations and so the
Gaussian fitting problem benefits substantially from the geodesic
acceleration correction.  The column marked ‘|a|/|v|’ above shows the
ratio of the acceleration term to the velocity term as the iteration
progresses.  Larger values of this ratio indicate that the geodesic
acceleration correction term is contributing substantial information to
the solver relative to the standard LM velocity step.

The data and fitted model are shown in Fig.  %s.

[gsl-ref-figures/nlfit2b]
Figure: Gaussian model fitted to data

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     struct data
     {
       double *t;
       double *y;
       size_t n;
     };

     /* model function: a * exp( -1/2 * [ (t - b) / c ]^2 ) */
     double
     gaussian(const double a, const double b, const double c, const double t)
     {
       const double z = (t - b) / c;
       return (a * exp(-0.5 * z * z));
     }

     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double yi = d->y[i];
           double y = gaussian(a, b, c, ti);

           gsl_vector_set(f, i, yi - y);
         }

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double zi = (ti - b) / c;
           double ei = exp(-0.5 * zi * zi);

           gsl_matrix_set(J, i, 0, -ei);
           gsl_matrix_set(J, i, 1, -(a / c) * ei * zi);
           gsl_matrix_set(J, i, 2, -(a / c) * ei * zi * zi);
         }

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       double va = gsl_vector_get(v, 0);
       double vb = gsl_vector_get(v, 1);
       double vc = gsl_vector_get(v, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double zi = (ti - b) / c;
           double ei = exp(-0.5 * zi * zi);
           double Dab = -zi * ei / c;
           double Dac = -zi * zi * ei / c;
           double Dbb = a * ei / (c * c) * (1.0 - zi*zi);
           double Dbc = a * zi * ei / (c * c) * (2.0 - zi*zi);
           double Dcc = a * zi * zi * ei / (c * c) * (3.0 - zi*zi);
           double sum;

           sum = 2.0 * va * vb * Dab +
                 2.0 * va * vc * Dac +
                       vb * vb * Dbb +
                 2.0 * vb * vc * Dbc +
                       vc * vc * Dcc;

           gsl_vector_set(fvv, i, sum);
         }

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector *f = gsl_multifit_nlinear_residual(w);
       gsl_vector *x = gsl_multifit_nlinear_position(w);
       double avratio = gsl_multifit_nlinear_avratio(w);
       double rcond;

       (void) params; /* not used */

       /* compute reciprocal condition number of J(x) */
       gsl_multifit_nlinear_rcond(&rcond, w);

       fprintf(stderr, "iter %2zu: a = %.4f, b = %.4f, c = %.4f, |a|/|v| = %.4f cond(J) = %8.4f, |f(x)| = %.4f\n",
               iter,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1),
               gsl_vector_get(x, 2),
               avratio,
               1.0 / rcond,
               gsl_blas_dnrm2(f));
     }

     void
     solve_system(gsl_vector *x, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * y = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       /* initialize solver */
       gsl_multifit_nlinear_init(x, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       gsl_vector_memcpy(x, y);

       /* print summary */

       fprintf(stderr, "NITER         = %zu\n", gsl_multifit_nlinear_niter(work));
       fprintf(stderr, "NFEV          = %zu\n", fdf->nevalf);
       fprintf(stderr, "NJEV          = %zu\n", fdf->nevaldf);
       fprintf(stderr, "NAEV          = %zu\n", fdf->nevalfvv);
       fprintf(stderr, "initial cost  = %.12e\n", chisq0);
       fprintf(stderr, "final cost    = %.12e\n", chisq);
       fprintf(stderr, "final x       = (%.12e, %.12e, %12e)\n",
               gsl_vector_get(x, 0), gsl_vector_get(x, 1), gsl_vector_get(x, 2));
       fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 300;  /* number of data points to fit */
       const size_t p = 3;    /* number of model parameters */
       const double a = 5.0;  /* amplitude */
       const double b = 0.4;  /* center */
       const double c = 0.15; /* width */
       const gsl_rng_type * T = gsl_rng_default;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       struct data fit_data;
       gsl_rng * r;
       size_t i;

       gsl_rng_env_setup ();
       r = gsl_rng_alloc (T);

       fit_data.t = malloc(n * sizeof(double));
       fit_data.y = malloc(n * sizeof(double));
       fit_data.n = n;

       /* generate synthetic data with noise */
       for (i = 0; i < n; ++i)
         {
           double t = (double)i / (double) n;
           double y0 = gaussian(a, b, c, t);
           double dy = gsl_ran_gaussian (r, 0.1 * y0);

           fit_data.t[i] = t;
           fit_data.y[i] = y0 + dy;
         }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &fit_data;

       /* starting point */
       gsl_vector_set(x, 0, 1.0);
       gsl_vector_set(x, 1, 0.0);
       gsl_vector_set(x, 2, 1.0);

       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       /* print data and model */
       {
         double A = gsl_vector_get(x, 0);
         double B = gsl_vector_get(x, 1);
         double C = gsl_vector_get(x, 2);

         for (i = 0; i < n; ++i)
           {
             double ti = fit_data.t[i];
             double yi = fit_data.y[i];
             double fi = gaussian(A, B, C, ti);

             printf("%f %f %f\n", ti, yi, fi);
           }
       }

       gsl_vector_free(f);
       gsl_vector_free(x);
       gsl_rng_free(r);

       return 0;
     }


File: gsl-ref.info,  Node: Comparing TRS Methods Example,  Next: Large Nonlinear Least Squares Example,  Prev: Geodesic Acceleration Example 2,  Up: Examples<32>

41.12.4 Comparing TRS Methods Example
-------------------------------------

The following program compares all available nonlinear least squares
trust-region subproblem (TRS) methods on the Branin function, a common
optimization test problem.  The cost function is

     \Phi(x) &= 1/2 (f_1^2 + f_2^2)
     f_1 &= x_2 + a_1 x_1^2 + a_2 x_1 + a_3
     f_2 &= sqrt(a_4) sqrt(1 + (1 - a_5) cos(x_1))

with a_1 = -{5.1 \over 4 \pi^2}, a_2 = {5 \over \pi}, a_3 = -6, a_4 =
10, a_5 = {1 \over 8\pi}.  There are three minima of this function in
the range (x_1,x_2) \in [-5,15] \times [-5,15].  The program below uses
the starting point (x_1,x_2) = (6,14.5) and calculates the solution with
all available nonlinear least squares TRS methods.  The program output
is shown below:

     Method                    NITER  NFEV  NJEV  Initial Cost  Final cost   Final cond(J) Final x
     levenberg-marquardt       20     27    21    1.9874e+02    3.9789e-01   6.1399e+07    (-3.14e+00, 1.23e+01)
     levenberg-marquardt+accel 27     36    28    1.9874e+02    3.9789e-01   1.4465e+07    (3.14e+00, 2.27e+00)
     dogleg                    23     64    23    1.9874e+02    3.9789e-01   5.0692e+08    (3.14e+00, 2.28e+00)
     double-dogleg             24     69    24    1.9874e+02    3.9789e-01   3.4879e+07    (3.14e+00, 2.27e+00)
     2D-subspace               23     54    24    1.9874e+02    3.9789e-01   2.5142e+07    (3.14e+00, 2.27e+00)

The first row of output above corresponds to standard
Levenberg-Marquardt, while the second row includes geodesic
acceleration.  We see that the standard LM method converges to the
minimum at (-\pi,12.275) and also uses the least number of iterations
and Jacobian evaluations.  All other methods converge to the minimum
(\pi,2.275) and perform similarly in terms of number of Jacobian
evaluations.  We see that J is fairly ill-conditioned at both minima,
indicating that the QR (or SVD) solver is the best choice for this
problem.  Since there are only two parameters in this optimization
problem, we can easily visualize the paths taken by each method, which
are shown in Fig.  %s.  The figure shows contours of the cost function
\Phi(x_1,x_2) which exhibits three global minima in the range [-5,15]
\times [-5,15].  The paths taken by each solver are shown as colored
lines.

[gsl-ref-figures/nlfit3]
Figure: Paths taken for different TRS methods for the Branin function

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     /* parameters to model */
     struct model_params
     {
       double a1;
       double a2;
       double a3;
       double a4;
       double a5;
     };

     /* Branin function */
     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);
       double f1 = x2 + par->a1 * x1 * x1 + par->a2 * x1 + par->a3;
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

       gsl_vector_set(f, 0, f1);
       gsl_vector_set(f, 1, f2);

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

       gsl_matrix_set(J, 0, 0, 2.0 * par->a1 * x1 + par->a2);
       gsl_matrix_set(J, 0, 1, 1.0);

       gsl_matrix_set(J, 1, 0, -0.5 * par->a4 / f2 * (1.0 - par->a5) * sin(x1));
       gsl_matrix_set(J, 1, 1, 0.0);

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double v1 = gsl_vector_get(v, 0);
       double c = cos(x1);
       double s = sin(x1);
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * c);
       double t = 0.5 * par->a4 * (1.0 - par->a5) / f2;

       gsl_vector_set(fvv, 0, 2.0 * par->a1 * v1 * v1);
       gsl_vector_set(fvv, 1, -t * (c + s*s/f2) * v1 * v1);

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector * x = gsl_multifit_nlinear_position(w);
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);

       /* print out current location */
       printf("%f %f\n", x1, x2);
     }

     void
     solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * x = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       printf("# %s/%s\n",
              gsl_multifit_nlinear_name(work),
              gsl_multifit_nlinear_trs_name(work));

       /* initialize solver */
       gsl_multifit_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       /* print summary */
       fprintf(stderr, "%-25s %-6zu %-5zu %-5zu %-13.4e %-12.4e %-13.4e (%.2e, %.2e)\n",
               gsl_multifit_nlinear_trs_name(work),
               gsl_multifit_nlinear_niter(work),
               fdf->nevalf,
               fdf->nevaldf,
               chisq0,
               chisq,
               1.0 / rcond,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1));

       printf("\n\n");

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 2;
       const size_t p = 2;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       struct model_params params;

       params.a1 = -5.1 / (4.0 * M_PI * M_PI);
       params.a2 = 5.0 / M_PI;
       params.a3 = -6.0;
       params.a4 = 10.0;
       params.a5 = 1.0 / (8.0 * M_PI);

       /* print map of Phi(x1, x2) */
       {
         double x1, x2, chisq;

         for (x1 = -5.0; x1 < 15.0; x1 += 0.1)
           {
             for (x2 = -5.0; x2 < 15.0; x2 += 0.1)
               {
                 gsl_vector_set(x, 0, x1);
                 gsl_vector_set(x, 1, x2);
                 func_f(x, &params, f);

                 gsl_blas_ddot(f, f, &chisq);

                 printf("%f %f %f\n", x1, x2, chisq);
               }
             printf("\n");
           }
         printf("\n\n");
       }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &params;

       /* starting point */
       gsl_vector_set(x, 0, 6.0);
       gsl_vector_set(x, 1, 14.5);

       fprintf(stderr, "%-25s %-6s %-5s %-5s %-13s %-12s %-13s %-15s\n",
               "Method", "NITER", "NFEV", "NJEV", "Initial Cost",
               "Final cost", "Final cond(J)", "Final x");

       fdf_params.trs = gsl_multifit_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_dogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_ddogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_subspace2D;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);

       return 0;
     }


File: gsl-ref.info,  Node: Large Nonlinear Least Squares Example,  Prev: Comparing TRS Methods Example,  Up: Examples<32>

41.12.5 Large Nonlinear Least Squares Example
---------------------------------------------

The following program illustrates the large nonlinear least squares
solvers on a system with significant sparse structure in the Jacobian.
The cost function is

     \Phi(x) &= 1/2 \sum_{i=1}^{p+1} f_i^2
     f_i &= \sqrt{\alpha} (x_i - 1), 1 \le i \le p
     f_{p+1} &= ||x||^2 - 1/4

with \alpha = 10^{-5}.  The residual f_{p+1} imposes a constraint on the
p parameters x, to ensure that ||x||^2 \approx {1 \over 4}.  The
(p+1)-by-p Jacobian for this system is

     J(x) = [ \sqrt{alpha} I_p; 2 x^T ]

and the normal equations matrix is

     J^T J = \alpha I_p + 4 x x^T

Finally, the second directional derivative of f for the geodesic
acceleration method is

     fvv = [     0     ]
           [ 2 ||v||^2 ]

Since the upper p-by-p block of J is diagonal, this sparse structure
should be exploited in the nonlinear solver.  For comparison, the
following program solves the system for p = 2000 using the dense direct
Cholesky solver based on the normal equations matrix J^T J, as well as
the iterative Steihaug-Toint solver, based on sparse matrix-vector
products J u and J^T u.  The program output is shown below:

     Method                    NITER NFEV NJUEV NJTJEV NAEV Init Cost  Final cost cond(J) Final |x|^2 Time (s)
     levenberg-marquardt       25    31   26    26     0    7.1218e+18 1.9555e-02 447.50  2.5044e-01  46.28
     levenberg-marquardt+accel 22    23   45    23     22   7.1218e+18 1.9555e-02 447.64  2.5044e-01  33.92
     dogleg                    37    87   36    36     0    7.1218e+18 1.9555e-02 447.59  2.5044e-01  56.05
     double-dogleg             35    88   34    34     0    7.1218e+18 1.9555e-02 447.62  2.5044e-01  52.65
     2D-subspace               37    88   36    36     0    7.1218e+18 1.9555e-02 447.71  2.5044e-01  59.75
     steihaug-toint            35    88   345   0      0    7.1218e+18 1.9555e-02 inf     2.5044e-01  0.09

The first five rows use methods based on factoring the dense J^T J
matrix while the last row uses the iterative Steihaug-Toint method.
While the number of Jacobian matrix-vector products (NJUEV) is less for
the dense methods, the added time to construct and factor the J^T J
matrix (NJTJEV) results in a much larger runtime than the iterative
method (see last column).

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <sys/time.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multilarge_nlinear.h>
     #include <gsl/gsl_spblas.h>
     #include <gsl/gsl_spmatrix.h>

     /* parameters for functions */
     struct model_params
     {
       double alpha;
       gsl_spmatrix *J;
     };

     /* penalty function */
     int
     penalty_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct model_params *par = (struct model_params *) params;
       const double sqrt_alpha = sqrt(par->alpha);
       const size_t p = x->size;
       size_t i;
       double sum = 0.0;

       for (i = 0; i < p; ++i)
         {
           double xi = gsl_vector_get(x, i);

           gsl_vector_set(f, i, sqrt_alpha*(xi - 1.0));

           sum += xi * xi;
         }

       gsl_vector_set(f, p, sum - 0.25);

       return GSL_SUCCESS;
     }

     int
     penalty_df (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x,
                 const gsl_vector * u, void * params, gsl_vector * v,
                 gsl_matrix * JTJ)
     {
       struct model_params *par = (struct model_params *) params;
       const size_t p = x->size;
       size_t j;

       /* store 2*x in last row of J */
       for (j = 0; j < p; ++j)
         {
           double xj = gsl_vector_get(x, j);
           gsl_spmatrix_set(par->J, p, j, 2.0 * xj);
         }

       /* compute v = op(J) u */
       if (v)
         gsl_spblas_dgemv(TransJ, 1.0, par->J, u, 0.0, v);

       if (JTJ)
         {
           gsl_vector_view diag = gsl_matrix_diagonal(JTJ);

           /* compute J^T J = [ alpha*I_p + 4 x x^T ] */
           gsl_matrix_set_zero(JTJ);

           /* store 4 x x^T in lower half of JTJ */
           gsl_blas_dsyr(CblasLower, 4.0, x, JTJ);

           /* add alpha to diag(JTJ) */
           gsl_vector_add_constant(&diag.vector, par->alpha);
         }

       return GSL_SUCCESS;
     }

     int
     penalty_fvv (const gsl_vector * x, const gsl_vector * v,
                  void *params, gsl_vector * fvv)
     {
       const size_t p = x->size;
       double normv = gsl_blas_dnrm2(v);

       gsl_vector_set_zero(fvv);
       gsl_vector_set(fvv, p, 2.0 * normv * normv);

       (void)params; /* avoid unused parameter warning */

       return GSL_SUCCESS;
     }

     void
     solve_system(const gsl_vector *x0, gsl_multilarge_nlinear_fdf *fdf,
                  gsl_multilarge_nlinear_parameters *params)
     {
       const gsl_multilarge_nlinear_type *T = gsl_multilarge_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multilarge_nlinear_workspace *work =
         gsl_multilarge_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multilarge_nlinear_residual(work);
       gsl_vector * x = gsl_multilarge_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond, xsq;
       struct timeval tv0, tv1;

       gettimeofday(&tv0, NULL);

       /* initialize solver */
       gsl_multilarge_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multilarge_nlinear_driver(max_iter, xtol, gtol, ftol,
                                     NULL, NULL, &info, work);

       gettimeofday(&tv1, NULL);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* compute final ||x||^2 */
       gsl_blas_ddot(x, x, &xsq);

       /* store cond(J(x)) */
       gsl_multilarge_nlinear_rcond(&rcond, work);

       /* print summary */
       fprintf(stderr, "%-25s %-5zu %-4zu %-5zu %-6zu %-4zu %-10.4e %-10.4e %-7.2f %-11.4e %.2f\n",
               gsl_multilarge_nlinear_trs_name(work),
               gsl_multilarge_nlinear_niter(work),
               fdf->nevalf,
               fdf->nevaldfu,
               fdf->nevaldf2,
               fdf->nevalfvv,
               chisq0,
               chisq,
               1.0 / rcond,
               xsq,
               (tv1.tv_sec - tv0.tv_sec) + 1.0e-6 * (tv1.tv_usec - tv0.tv_usec));

       gsl_multilarge_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t p = 2000;
       const size_t n = p + 1;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);

       /* allocate sparse Jacobian matrix with 2*p non-zero elements in triplet format */
       gsl_spmatrix *J = gsl_spmatrix_alloc_nzmax(n, p, 2 * p, GSL_SPMATRIX_TRIPLET);

       gsl_multilarge_nlinear_fdf fdf;
       gsl_multilarge_nlinear_parameters fdf_params =
         gsl_multilarge_nlinear_default_parameters();
       struct model_params params;
       size_t i;

       params.alpha = 1.0e-5;
       params.J = J;

       /* define function to be minimized */
       fdf.f = penalty_f;
       fdf.df = penalty_df;
       fdf.fvv = penalty_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &params;

       for (i = 0; i < p; ++i)
         {
           /* starting point */
           gsl_vector_set(x, i, i + 1.0);

           /* store sqrt(alpha)*I_p in upper p-by-p block of J */
           gsl_spmatrix_set(J, i, i, sqrt(params.alpha));
         }

       fprintf(stderr, "%-25s %-4s %-4s %-5s %-6s %-4s %-10s %-10s %-7s %-11s %-10s\n",
               "Method", "NITER", "NFEV", "NJUEV", "NJTJEV", "NAEV", "Init Cost",
               "Final cost", "cond(J)", "Final |x|^2", "Time (s)");

       fdf_params.scale = gsl_multilarge_nlinear_scale_levenberg;

       fdf_params.trs = gsl_multilarge_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_dogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_ddogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_subspace2D;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_cgst;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);
       gsl_spmatrix_free(J);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<34>,  Prev: Examples<32>,  Up: Nonlinear Least-Squares Fitting

41.13 References and Further Reading
====================================

The following publications are relevant to the algorithms described in
this section,

   * J.J. Moré, `The Levenberg-Marquardt Algorithm: Implementation and
     Theory', Lecture Notes in Mathematics, v630 (1978), ed G. Watson.

   * H. B. Nielsen, “Damping Parameter in Marquardt’s Method”, IMM
     Department of Mathematical Modeling, DTU, Tech.  Report
     IMM-REP-1999-05 (1999).

   * K. Madsen and H. B. Nielsen, “Introduction to Optimization and Data
     Fitting”, IMM Department of Mathematical Modeling, DTU, 2010.

   * J. E. Dennis and R. B. Schnabel, Numerical Methods for
     Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.

   * M. K. Transtrum, B. B. Machta, and J. P. Sethna, Geometry of
     nonlinear least squares with applications to sloppy models and
     optimization, Phys.  Rev.  E 83, 036701, 2011.

   * M. K. Transtrum and J. P. Sethna, Improvements to the
     Levenberg-Marquardt algorithm for nonlinear least-squares
     minimization, arXiv:1201.5885, 2012.

   * J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained
     Optimization Software”, ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17–41.

   * H. B. Nielsen, “UCTP Test Problems for Unconstrained Optimization”,
     IMM Department of Mathematical Modeling, DTU, Tech.  Report
     IMM-REP-2000-17 (2000).


File: gsl-ref.info,  Node: Basis Splines,  Next: Sparse Matrices,  Prev: Nonlinear Least-Squares Fitting,  Up: Top

42 Basis Splines
****************

This chapter describes functions for the computation of smoothing basis
splines (B-splines).  A smoothing spline differs from an interpolating
spline in that the resulting curve is not required to pass through each
datapoint.  For information about interpolating splines, see *note
Interpolation: 9b0.

The header file ‘gsl_bspline.h’ contains the prototypes for the bspline
functions and related declarations.

* Menu:

* Overview: Overview<7>.
* Initializing the B-splines solver::
* Constructing the knots vector::
* Evaluation of B-splines::
* Evaluation of B-spline derivatives::
* Working with the Greville abscissae::
* Examples: Examples<33>.
* References and Further Reading: References and Further Reading<35>.


File: gsl-ref.info,  Node: Overview<7>,  Next: Initializing the B-splines solver,  Up: Basis Splines

42.1 Overview
=============

B-splines are commonly used as basis functions to fit smoothing curves
to large data sets.  To do this, the abscissa axis is broken up into
some number of intervals, where the endpoints of each interval are
called `breakpoints'.  These breakpoints are then converted to `knots'
by imposing various continuity and smoothness conditions at each
interface.  Given a nondecreasing knot vector

     t = \{t_0, t_1, \dots, t_{n+k-1}\}

the n basis splines of order k are defined by

     B_(i,1)(x) = (1, t_i <= x < t_(i+1)
                  (0, else
     B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x)
                   + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)

for i = 0, \ldots, n-1.  The common case of cubic B-splines is given by
k = 4.  The above recurrence relation can be evaluated in a numerically
stable way by the de Boor algorithm.

If we define appropriate knots on an interval [a,b] then the B-spline
basis functions form a complete set on that interval.  Therefore we can
expand a smoothing function as

     f(x) = \sum_{i=0}^{n-1} c_i B_{i,k}(x)

given enough (x_j, f(x_j)) data pairs.  The coefficients c_i can be
readily obtained from a least-squares fit.


File: gsl-ref.info,  Node: Initializing the B-splines solver,  Next: Constructing the knots vector,  Prev: Overview<7>,  Up: Basis Splines

42.2 Initializing the B-splines solver
======================================

 -- Type: gsl_bspline_workspace

     The computation of B-spline functions requires a preallocated
     workspace.

 -- Function: *note gsl_bspline_workspace: c01. *gsl_bspline_alloc
          (const size_t k, const size_t nbreak)

     This function allocates a workspace for computing B-splines of
     order *note k: c02.  The number of breakpoints is given by *note
     nbreak: c02.  This leads to n = nbreak + k - 2 basis functions.
     Cubic B-splines are specified by k = 4.  The size of the workspace
     is O(2k^2 + 5k + nbreak).

 -- Function: void gsl_bspline_free (gsl_bspline_workspace *w)

     This function frees the memory associated with the workspace *note
     w: c03.


File: gsl-ref.info,  Node: Constructing the knots vector,  Next: Evaluation of B-splines,  Prev: Initializing the B-splines solver,  Up: Basis Splines

42.3 Constructing the knots vector
==================================

 -- Function: int gsl_bspline_knots (const gsl_vector *breakpts,
          gsl_bspline_workspace *w)

     This function computes the knots associated with the given
     breakpoints and stores them internally in ‘w->knots’.

 -- Function: int gsl_bspline_knots_uniform (const double a, const
          double b, gsl_bspline_workspace *w)

     This function assumes uniformly spaced breakpoints on [a,b] and
     constructs the corresponding knot vector using the previously
     specified ‘nbreak’ parameter.  The knots are stored in ‘w->knots’.


File: gsl-ref.info,  Node: Evaluation of B-splines,  Next: Evaluation of B-spline derivatives,  Prev: Constructing the knots vector,  Up: Basis Splines

42.4 Evaluation of B-splines
============================

 -- Function: int gsl_bspline_eval (const double x, gsl_vector *B,
          gsl_bspline_workspace *w)

     This function evaluates all B-spline basis functions at the
     position *note x: c08. and stores them in the vector *note B: c08,
     so that the i-th element is B_i(x).  The vector *note B: c08. must
     be of length n = nbreak + k - 2.  This value may also be obtained
     by calling *note gsl_bspline_ncoeffs(): c09.  Computing all the
     basis functions at once is more efficient than computing them
     individually, due to the nature of the defining recurrence
     relation.

 -- Function: int gsl_bspline_eval_nonzero (const double x, gsl_vector
          *Bk, size_t *istart, size_t *iend, gsl_bspline_workspace *w)

     This function evaluates all potentially nonzero B-spline basis
     functions at the position *note x: c0a. and stores them in the
     vector *note Bk: c0a, so that the i-th element is
     B_{(istart+i)}(x).  The last element of *note Bk: c0a. is
     B_{iend}(x).  The vector *note Bk: c0a. must be of length k.  By
     returning only the nonzero basis functions, this function allows
     quantities involving linear combinations of the B_i(x) to be
     computed without unnecessary terms (such linear combinations occur,
     for example, when evaluating an interpolated function).

 -- Function: size_t gsl_bspline_ncoeffs (gsl_bspline_workspace *w)

     This function returns the number of B-spline coefficients given by
     n = nbreak + k - 2.


File: gsl-ref.info,  Node: Evaluation of B-spline derivatives,  Next: Working with the Greville abscissae,  Prev: Evaluation of B-splines,  Up: Basis Splines

42.5 Evaluation of B-spline derivatives
=======================================

 -- Function: int gsl_bspline_deriv_eval (const double x, const size_t
          nderiv, gsl_matrix *dB, gsl_bspline_workspace *w)

     This function evaluates all B-spline basis function derivatives of
     orders 0 through *note nderiv: c0c. (inclusive) at the position
     *note x: c0c. and stores them in the matrix *note dB: c0c.  The
     (i,j)-th element of *note dB: c0c. is d^jB_i(x)/dx^j.  The matrix
     *note dB: c0c. must be of size n = nbreak + k - 2 by nderiv + 1.
     The value n may also be obtained by calling *note
     gsl_bspline_ncoeffs(): c09.  Note that function evaluations are
     included as the zeroth order derivatives in *note dB: c0c.
     Computing all the basis function derivatives at once is more
     efficient than computing them individually, due to the nature of
     the defining recurrence relation.

 -- Function: int gsl_bspline_deriv_eval_nonzero (const double x, const
          size_t nderiv, gsl_matrix *dB, size_t *istart, size_t *iend,
          gsl_bspline_workspace *w)

     This function evaluates all potentially nonzero B-spline basis
     function derivatives of orders 0 through *note nderiv: c0d.
     (inclusive) at the position *note x: c0d. and stores them in the
     matrix *note dB: c0d.  The (i,j)-th element of *note dB: c0d. is
     d^jB_{(istart+i)}(x)/dx^j.  The last row of *note dB: c0d. contains
     d^jB_{iend}(x)/dx^j.  The matrix *note dB: c0d. must be of size k
     by at least nderiv + 1.  Note that function evaluations are
     included as the zeroth order derivatives in *note dB: c0d.  By
     returning only the nonzero basis functions, this function allows
     quantities involving linear combinations of the B_i(x) and their
     derivatives to be computed without unnecessary terms.


File: gsl-ref.info,  Node: Working with the Greville abscissae,  Next: Examples<33>,  Prev: Evaluation of B-spline derivatives,  Up: Basis Splines

42.6 Working with the Greville abscissae
========================================

The Greville abscissae are defined to be the mean location of k-1
consecutive knots in the knot vector for each basis spline function of
order k.  With the first and last knots in the *note
gsl_bspline_workspace: c01. knot vector excluded, there are *note
gsl_bspline_ncoeffs(): c09. Greville abscissae for any given B-spline
basis.  These values are often used in B-spline collocation applications
and may also be called Marsden-Schoenberg points.

 -- Function: double gsl_bspline_greville_abscissa (size_t i,
          gsl_bspline_workspace *w)

     Returns the location of the i-th Greville abscissa for the given
     B-spline basis.  For the ill-defined case when k = 1, the
     implementation chooses to return breakpoint interval midpoints.


File: gsl-ref.info,  Node: Examples<33>,  Next: References and Further Reading<35>,  Prev: Working with the Greville abscissae,  Up: Basis Splines

42.7 Examples
=============

The following program computes a linear least squares fit to data using
cubic B-spline basis functions with uniform breakpoints.  The data is
generated from the curve y(x) = \cos{(x)} \exp{(-x/10)} on the interval
[0, 15] with Gaussian noise added.

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>
     #include <gsl/gsl_bspline.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_statistics.h>

     /* number of data points to fit */
     #define N        200

     /* number of fit coefficients */
     #define NCOEFFS  12

     /* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
     #define NBREAK   (NCOEFFS - 2)

     int
     main (void)
     {
       const size_t n = N;
       const size_t ncoeffs = NCOEFFS;
       const size_t nbreak = NBREAK;
       size_t i, j;
       gsl_bspline_workspace *bw;
       gsl_vector *B;
       double dy;
       gsl_rng *r;
       gsl_vector *c, *w;
       gsl_vector *x, *y;
       gsl_matrix *X, *cov;
       gsl_multifit_linear_workspace *mw;
       double chisq, Rsq, dof, tss;

       gsl_rng_env_setup();
       r = gsl_rng_alloc(gsl_rng_default);

       /* allocate a cubic bspline workspace (k = 4) */
       bw = gsl_bspline_alloc(4, nbreak);
       B = gsl_vector_alloc(ncoeffs);

       x = gsl_vector_alloc(n);
       y = gsl_vector_alloc(n);
       X = gsl_matrix_alloc(n, ncoeffs);
       c = gsl_vector_alloc(ncoeffs);
       w = gsl_vector_alloc(n);
       cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
       mw = gsl_multifit_linear_alloc(n, ncoeffs);

       /* this is the data to be fitted */
       for (i = 0; i < n; ++i)
         {
           double sigma;
           double xi = (15.0 / (N - 1)) * i;
           double yi = cos(xi) * exp(-0.1 * xi);

           sigma = 0.1 * yi;
           dy = gsl_ran_gaussian(r, sigma);
           yi += dy;

           gsl_vector_set(x, i, xi);
           gsl_vector_set(y, i, yi);
           gsl_vector_set(w, i, 1.0 / (sigma * sigma));

           printf("%f %f\n", xi, yi);
         }

       /* use uniform breakpoints on [0, 15] */
       gsl_bspline_knots_uniform(0.0, 15.0, bw);

       /* construct the fit matrix X */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);

           /* compute B_j(xi) for all j */
           gsl_bspline_eval(xi, B, bw);

           /* fill in row i of X */
           for (j = 0; j < ncoeffs; ++j)
             {
               double Bj = gsl_vector_get(B, j);
               gsl_matrix_set(X, i, j, Bj);
             }
         }

       /* do the fit */
       gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);

       dof = n - ncoeffs;
       tss = gsl_stats_wtss(w->data, 1, y->data, 1, y->size);
       Rsq = 1.0 - chisq / tss;

       fprintf(stderr, "chisq/dof = %e, Rsq = %f\n",
                        chisq / dof, Rsq);

       printf("\n\n");

       /* output the smoothed curve */
       {
         double xi, yi, yerr;

         for (xi = 0.0; xi < 15.0; xi += 0.1)
           {
             gsl_bspline_eval(xi, B, bw);
             gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
             printf("%f %f\n", xi, yi);
           }
       }

       gsl_rng_free(r);
       gsl_bspline_free(bw);
       gsl_vector_free(B);
       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_matrix_free(X);
       gsl_vector_free(c);
       gsl_vector_free(w);
       gsl_matrix_free(cov);
       gsl_multifit_linear_free(mw);

       return 0;
     } /* main() */

The output is shown below:

     $ ./a.out > bspline.txt
     chisq/dof = 1.118217e+00, Rsq = 0.989771

The data and fitted model are shown in Fig.  %s.

[gsl-ref-figures/bspline]
Figure: Data (black) and fitted model (red)


File: gsl-ref.info,  Node: References and Further Reading<35>,  Prev: Examples<33>,  Up: Basis Splines

42.8 References and Further Reading
===================================

Further information on the algorithms described in this section can be
found in the following book,

   * C. de Boor, `A Practical Guide to Splines' (1978), Springer-Verlag,
     ISBN 0-387-90356-9.

Further information of Greville abscissae and B-spline collocation can
be found in the following paper,

   * Richard W. Johnson, Higher order B-spline collocation at the
     Greville abscissae.  `Applied Numerical Mathematics'.  vol.: 52,
     2005, 63–75.

A large collection of B-spline routines is available in the PPPACK
library available at ‘http://www.netlib.org/pppack’, which is also part
of SLATEC.


File: gsl-ref.info,  Node: Sparse Matrices,  Next: Sparse BLAS Support,  Prev: Basis Splines,  Up: Top

43 Sparse Matrices
******************

This chapter describes functions for the construction and manipulation
of sparse matrices, matrices which are populated primarily with zeros
and contain only a few non-zero elements.  Sparse matrices often appear
in the solution of partial differential equations.  It is beneficial to
use specialized data structures and algorithms for storing and working
with sparse matrices, since dense matrix algorithms and structures can
be prohibitively slow and use huge amounts of memory when applied to
sparse matrices.

The header file ‘gsl_spmatrix.h’ contains the prototypes for the sparse
matrix functions and related declarations.

* Menu:

* Data types: Data types<2>.
* Sparse Matrix Storage Formats::
* Overview: Overview<8>.
* Allocation::
* Accessing Matrix Elements::
* Initializing Matrix Elements::
* Reading and Writing Matrices::
* Copying Matrices::
* Exchanging Rows and Columns::
* Matrix Operations::
* Matrix Properties::
* Finding Maximum and Minimum Elements::
* Compressed Format::
* Conversion Between Sparse and Dense Matrices::
* Examples: Examples<34>.
* References and Further Reading: References and Further Reading<36>.


File: gsl-ref.info,  Node: Data types<2>,  Next: Sparse Matrix Storage Formats,  Up: Sparse Matrices

43.1 Data types
===============

All the functions are available for each of the standard data-types.
The versions for ‘double’ have the prefix ‘gsl_spmatrix’, Similarly the
versions for single-precision ‘float’ arrays have the prefix
‘gsl_spmatrix_float’.  The full list of available types is given below,

Prefix                               Type
                                     
--------------------------------------------------------------
                                     
gsl_spmatrix                         double
                                     
                                     
gsl_spmatrix_float                   float
                                     
                                     
gsl_spmatrix_long_double             long double
                                     
                                     
gsl_spmatrix_int                     int
                                     
                                     
gsl_spmatrix_uint                    unsigned int
                                     
                                     
gsl_spmatrix_long                    long
                                     
                                     
gsl_spmatrix_ulong                   unsigned long
                                     
                                     
gsl_spmatrix_short                   short
                                     
                                     
gsl_spmatrix_ushort                  unsigned short
                                     
                                     
gsl_spmatrix_char                    char
                                     
                                     
gsl_spmatrix_uchar                   unsigned char
                                     
                                     
gsl_spmatrix_complex                 complex double
                                     
                                     
gsl_spmatrix_complex_float           complex float
                                     
                                     
gsl_spmatrix_complex_long_double     complex long double
                                     


File: gsl-ref.info,  Node: Sparse Matrix Storage Formats,  Next: Overview<8>,  Prev: Data types<2>,  Up: Sparse Matrices

43.2 Sparse Matrix Storage Formats
==================================

GSL currently supports three storage formats for sparse matrices: the
coordinate (COO) representation, compressed sparse column (CSC) and
compressed sparse row (CSR) formats.  These are discussed in more detail
below.  In order to illustrate the different storage formats, the
following sections will reference this M-by-N sparse matrix, with M=4
and N=5:

     \begin{pmatrix} 9 & 0 & 0 & 0 & -3 \\ 4 & 7 & 0 & 0 & 0 \\ 0 & 8 & -1 & 8 & 0 \\ 4 & 0 & 5 & 6 & 0 \end{pmatrix}

The number of non-zero elements in the matrix, also abbreviated as ‘nnz’
is equal to 10 in this case.

* Menu:

* Coordinate Storage (COO): Coordinate Storage COO.
* Compressed Sparse Column (CSC): Compressed Sparse Column CSC.
* Compressed Sparse Row (CSR): Compressed Sparse Row CSR.


File: gsl-ref.info,  Node: Coordinate Storage COO,  Next: Compressed Sparse Column CSC,  Up: Sparse Matrix Storage Formats

43.2.1 Coordinate Storage (COO)
-------------------------------

The coordinate storage format, also known as `triplet format', stores
triplets (i,j,x) for each non-zero element of the matrix.  This notation
means that the (i,j) element of the matrix A is A_{ij} = x.  The matrix
is stored using three arrays of the same length, representing the row
indices, column indices, and matrix data.  For the reference matrix
above, one possible storage format is:

data     9      7      4      8      -3     -1     8      5      6      4
                                                                        
                                                                        
row      0      1      1      2      0      2      2      3      3      3
                                                                        
                                                                        
col      0      1      0      1      4      2      3      2      3      0
                                                                        

Note that this representation is not unique - the coordinate triplets
may appear in any ordering and would still represent the same sparse
matrix.  The length of the three arrays is equal to the number of
non-zero elements in the matrix, ‘nnz’, which in this case is 10.  The
coordinate format is extremely convenient for sparse matrix `assembly',
the process of adding new elements, or changing existing elements, in a
sparse matrix.  However, it is generally not suitable for the efficient
implementation of matrix-matrix products, or matrix factorization
algorithms.  For these applications it is better to use one of the
compressed formats discussed below.

In order to faciliate efficient sparse matrix assembly, GSL stores the
coordinate data in a balanced binary search tree, specifically an AVL
tree, in addition to the three arrays discussed above.  This allows GSL
to efficiently determine whether an entry (i,j) already exists in the
matrix, and to replace an existing matrix entry with a new value,
without needing to search unsorted arrays.


File: gsl-ref.info,  Node: Compressed Sparse Column CSC,  Next: Compressed Sparse Row CSR,  Prev: Coordinate Storage COO,  Up: Sparse Matrix Storage Formats

43.2.2 Compressed Sparse Column (CSC)
-------------------------------------

Compressed sparse column storage stores each column of non-zero values
in the sparse matrix in a continuous memory block, keeping pointers to
the beginning of each column in that memory block, and storing the row
indices of each non-zero element.  For the reference matrix above, these
arrays look like

data        9      4      4      7      8      -1     5      8      6      -3
                                                                           
                                                                           
row         0      1      3      1      2      2      3      2      3      0
                                                                           
                                                                           
col_ptr     0      3      5      7      9      10
                                               

The ‘data’ and ‘row’ arrays are of length ‘nnz’ and are the same as the
COO storage format.  The ‘col_ptr’ array has length N+1, and
‘col_ptr[j]’ gives the index in ‘data’ of the start of column ‘j’.
Therefore, the j-th column of the matrix is stored in
‘data[col_ptr[j]]’, ‘data[col_ptr[j] + 1]’, …, ‘data[col_ptr[j+1] - 1]’.
The last element of ‘col_ptr’ is ‘nnz’.


File: gsl-ref.info,  Node: Compressed Sparse Row CSR,  Prev: Compressed Sparse Column CSC,  Up: Sparse Matrix Storage Formats

43.2.3 Compressed Sparse Row (CSR)
----------------------------------

Compressed row storage stores each row of non-zero values in a
continuous memory block, keeping pointers to the beginning of each row
in the block and storing the column indices of each non-zero element.
For the reference matrix above, these arrays look like

data        9      -3     4      7      8      -1     8      4      5      6
                                                                           
                                                                           
col         0      4      0      1      1      2      3      0      2      3
                                                                           
                                                                           
row_ptr     0      2      4      7      10
                                        

The ‘data’ and ‘col’ arrays are of length ‘nnz’ and are the same as the
COO storage format.  The ‘row_ptr’ array has length M+1, and
‘row_ptr[i]’ gives the index in ‘data’ of the start of row ‘i’.
Therefore, the i-th row of the matrix is stored in ‘data[row_ptr[i]]’,
‘data[row_ptr[i] + 1]’, …, ‘data[row_ptr[i+1] - 1]’.  The last element
of ‘row_ptr’ is ‘nnz’.


File: gsl-ref.info,  Node: Overview<8>,  Next: Allocation,  Prev: Sparse Matrix Storage Formats,  Up: Sparse Matrices

43.3 Overview
=============

These routines provide support for constructing and manipulating sparse
matrices in GSL, using an API similar to *note gsl_matrix: 3a2.  The
basic structure is called *note gsl_spmatrix: c1e.

 -- Type: gsl_spmatrix

     This structure is defined as:

          typedef struct
          {
            size_t size1;
            size_t size2;
            int *i;
            double *data;
            int *p;
            size_t nzmax;
            size_t nz;
            [ ... variables for binary tree and memory management ... ]
            size_t sptype;
          } gsl_spmatrix;

     This defines a ‘size1’-by-‘size2’ sparse matrix.  The number of
     non-zero elements currently in the matrix is given by ‘nz’.  For
     the triplet representation, ‘i’, ‘p’, and ‘data’ are arrays of size
     ‘nz’ which contain the row indices, column indices, and element
     value, respectively.  So if data[k] = A(i,j), then i = i[k] and j =
     p[k].

     For compressed column storage, ‘i’ and ‘data’ are arrays of size
     ‘nz’ containing the row indices and element values, identical to
     the triplet case.  ‘p’ is an array of size ‘size2’ + 1 where ‘p[j]’
     points to the index in ‘data’ of the start of column ‘j’.  Thus, if
     data[k] = A(i,j), then i = i[k] and p[j] <= k < p[j+1].

     For compressed row storage, ‘i’ and ‘data’ are arrays of size ‘nz’
     containing the column indices and element values, identical to the
     triplet case.  ‘p’ is an array of size ‘size1’ + 1 where ‘p[i]’
     points to the index in ‘data’ of the start of row ‘i’.  Thus, if
     data[k] = A(i,j), then j = i[k] and p[i] <= k < p[i+1].

     There are additional variables in the *note gsl_spmatrix: c1e.
     structure related to binary tree storage and memory management.
     The GSL implementation of sparse matrices uses balanced AVL trees
     to sort matrix elements in the triplet representation.  This speeds
     up element searches and duplicate detection during the matrix
     assembly process.  The *note gsl_spmatrix: c1e. structure also
     contains additional workspace variables needed for various
     operations like converting from triplet to compressed storage.
     ‘sptype’ indicates the type of storage format being used (COO, CSC
     or CSR).

     The compressed storage format defined above makes it very simple to
     interface with sophisticated external linear solver libraries which
     accept compressed storage input.  The user can simply pass the
     arrays ‘i’, ‘p’, and ‘data’ as the inputs to external libraries.


File: gsl-ref.info,  Node: Allocation,  Next: Accessing Matrix Elements,  Prev: Overview<8>,  Up: Sparse Matrices

43.4 Allocation
===============

The functions for allocating memory for a sparse matrix follow the style
of ‘malloc()’ and ‘free()’.  They also perform their own error checking.
If there is insufficient memory available to allocate a matrix then the
functions call the GSL error handler with an error code of *note
GSL_ENOMEM: 2a. in addition to returning a null pointer.

 -- Function: *note gsl_spmatrix: c1e. *gsl_spmatrix_alloc (const size_t
          n1, const size_t n2)

     This function allocates a sparse matrix of size *note n1:
     c20.-by-*note n2: c20. and initializes it to all zeros.  If the
     size of the matrix is not known at allocation time, both *note n1:
     c20. and *note n2: c20. may be set to 1, and they will
     automatically grow as elements are added to the matrix.  This
     function sets the matrix to the triplet representation, which is
     the easiest for adding and accessing matrix elements.  This
     function tries to make a reasonable guess for the number of
     non-zero elements (‘nzmax’) which will be added to the matrix by
     assuming a sparse density of 10\%.  The function *note
     gsl_spmatrix_alloc_nzmax(): c21. can be used if this number is
     known more accurately.  The workspace is of size O(nzmax).

 -- Function: *note gsl_spmatrix: c1e. *gsl_spmatrix_alloc_nzmax (const
          size_t n1, const size_t n2, const size_t nzmax, const size_t
          sptype)

     This function allocates a sparse matrix of size *note n1:
     c21.-by-*note n2: c21. and initializes it to all zeros.  If the
     size of the matrix is not known at allocation time, both *note n1:
     c21. and *note n2: c21. may be set to 1, and they will
     automatically grow as elements are added to the matrix.  The
     parameter *note nzmax: c21. specifies the maximum number of
     non-zero elements which will be added to the matrix.  It does not
     need to be precisely known in advance, since storage space will
     automatically grow using *note gsl_spmatrix_realloc(): c22. if
     *note nzmax: c21. is not large enough.  Accurate knowledge of this
     parameter reduces the number of reallocation calls required.  The
     parameter *note sptype: c21. specifies the storage format of the
     sparse matrix.  Possible values are

      -- Macro: GSL_SPMATRIX_COO

          This flag specifies coordinate (triplet) storage.

      -- Macro: GSL_SPMATRIX_CSC

          This flag specifies compressed sparse column storage.

      -- Macro: GSL_SPMATRIX_CSR

          This flag specifies compressed sparse row storage.

     The allocated *note gsl_spmatrix: c1e. structure is of size
     O(nzmax).

 -- Function: int gsl_spmatrix_realloc (const size_t nzmax, gsl_spmatrix
          *m)

     This function reallocates the storage space for *note m: c22. to
     accomodate *note nzmax: c22. non-zero elements.  It is typically
     called internally by *note gsl_spmatrix_set(): c26. if the user
     wants to add more elements to the sparse matrix than the previously
     specified *note nzmax: c22.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: void gsl_spmatrix_free (gsl_spmatrix *m)

     This function frees the memory associated with the sparse matrix
     *note m: c27.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Accessing Matrix Elements,  Next: Initializing Matrix Elements,  Prev: Allocation,  Up: Sparse Matrices

43.5 Accessing Matrix Elements
==============================

 -- Function: double gsl_spmatrix_get (const gsl_spmatrix *m, const
          size_t i, const size_t j)

     This function returns element (*note i: c29, *note j: c29.) of the
     matrix *note m: c29.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_set (gsl_spmatrix *m, const size_t i,
          const size_t j, const double x)

     This function sets element (*note i: c26, *note j: c26.) of the
     matrix *note m: c26. to the value *note x: c26.

     Input matrix formats supported: *note COO: c18.

 -- Function: double *gsl_spmatrix_ptr (gsl_spmatrix *m, const size_t i,
          const size_t j)

     This function returns a pointer to the (*note i: c2a, *note j:
     c2a.) element of the matrix *note m: c2a.  If the (*note i: c2a,
     *note j: c2a.) element is not explicitly stored in the matrix, a
     null pointer is returned.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Initializing Matrix Elements,  Next: Reading and Writing Matrices,  Prev: Accessing Matrix Elements,  Up: Sparse Matrices

43.6 Initializing Matrix Elements
=================================

Since the sparse matrix format only stores the non-zero elements, it is
automatically initialized to zero upon allocation.  The function *note
gsl_spmatrix_set_zero(): c2c. may be used to re-initialize a matrix to
zero after elements have been added to it.

 -- Function: int gsl_spmatrix_set_zero (gsl_spmatrix *m)

     This function sets (or resets) all the elements of the matrix *note
     m: c2c. to zero.  For CSC and CSR matrices, the cost of this
     operation is O(1).  For COO matrices, the binary tree structure
     must be dismantled, so the cost is O(nz).

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Reading and Writing Matrices,  Next: Copying Matrices,  Prev: Initializing Matrix Elements,  Up: Sparse Matrices

43.7 Reading and Writing Matrices
=================================

 -- Function: int gsl_spmatrix_fwrite (FILE *stream, const gsl_spmatrix
          *m)

     This function writes the elements of the matrix *note m: c2e. to
     the stream *note stream: c2e. in binary format.  The return value
     is 0 for success and ‘GSL_EFAILED’ if there was a problem writing
     to the file.  Since the data is written in the native binary format
     it may not be portable between different architectures.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_fread (FILE *stream, gsl_spmatrix *m)

     This function reads into the matrix *note m: c2f. from the open
     stream *note stream: c2f. in binary format.  The matrix *note m:
     c2f. must be preallocated with the correct storage format,
     dimensions and have a sufficiently large ‘nzmax’ in order to read
     in all matrix elements, otherwise ‘GSL_EBADLEN’ is returned.  The
     return value is 0 for success and ‘GSL_EFAILED’ if there was a
     problem reading from the file.  The data is assumed to have been
     written in the native binary format on the same architecture.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_fprintf (FILE *stream, const gsl_spmatrix
          *m, const char *format)

     This function writes the elements of the matrix *note m: c30.
     line-by-line to the stream *note stream: c30. using the format
     specifier *note format: c30, which should be one of the ‘%g’, ‘%e’
     or ‘%f’ formats for floating point numbers.  The function returns 0
     for success and ‘GSL_EFAILED’ if there was a problem writing to the
     file.  The input matrix *note m: c30. may be in any storage format,
     and the output file will be written in MatrixMarket format.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: *note gsl_spmatrix: c1e. *gsl_spmatrix_fscanf (FILE
          *stream)

     This function reads sparse matrix data in the MatrixMarket format
     from the stream *note stream: c31. and stores it in a newly
     allocated matrix which is returned in *note COO: c18. format.  The
     function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem reading from the file.  The user should free the returned
     matrix when it is no longer needed.


File: gsl-ref.info,  Node: Copying Matrices,  Next: Exchanging Rows and Columns,  Prev: Reading and Writing Matrices,  Up: Sparse Matrices

43.8 Copying Matrices
=====================

 -- Function: int gsl_spmatrix_memcpy (gsl_spmatrix *dest, const
          gsl_spmatrix *src)

     This function copies the elements of the sparse matrix *note src:
     c33. into *note dest: c33.  The two matrices must have the same
     dimensions and be in the same storage format.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Exchanging Rows and Columns,  Next: Matrix Operations,  Prev: Copying Matrices,  Up: Sparse Matrices

43.9 Exchanging Rows and Columns
================================

 -- Function: int gsl_spmatrix_transpose_memcpy (gsl_spmatrix *dest,
          const gsl_spmatrix *src)

     This function copies the transpose of the sparse matrix *note src:
     c35. into *note dest: c35.  The dimensions of *note dest: c35. must
     match the transpose of the matrix *note src: c35.  Also, both
     matrices must use the same sparse storage format.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_transpose (gsl_spmatrix *m)

     This function replaces the matrix *note m: c36. by its transpose,
     but changes the storage format for compressed matrix inputs.  Since
     compressed column storage is the transpose of compressed row
     storage, this function simply converts a CSC matrix to CSR and vice
     versa.  This is the most efficient way to transpose a compressed
     storage matrix, but the user should note that the storage format of
     their compressed matrix will change on output.  For COO matrix
     inputs, the output matrix is also in COO storage.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Matrix Operations,  Next: Matrix Properties,  Prev: Exchanging Rows and Columns,  Up: Sparse Matrices

43.10 Matrix Operations
=======================

 -- Function: int gsl_spmatrix_scale (gsl_spmatrix *m, const double x)

     This function scales all elements of the matrix *note m: c38. by
     the constant factor *note x: c38.  The result m(i,j) \leftarrow x
     m(i,j) is stored in *note m: c38.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_scale_columns (gsl_spmatrix *A, const
          gsl_vector *x)

     This function scales the columns of the M-by-N sparse matrix *note
     A: c39. by the elements of the vector *note x: c39, of length N.
     The j-th column of *note A: c39. is multiplied by ‘x[j]’.  This is
     equivalent to forming

          A \rightarrow A X

     where X = \textrm{diag}(x).

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_scale_rows (gsl_spmatrix *A, const
          gsl_vector *x)

     This function scales the rows of the M-by-N sparse matrix *note A:
     c3a. by the elements of the vector *note x: c3a, of length M. The
     i-th row of *note A: c3a. is multiplied by ‘x[i]’.  This is
     equivalent to forming

          A \rightarrow X A

     where X = \textrm{diag}(x).

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_add (gsl_spmatrix *c, const gsl_spmatrix
          *a, const gsl_spmatrix *b)

     This function computes the sum c = a + b.  The three matrices must
     have the same dimensions.

     Input matrix formats supported: *note CSC: c1a, *note CSR: c1c.

 -- Function: int gsl_spmatrix_dense_add (gsl_matrix *a, const
          gsl_spmatrix *b)

     This function adds the elements of the sparse matrix *note b: c3c.
     to the elements of the dense matrix *note a: c3c.  The result
     a(i,j) \leftarrow a(i,j) + b(i,j) is stored in *note a: c3c. and
     *note b: c3c. remains unchanged.  The two matrices must have the
     same dimensions.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_dense_sub (gsl_matrix *a, const
          gsl_spmatrix *b)

     This function subtracts the elements of the sparse matrix *note b:
     c3d. from the elements of the dense matrix *note a: c3d.  The
     result a(i,j) \leftarrow a(i,j) - b(i,j) is stored in *note a: c3d.
     and *note b: c3d. remains unchanged.  The two matrices must have
     the same dimensions.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Matrix Properties,  Next: Finding Maximum and Minimum Elements,  Prev: Matrix Operations,  Up: Sparse Matrices

43.11 Matrix Properties
=======================

 -- Function: const char *gsl_spmatrix_type (const gsl_spmatrix *m)

     This function returns a string describing the sparse storage format
     of the matrix *note m: c3f.  For example:

          printf ("matrix is '%s' format.\n", gsl_spmatrix_type (m));

     would print something like:

          matrix is 'CSR' format.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: size_t gsl_spmatrix_nnz (const gsl_spmatrix *m)

     This function returns the number of non-zero elements in *note m:
     c40.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_equal (const gsl_spmatrix *a, const
          gsl_spmatrix *b)

     This function returns 1 if the matrices *note a: c41. and *note b:
     c41. are equal (by comparison of element values) and 0 otherwise.
     The matrices *note a: c41. and *note b: c41. must be in the same
     sparse storage format for comparison.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: double gsl_spmatrix_norm1 (const gsl_spmatrix *A)

     This function returns the 1-norm of the m-by-n matrix *note A: c42,
     defined as the maximum column sum,

          ||A||_1 = \textrm{max}_{1 \le j \le n} \sum_{i=1}^m |A_{ij}|

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Finding Maximum and Minimum Elements,  Next: Compressed Format,  Prev: Matrix Properties,  Up: Sparse Matrices

43.12 Finding Maximum and Minimum Elements
==========================================

 -- Function: int gsl_spmatrix_minmax (const gsl_spmatrix *m, double
          *min_out, double *max_out)

     This function returns the minimum and maximum elements of the
     matrix *note m: c44, storing them in *note min_out: c44. and *note
     max_out: c44, and searching only the non-zero values.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.

 -- Function: int gsl_spmatrix_min_index (const gsl_spmatrix *m, size_t
          *imin, size_t *jmin)

     This function returns the indices of the minimum value in the
     matrix *note m: c45, searching only the non-zero values, and
     storing them in *note imin: c45. and *note jmin: c45.  When there
     are several equal minimum elements then the first element found is
     returned.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Compressed Format,  Next: Conversion Between Sparse and Dense Matrices,  Prev: Finding Maximum and Minimum Elements,  Up: Sparse Matrices

43.13 Compressed Format
=======================

These routines calculate a compressed matrix from a coordinate
representation.

 -- Function: int gsl_spmatrix_csc (gsl_spmatrix *dest, const
          gsl_spmatrix *src)

     This function creates a sparse matrix in *note compressed sparse
     column: c1a. format from the input sparse matrix *note src: c47.
     which must be in COO format.  The compressed matrix is stored in
     *note dest: c47.

     Input matrix formats supported: *note COO: c18.

 -- Function: int gsl_spmatrix_csr (gsl_spmatrix *dest, const
          gsl_spmatrix *src)

     This function creates a sparse matrix in *note compressed sparse
     row: c1c. format from the input sparse matrix *note src: c48. which
     must be in COO format.  The compressed matrix is stored in *note
     dest: c48.

     Input matrix formats supported: *note COO: c18.

 -- Function: *note gsl_spmatrix: c1e. *gsl_spmatrix_compress (const
          gsl_spmatrix *src, const int sptype)

     This function allocates a new sparse matrix, and stores *note src:
     c49. into it using the format specified by *note sptype: c49.  The
     input *note sptype: c49. can be one of ‘GSL_SPMATRIX_COO’,
     ‘GSL_SPMATRIX_CSC’, or ‘GSL_SPMATRIX_CSR’.  A pointer to the newly
     allocated matrix is returned, and must be freed by the caller when
     no longer needed.


File: gsl-ref.info,  Node: Conversion Between Sparse and Dense Matrices,  Next: Examples<34>,  Prev: Compressed Format,  Up: Sparse Matrices

43.14 Conversion Between Sparse and Dense Matrices
==================================================

The *note gsl_spmatrix: c1e. structure can be converted into the dense
*note gsl_matrix: 3a2. format and vice versa with the following
routines.

 -- Function: int gsl_spmatrix_d2sp (gsl_spmatrix *S, const gsl_matrix
          *A)

     This function converts the dense matrix *note A: c4b. into sparse
     COO format and stores the result in *note S: c4b.

     Input matrix formats supported: *note COO: c18.

 -- Function: int gsl_spmatrix_sp2d (gsl_matrix *A, const gsl_spmatrix
          *S)

     This function converts the sparse matrix *note S: c4c. into a dense
     matrix and stores the result in *note A: c4c.

     Input matrix formats supported: *note COO: c18, *note CSC: c1a,
     *note CSR: c1c.


File: gsl-ref.info,  Node: Examples<34>,  Next: References and Further Reading<36>,  Prev: Conversion Between Sparse and Dense Matrices,  Up: Sparse Matrices

43.15 Examples
==============

The following example program builds a 5-by-4 sparse matrix and prints
it in coordinate, compressed column, and compressed row format.  The
matrix which is constructed is

     [ 0    0  3.1  4.6 ]
     [ 1    0  7.2   0  ]
     [ 0    0   0    0  ]
     [ 2.1 2.9  0   8.5 ]
     [ 4.1  0   0    0  ]

The output of the program is:

     printing all matrix elements:
     A(0,0) = 0
     A(0,1) = 0
     A(0,2) = 3.1
     A(0,3) = 4.6
     A(1,0) = 1
     .
     .
     .
     A(4,0) = 4.1
     A(4,1) = 0
     A(4,2) = 0
     A(4,3) = 0
     matrix in triplet format (i,j,Aij):
     (0, 2, 3.1)
     (0, 3, 4.6)
     (1, 0, 1.0)
     (1, 2, 7.2)
     (3, 0, 2.1)
     (3, 1, 2.9)
     (3, 3, 8.5)
     (4, 0, 4.1)
     matrix in compressed column format:
     i = [ 1, 3, 4, 3, 0, 1, 0, 3, ]
     p = [ 0, 3, 4, 6, 8, ]
     d = [ 1, 2.1, 4.1, 2.9, 3.1, 7.2, 4.6, 8.5, ]
     matrix in compressed row format:
     i = [ 2, 3, 0, 2, 0, 1, 3, 0, ]
     p = [ 0, 2, 4, 4, 7, 8, ]
     d = [ 3.1, 4.6, 1, 7.2, 2.1, 2.9, 8.5, 4.1, ]

We see in the compressed column output, the data array stores each
column contiguously, the array i stores the row index of the
corresponding data element, and the array p stores the index of the
start of each column in the data array.  Similarly, for the compressed
row output, the data array stores each row contiguously, the array i
stores the column index of the corresponding data element, and the p
array stores the index of the start of each row in the data array.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_spmatrix.h>

     int
     main()
     {
       gsl_spmatrix *A = gsl_spmatrix_alloc(5, 4); /* triplet format */
       gsl_spmatrix *B, *C;
       size_t i, j;

       /* build the sparse matrix */
       gsl_spmatrix_set(A, 0, 2, 3.1);
       gsl_spmatrix_set(A, 0, 3, 4.6);
       gsl_spmatrix_set(A, 1, 0, 1.0);
       gsl_spmatrix_set(A, 1, 2, 7.2);
       gsl_spmatrix_set(A, 3, 0, 2.1);
       gsl_spmatrix_set(A, 3, 1, 2.9);
       gsl_spmatrix_set(A, 3, 3, 8.5);
       gsl_spmatrix_set(A, 4, 0, 4.1);

       printf("printing all matrix elements:\n");
       for (i = 0; i < 5; ++i)
         for (j = 0; j < 4; ++j)
           printf("A(%zu,%zu) = %g\n", i, j,
                  gsl_spmatrix_get(A, i, j));

       /* print out elements in triplet format */
       printf("matrix in triplet format (i,j,Aij):\n");
       gsl_spmatrix_fprintf(stdout, A, "%.1f");

       /* convert to compressed column format */
       B = gsl_spmatrix_ccs(A);

       printf("matrix in compressed column format:\n");
       printf("i = [ ");
       for (i = 0; i < B->nz; ++i)
         printf("%d, ", B->i[i]);
       printf("]\n");

       printf("p = [ ");
       for (i = 0; i < B->size2 + 1; ++i)
         printf("%d, ", B->p[i]);
       printf("]\n");

       printf("d = [ ");
       for (i = 0; i < B->nz; ++i)
         printf("%g, ", B->data[i]);
       printf("]\n");

       /* convert to compressed row format */
       C = gsl_spmatrix_crs(A);

       printf("matrix in compressed row format:\n");
       printf("i = [ ");
       for (i = 0; i < C->nz; ++i)
         printf("%d, ", C->i[i]);
       printf("]\n");

       printf("p = [ ");
       for (i = 0; i < C->size1 + 1; ++i)
         printf("%d, ", C->p[i]);
       printf("]\n");

       printf("d = [ ");
       for (i = 0; i < C->nz; ++i)
         printf("%g, ", C->data[i]);
       printf("]\n");

       gsl_spmatrix_free(A);
       gsl_spmatrix_free(B);
       gsl_spmatrix_free(C);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<36>,  Prev: Examples<34>,  Up: Sparse Matrices

43.16 References and Further Reading
====================================

The algorithms used by these functions are described in the following
sources,

   * Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.

   * CSparse software library,
     ‘https://www.cise.ufl.edu/research/sparse/CSparse’


File: gsl-ref.info,  Node: Sparse BLAS Support,  Next: Sparse Linear Algebra,  Prev: Sparse Matrices,  Up: Top

44 Sparse BLAS Support
**********************

The Sparse Basic Linear Algebra Subprograms (BLAS) define a set of
fundamental operations on vectors and sparse matrices which can be used
to create optimized higher-level linear algebra functionality.  GSL
supports a limited number of BLAS operations for sparse matrices.

The header file ‘gsl_spblas.h’ contains the prototypes for the sparse
BLAS functions and related declarations.

* Menu:

* Sparse BLAS operations::
* References and Further Reading: References and Further Reading<37>.


File: gsl-ref.info,  Node: Sparse BLAS operations,  Next: References and Further Reading<37>,  Up: Sparse BLAS Support

44.1 Sparse BLAS operations
===========================

 -- Function: int gsl_spblas_dgemv (const CBLAS_TRANSPOSE_t TransA,
          const double alpha, const gsl_spmatrix *A, const gsl_vector
          *x, const double beta, gsl_vector *y)

     This function computes the matrix-vector product and sum y
     \leftarrow \alpha op(A) x + \beta y, where op(A) = A, A^T for *note
     TransA: c52. = ‘CblasNoTrans’, ‘CblasTrans’.  In-place computations
     are not supported, so *note x: c52. and *note y: c52. must be
     distinct vectors.  The matrix *note A: c52. may be in triplet or
     compressed format.

 -- Function: int gsl_spblas_dgemm (const double alpha, const
          gsl_spmatrix *A, const gsl_spmatrix *B, gsl_spmatrix *C)

     This function computes the sparse matrix-matrix product C = \alpha
     A B. The matrices must be in compressed format.


File: gsl-ref.info,  Node: References and Further Reading<37>,  Prev: Sparse BLAS operations,  Up: Sparse BLAS Support

44.2 References and Further Reading
===================================

The algorithms used by these functions are described in the following
sources:

   * Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.

   * CSparse software library,
     ‘https://www.cise.ufl.edu/research/sparse/CSparse’


File: gsl-ref.info,  Node: Sparse Linear Algebra,  Next: Physical Constants,  Prev: Sparse BLAS Support,  Up: Top

45 Sparse Linear Algebra
************************

This chapter describes functions for solving sparse linear systems of
equations.  The library provides linear algebra routines which operate
directly on the *note gsl_spmatrix: c1e. and *note gsl_vector: 35f.
objects.

The functions described in this chapter are declared in the header file
‘gsl_splinalg.h’.

* Menu:

* Overview: Overview<9>.
* Sparse Iterative Solvers::
* Examples: Examples<35>.
* References and Further Reading: References and Further Reading<38>.


File: gsl-ref.info,  Node: Overview<9>,  Next: Sparse Iterative Solvers,  Up: Sparse Linear Algebra

45.1 Overview
=============

This chapter is primarily concerned with the solution of the linear
system

     A x = b

where A is a general square n-by-n non-singular sparse matrix, x is an
unknown n-by-1 vector, and b is a given n-by-1 right hand side vector.
There exist many methods for solving such sparse linear systems, which
broadly fall into either direct or iterative categories.  Direct methods
include LU and QR decompositions, while iterative methods start with an
initial guess for the vector x and update the guess through iteration
until convergence.  GSL does not currently provide any direct sparse
solvers.


File: gsl-ref.info,  Node: Sparse Iterative Solvers,  Next: Examples<35>,  Prev: Overview<9>,  Up: Sparse Linear Algebra

45.2 Sparse Iterative Solvers
=============================

* Menu:

* Overview: Overview<10>.
* Types of Sparse Iterative Solvers::
* Iterating the Sparse Linear System::


File: gsl-ref.info,  Node: Overview<10>,  Next: Types of Sparse Iterative Solvers,  Up: Sparse Iterative Solvers

45.2.1 Overview
---------------

Many practical iterative methods of solving large n-by-n sparse linear
systems involve projecting an approximate solution for ‘x’ onto a
subspace of {\bf R}^n.  If we define a m-dimensional subspace {\cal K}
as the subspace of approximations to the solution ‘x’, then m
constraints must be imposed to determine the next approximation.  These
m constraints define another m-dimensional subspace denoted by {\cal L}.
The subspace dimension m is typically chosen to be much smaller than n
in order to reduce the computational effort needed to generate the next
approximate solution vector.  The many iterative algorithms which exist
differ mainly in their choice of {\cal K} and {\cal L}.


File: gsl-ref.info,  Node: Types of Sparse Iterative Solvers,  Next: Iterating the Sparse Linear System,  Prev: Overview<10>,  Up: Sparse Iterative Solvers

45.2.2 Types of Sparse Iterative Solvers
----------------------------------------

The sparse linear algebra library provides the following types of
iterative solvers:

 -- Type: gsl_splinalg_itersolve_type

      -- Variable: *note gsl_splinalg_itersolve_type: c5b.
               *gsl_splinalg_itersolve_gmres

          This specifies the Generalized Minimum Residual Method
          (GMRES). This is a projection method using {\cal K} = {\cal
          K}_m and {\cal L} = A {\cal K}_m where {\cal K}_m is the m-th
          Krylov subspace

               K_m = span( r_0, A r_0, ..., A^(m-1) r_0)

          and r_0 = b - A x_0 is the residual vector of the initial
          guess x_0.  If m is set equal to n, then the Krylov subspace
          is {\bf R}^n and GMRES will provide the exact solution ‘x’.
          However, the goal is for the method to arrive at a very good
          approximation to ‘x’ using a much smaller subspace {\cal K}_m.
          By default, the GMRES method selects m = MIN(n,10) but the
          user may specify a different value for m.  The GMRES storage
          requirements grow as O(n(m+1)) and the number of flops grow as
          O(4 m^2 n - 4 m^3 / 3).

          In the below function *note gsl_splinalg_itersolve_iterate():
          c5d, one GMRES iteration is defined as projecting the
          approximate solution vector onto each Krylov subspace {\cal
          K}_1, ..., {\cal K}_m, and so m can be kept smaller by
          “restarting” the method and calling *note
          gsl_splinalg_itersolve_iterate(): c5d. multiple times,
          providing the updated approximation ‘x’ to each new call.  If
          the method is not adequately converging, the user may try
          increasing the parameter m.

          GMRES is considered a robust general purpose iterative solver,
          however there are cases where the method stagnates if the
          matrix is not positive-definite and fails to reduce the
          residual until the very last projection onto the subspace
          {\cal K}_n = {\bf R}^n.  In these cases, preconditioning the
          linear system can help, but GSL does not currently provide any
          preconditioners.


File: gsl-ref.info,  Node: Iterating the Sparse Linear System,  Prev: Types of Sparse Iterative Solvers,  Up: Sparse Iterative Solvers

45.2.3 Iterating the Sparse Linear System
-----------------------------------------

The following functions are provided to allocate storage for the sparse
linear solvers and iterate the system to a solution.

 -- Function: gsl_splinalg_itersolve *gsl_splinalg_itersolve_alloc
          (const gsl_splinalg_itersolve_type *T, const size_t n, const
          size_t m)

     This function allocates a workspace for the iterative solution of
     *note n: c5f.-by-*note n: c5f. sparse matrix systems.  The
     iterative solver type is specified by *note T: c5f.  The argument
     *note m: c5f. specifies the size of the solution candidate subspace
     {\cal K}_m.  The dimension *note m: c5f. may be set to 0 in which
     case a reasonable default value is used.

 -- Function: void gsl_splinalg_itersolve_free (gsl_splinalg_itersolve
          *w)

     This function frees the memory associated with the workspace *note
     w: c60.

 -- Function: const char *gsl_splinalg_itersolve_name (const
          gsl_splinalg_itersolve *w)

     This function returns a string pointer to the name of the solver.

 -- Function: int gsl_splinalg_itersolve_iterate (const gsl_spmatrix *A,
          const gsl_vector *b, const double tol, gsl_vector *x,
          gsl_splinalg_itersolve *w)

     This function performs one iteration of the iterative method for
     the sparse linear system specfied by the matrix *note A: c5d, right
     hand side vector *note b: c5d. and solution vector *note x: c5d.
     On input, *note x: c5d. must be set to an initial guess for the
     solution.  On output, *note x: c5d. is updated to give the current
     solution estimate.  The parameter *note tol: c5d. specifies the
     relative tolerance between the residual norm and norm of *note b:
     c5d. in order to check for convergence.  When the following
     condition is satisfied:

          || A x - b || <= tol * || b ||

     the method has converged, the function returns ‘GSL_SUCCESS’ and
     the final solution is provided in *note x: c5d.  Otherwise, the
     function returns ‘GSL_CONTINUE’ to signal that more iterations are
     required.  Here, || \cdot || represents the Euclidean norm.  The
     input matrix *note A: c5d. may be in triplet or compressed format.

 -- Function: double gsl_splinalg_itersolve_normr (const
          gsl_splinalg_itersolve *w)

     This function returns the current residual norm ||r|| = ||A x -
     b||, which is updated after each call to *note
     gsl_splinalg_itersolve_iterate(): c5d.


File: gsl-ref.info,  Node: Examples<35>,  Next: References and Further Reading<38>,  Prev: Sparse Iterative Solvers,  Up: Sparse Linear Algebra

45.3 Examples
=============

This example program demonstrates the sparse linear algebra routines on
the solution of a simple 1D Poisson equation on [0,1]:

     u''(x) = f(x) = -\pi^2 \sin(\pi x)

with boundary conditions u(0) = u(1) = 0.  The analytic solution of this
simple problem is u(x) = \sin{\pi x}.  We will solve this problem by
finite differencing the left hand side to give

     1/h^2 ( u_(i+1) - 2 u_i + u_(i-1) ) = f_i

Defining a grid of N points, h = 1/(N-1).  In the finite difference
equation above, u_0 = u_{N-1} = 0 are known from the boundary
conditions, so we will only put the equations for i = 1, ..., N-2 into
the matrix system.  The resulting (N-2) \times (N-2) matrix equation is

An example program which constructs and solves this system is given
below.  The system is solved using the iterative GMRES solver.  Here is
the output of the program:

     iter 0 residual = 4.297275996844e-11
     Converged

showing that the method converged in a single iteration.  The calculated
solution is shown in Fig.  %s.

[gsl-ref-figures/sparse_poisson]
Figure: Solution of PDE

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_spmatrix.h>
     #include <gsl/gsl_splinalg.h>

     int
     main()
     {
       const size_t N = 100;                       /* number of grid points */
       const size_t n = N - 2;                     /* subtract 2 to exclude boundaries */
       const double h = 1.0 / (N - 1.0);           /* grid spacing */
       gsl_spmatrix *A = gsl_spmatrix_alloc(n ,n); /* triplet format */
       gsl_spmatrix *C;                            /* compressed format */
       gsl_vector *f = gsl_vector_alloc(n);        /* right hand side vector */
       gsl_vector *u = gsl_vector_alloc(n);        /* solution vector */
       size_t i;

       /* construct the sparse matrix for the finite difference equation */

       /* construct first row */
       gsl_spmatrix_set(A, 0, 0, -2.0);
       gsl_spmatrix_set(A, 0, 1, 1.0);

       /* construct rows [1:n-2] */
       for (i = 1; i < n - 1; ++i)
         {
           gsl_spmatrix_set(A, i, i + 1, 1.0);
           gsl_spmatrix_set(A, i, i, -2.0);
           gsl_spmatrix_set(A, i, i - 1, 1.0);
         }

       /* construct last row */
       gsl_spmatrix_set(A, n - 1, n - 1, -2.0);
       gsl_spmatrix_set(A, n - 1, n - 2, 1.0);

       /* scale by h^2 */
       gsl_spmatrix_scale(A, 1.0 / (h * h));

       /* construct right hand side vector */
       for (i = 0; i < n; ++i)
         {
           double xi = (i + 1) * h;
           double fi = -M_PI * M_PI * sin(M_PI * xi);
           gsl_vector_set(f, i, fi);
         }

       /* convert to compressed column format */
       C = gsl_spmatrix_ccs(A);

       /* now solve the system with the GMRES iterative solver */
       {
         const double tol = 1.0e-6;  /* solution relative tolerance */
         const size_t max_iter = 10; /* maximum iterations */
         const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
         gsl_splinalg_itersolve *work =
           gsl_splinalg_itersolve_alloc(T, n, 0);
         size_t iter = 0;
         double residual;
         int status;

         /* initial guess u = 0 */
         gsl_vector_set_zero(u);

         /* solve the system A u = f */
         do
           {
             status = gsl_splinalg_itersolve_iterate(C, f, tol, u, work);

             /* print out residual norm ||A*u - f|| */
             residual = gsl_splinalg_itersolve_normr(work);
             fprintf(stderr, "iter %zu residual = %.12e\n", iter, residual);

             if (status == GSL_SUCCESS)
               fprintf(stderr, "Converged\n");
           }
         while (status == GSL_CONTINUE && ++iter < max_iter);

         /* output solution */
         for (i = 0; i < n; ++i)
           {
             double xi = (i + 1) * h;
             double u_exact = sin(M_PI * xi);
             double u_gsl = gsl_vector_get(u, i);

             printf("%f %.12e %.12e\n", xi, u_gsl, u_exact);
           }

         gsl_splinalg_itersolve_free(work);
       }

       gsl_spmatrix_free(A);
       gsl_spmatrix_free(C);
       gsl_vector_free(f);
       gsl_vector_free(u);

       return 0;
     } /* main() */


File: gsl-ref.info,  Node: References and Further Reading<38>,  Prev: Examples<35>,  Up: Sparse Linear Algebra

45.4 References and Further Reading
===================================

The implementation of the GMRES iterative solver closely follows the
publications

   * H. F. Walker, Implementation of the GMRES method using Householder
     transformations, SIAM J. Sci.  Stat.  Comput.  9(1), 1988.

   * Y. Saad, Iterative methods for sparse linear systems, 2nd edition,
     SIAM, 2003.


File: gsl-ref.info,  Node: Physical Constants,  Next: IEEE floating-point arithmetic,  Prev: Sparse Linear Algebra,  Up: Top

46 Physical Constants
*********************

This chapter describes macros for the values of physical constants, such
as the speed of light, c, and gravitational constant, G. The values are
available in different unit systems, including the standard MKSA system
(meters, kilograms, seconds, amperes) and the CGSM system (centimeters,
grams, seconds, gauss), which is commonly used in Astronomy.

The definitions of constants in the MKSA system are available in the
file ‘gsl_const_mksa.h’.  The constants in the CGSM system are defined
in ‘gsl_const_cgsm.h’.  Dimensionless constants, such as the fine
structure constant, which are pure numbers are defined in
‘gsl_const_num.h’.

The full list of constants is described briefly below.  Consult the
header files themselves for the values of the constants used in the
library.

* Menu:

* Fundamental Constants::
* Astronomy and Astrophysics::
* Atomic and Nuclear Physics::
* Measurement of Time::
* Imperial Units::
* Speed and Nautical Units::
* Printers Units::
* Volume, Area and Length: Volume Area and Length.
* Mass and Weight::
* Thermal Energy and Power::
* Pressure::
* Viscosity::
* Light and Illumination::
* Radioactivity::
* Force and Energy::
* Prefixes::
* Examples: Examples<36>.
* References and Further Reading: References and Further Reading<39>.


File: gsl-ref.info,  Node: Fundamental Constants,  Next: Astronomy and Astrophysics,  Up: Physical Constants

46.1 Fundamental Constants
==========================

 -- Macro: GSL_CONST_MKSA_SPEED_OF_LIGHT

     The speed of light in vacuum, c.

 -- Macro: GSL_CONST_MKSA_VACUUM_PERMEABILITY

     The permeability of free space, \mu_0.  This constant is defined in
     the MKSA system only.

 -- Macro: GSL_CONST_MKSA_VACUUM_PERMITTIVITY

     The permittivity of free space, \epsilon_0.  This constant is
     defined in the MKSA system only.

 -- Macro: GSL_CONST_MKSA_PLANCKS_CONSTANT_H

     Planck’s constant, h.

 -- Macro: GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR

     Planck’s constant divided by 2\pi, \hbar.

 -- Macro: GSL_CONST_NUM_AVOGADRO

     Avogadro’s number, N_a.

 -- Macro: GSL_CONST_MKSA_FARADAY

     The molar charge of 1 Faraday.

 -- Macro: GSL_CONST_MKSA_BOLTZMANN

     The Boltzmann constant, k.

 -- Macro: GSL_CONST_MKSA_MOLAR_GAS

     The molar gas constant, R_0.

 -- Macro: GSL_CONST_MKSA_STANDARD_GAS_VOLUME

     The standard gas volume, V_0.

 -- Macro: GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT

     The Stefan-Boltzmann radiation constant, \sigma.

 -- Macro: GSL_CONST_MKSA_GAUSS

     The magnetic field of 1 Gauss.


File: gsl-ref.info,  Node: Astronomy and Astrophysics,  Next: Atomic and Nuclear Physics,  Prev: Fundamental Constants,  Up: Physical Constants

46.2 Astronomy and Astrophysics
===============================

 -- Macro: GSL_CONST_MKSA_ASTRONOMICAL_UNIT

     The length of 1 astronomical unit (mean earth-sun distance), au.

 -- Macro: GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT

     The gravitational constant, G.

 -- Macro: GSL_CONST_MKSA_LIGHT_YEAR

     The distance of 1 light-year, ly.

 -- Macro: GSL_CONST_MKSA_PARSEC

     The distance of 1 parsec, pc.

 -- Macro: GSL_CONST_MKSA_GRAV_ACCEL

     The standard gravitational acceleration on Earth, g.

 -- Macro: GSL_CONST_MKSA_SOLAR_MASS

     The mass of the Sun.


File: gsl-ref.info,  Node: Atomic and Nuclear Physics,  Next: Measurement of Time,  Prev: Astronomy and Astrophysics,  Up: Physical Constants

46.3 Atomic and Nuclear Physics
===============================

 -- Macro: GSL_CONST_MKSA_ELECTRON_CHARGE

     The charge of the electron, e.

 -- Macro: GSL_CONST_MKSA_ELECTRON_VOLT

     The energy of 1 electron volt, eV.

 -- Macro: GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS

     The unified atomic mass, amu.

 -- Macro: GSL_CONST_MKSA_MASS_ELECTRON

     The mass of the electron, m_e.

 -- Macro: GSL_CONST_MKSA_MASS_MUON

     The mass of the muon, m_\mu.

 -- Macro: GSL_CONST_MKSA_MASS_PROTON

     The mass of the proton, m_p.

 -- Macro: GSL_CONST_MKSA_MASS_NEUTRON

     The mass of the neutron, m_n.

 -- Macro: GSL_CONST_NUM_FINE_STRUCTURE

     The electromagnetic fine structure constant \alpha.

 -- Macro: GSL_CONST_MKSA_RYDBERG

     The Rydberg constant, Ry, in units of energy.  This is related to
     the Rydberg inverse wavelength R_\infty by Ry = h c R_\infty.

 -- Macro: GSL_CONST_MKSA_BOHR_RADIUS

     The Bohr radius, a_0.

 -- Macro: GSL_CONST_MKSA_ANGSTROM

     The length of 1 angstrom.

 -- Macro: GSL_CONST_MKSA_BARN

     The area of 1 barn.

 -- Macro: GSL_CONST_MKSA_BOHR_MAGNETON

     The Bohr Magneton, \mu_B.

 -- Macro: GSL_CONST_MKSA_NUCLEAR_MAGNETON

     The Nuclear Magneton, \mu_N.

 -- Macro: GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT

     The absolute value of the magnetic moment of the electron, \mu_e.
     The physical magnetic moment of the electron is negative.

 -- Macro: GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT

     The magnetic moment of the proton, \mu_p.

 -- Macro: GSL_CONST_MKSA_THOMSON_CROSS_SECTION

     The Thomson cross section, \sigma_T.

 -- Macro: GSL_CONST_MKSA_DEBYE

     The electric dipole moment of 1 Debye, D.


File: gsl-ref.info,  Node: Measurement of Time,  Next: Imperial Units,  Prev: Atomic and Nuclear Physics,  Up: Physical Constants

46.4 Measurement of Time
========================

 -- Macro: GSL_CONST_MKSA_MINUTE

     The number of seconds in 1 minute.

 -- Macro: GSL_CONST_MKSA_HOUR

     The number of seconds in 1 hour.

 -- Macro: GSL_CONST_MKSA_DAY

     The number of seconds in 1 day.

 -- Macro: GSL_CONST_MKSA_WEEK

     The number of seconds in 1 week.


File: gsl-ref.info,  Node: Imperial Units,  Next: Speed and Nautical Units,  Prev: Measurement of Time,  Up: Physical Constants

46.5 Imperial Units
===================

 -- Macro: GSL_CONST_MKSA_INCH

     The length of 1 inch.

 -- Macro: GSL_CONST_MKSA_FOOT

     The length of 1 foot.

 -- Macro: GSL_CONST_MKSA_YARD

     The length of 1 yard.

 -- Macro: GSL_CONST_MKSA_MILE

     The length of 1 mile.

 -- Macro: GSL_CONST_MKSA_MIL

     The length of 1 mil (1/1000th of an inch).


File: gsl-ref.info,  Node: Speed and Nautical Units,  Next: Printers Units,  Prev: Imperial Units,  Up: Physical Constants

46.6 Speed and Nautical Units
=============================

 -- Macro: GSL_CONST_MKSA_KILOMETERS_PER_HOUR

     The speed of 1 kilometer per hour.

 -- Macro: GSL_CONST_MKSA_MILES_PER_HOUR

     The speed of 1 mile per hour.

 -- Macro: GSL_CONST_MKSA_NAUTICAL_MILE

     The length of 1 nautical mile.

 -- Macro: GSL_CONST_MKSA_FATHOM

     The length of 1 fathom.

 -- Macro: GSL_CONST_MKSA_KNOT

     The speed of 1 knot.


File: gsl-ref.info,  Node: Printers Units,  Next: Volume Area and Length,  Prev: Speed and Nautical Units,  Up: Physical Constants

46.7 Printers Units
===================

 -- Macro: GSL_CONST_MKSA_POINT

     The length of 1 printer’s point (1/72 inch).

 -- Macro: GSL_CONST_MKSA_TEXPOINT

     The length of 1 TeX point (1/72.27 inch).


File: gsl-ref.info,  Node: Volume Area and Length,  Next: Mass and Weight,  Prev: Printers Units,  Up: Physical Constants

46.8 Volume, Area and Length
============================

 -- Macro: GSL_CONST_MKSA_MICRON

     The length of 1 micron.

 -- Macro: GSL_CONST_MKSA_HECTARE

     The area of 1 hectare.

 -- Macro: GSL_CONST_MKSA_ACRE

     The area of 1 acre.

 -- Macro: GSL_CONST_MKSA_LITER

     The volume of 1 liter.

 -- Macro: GSL_CONST_MKSA_US_GALLON

     The volume of 1 US gallon.

 -- Macro: GSL_CONST_MKSA_CANADIAN_GALLON

     The volume of 1 Canadian gallon.

 -- Macro: GSL_CONST_MKSA_UK_GALLON

     The volume of 1 UK gallon.

 -- Macro: GSL_CONST_MKSA_QUART

     The volume of 1 quart.

 -- Macro: GSL_CONST_MKSA_PINT

     The volume of 1 pint.


File: gsl-ref.info,  Node: Mass and Weight,  Next: Thermal Energy and Power,  Prev: Volume Area and Length,  Up: Physical Constants

46.9 Mass and Weight
====================

 -- Macro: GSL_CONST_MKSA_POUND_MASS

     The mass of 1 pound.

 -- Macro: GSL_CONST_MKSA_OUNCE_MASS

     The mass of 1 ounce.

 -- Macro: GSL_CONST_MKSA_TON

     The mass of 1 ton.

 -- Macro: GSL_CONST_MKSA_METRIC_TON

     The mass of 1 metric ton (1000 kg).

 -- Macro: GSL_CONST_MKSA_UK_TON

     The mass of 1 UK ton.

 -- Macro: GSL_CONST_MKSA_TROY_OUNCE

     The mass of 1 troy ounce.

 -- Macro: GSL_CONST_MKSA_CARAT

     The mass of 1 carat.

 -- Macro: GSL_CONST_MKSA_GRAM_FORCE

     The force of 1 gram weight.

 -- Macro: GSL_CONST_MKSA_POUND_FORCE

     The force of 1 pound weight.

 -- Macro: GSL_CONST_MKSA_KILOPOUND_FORCE

     The force of 1 kilopound weight.

 -- Macro: GSL_CONST_MKSA_POUNDAL

     The force of 1 poundal.


File: gsl-ref.info,  Node: Thermal Energy and Power,  Next: Pressure,  Prev: Mass and Weight,  Up: Physical Constants

46.10 Thermal Energy and Power
==============================

 -- Macro: GSL_CONST_MKSA_CALORIE

     The energy of 1 calorie.

 -- Macro: GSL_CONST_MKSA_BTU

     The energy of 1 British Thermal Unit, btu.

 -- Macro: GSL_CONST_MKSA_THERM

     The energy of 1 Therm.

 -- Macro: GSL_CONST_MKSA_HORSEPOWER

     The power of 1 horsepower.


File: gsl-ref.info,  Node: Pressure,  Next: Viscosity,  Prev: Thermal Energy and Power,  Up: Physical Constants

46.11 Pressure
==============

 -- Macro: GSL_CONST_MKSA_BAR

     The pressure of 1 bar.

 -- Macro: GSL_CONST_MKSA_STD_ATMOSPHERE

     The pressure of 1 standard atmosphere.

 -- Macro: GSL_CONST_MKSA_TORR

     The pressure of 1 torr.

 -- Macro: GSL_CONST_MKSA_METER_OF_MERCURY

     The pressure of 1 meter of mercury.

 -- Macro: GSL_CONST_MKSA_INCH_OF_MERCURY

     The pressure of 1 inch of mercury.

 -- Macro: GSL_CONST_MKSA_INCH_OF_WATER

     The pressure of 1 inch of water.

 -- Macro: GSL_CONST_MKSA_PSI

     The pressure of 1 pound per square inch.


File: gsl-ref.info,  Node: Viscosity,  Next: Light and Illumination,  Prev: Pressure,  Up: Physical Constants

46.12 Viscosity
===============

 -- Macro: GSL_CONST_MKSA_POISE

     The dynamic viscosity of 1 poise.

 -- Macro: GSL_CONST_MKSA_STOKES

     The kinematic viscosity of 1 stokes.


File: gsl-ref.info,  Node: Light and Illumination,  Next: Radioactivity,  Prev: Viscosity,  Up: Physical Constants

46.13 Light and Illumination
============================

 -- Macro: GSL_CONST_MKSA_STILB

     The luminance of 1 stilb.

 -- Macro: GSL_CONST_MKSA_LUMEN

     The luminous flux of 1 lumen.

 -- Macro: GSL_CONST_MKSA_LUX

     The illuminance of 1 lux.

 -- Macro: GSL_CONST_MKSA_PHOT

     The illuminance of 1 phot.

 -- Macro: GSL_CONST_MKSA_FOOTCANDLE

     The illuminance of 1 footcandle.

 -- Macro: GSL_CONST_MKSA_LAMBERT

     The luminance of 1 lambert.

 -- Macro: GSL_CONST_MKSA_FOOTLAMBERT

     The luminance of 1 footlambert.


File: gsl-ref.info,  Node: Radioactivity,  Next: Force and Energy,  Prev: Light and Illumination,  Up: Physical Constants

46.14 Radioactivity
===================

 -- Macro: GSL_CONST_MKSA_CURIE

     The activity of 1 curie.

 -- Macro: GSL_CONST_MKSA_ROENTGEN

     The exposure of 1 roentgen.

 -- Macro: GSL_CONST_MKSA_RAD

     The absorbed dose of 1 rad.


File: gsl-ref.info,  Node: Force and Energy,  Next: Prefixes,  Prev: Radioactivity,  Up: Physical Constants

46.15 Force and Energy
======================

 -- Macro: GSL_CONST_MKSA_NEWTON

     The SI unit of force, 1 Newton.

 -- Macro: GSL_CONST_MKSA_DYNE

     The force of 1 Dyne = 10^{-5} Newton.

 -- Macro: GSL_CONST_MKSA_JOULE

     The SI unit of energy, 1 Joule.

 -- Macro: GSL_CONST_MKSA_ERG

     The energy 1 erg = 10^{-7} Joule.


File: gsl-ref.info,  Node: Prefixes,  Next: Examples<36>,  Prev: Force and Energy,  Up: Physical Constants

46.16 Prefixes
==============

These constants are dimensionless scaling factors.

 -- Macro: GSL_CONST_NUM_YOTTA

     10^{24}

 -- Macro: GSL_CONST_NUM_ZETTA

     10^{21}

 -- Macro: GSL_CONST_NUM_EXA

     10^{18}

 -- Macro: GSL_CONST_NUM_PETA

     10^{15}

 -- Macro: GSL_CONST_NUM_TERA

     10^{12}

 -- Macro: GSL_CONST_NUM_GIGA

     10^9

 -- Macro: GSL_CONST_NUM_MEGA

     10^6

 -- Macro: GSL_CONST_NUM_KILO

     10^3

 -- Macro: GSL_CONST_NUM_MILLI

     10^{-3}

 -- Macro: GSL_CONST_NUM_MICRO

     10^{-6}

 -- Macro: GSL_CONST_NUM_NANO

     10^{-9}

 -- Macro: GSL_CONST_NUM_PICO

     10^{-12}

 -- Macro: GSL_CONST_NUM_FEMTO

     10^{-15}

 -- Macro: GSL_CONST_NUM_ATTO

     10^{-18}

 -- Macro: GSL_CONST_NUM_ZEPTO

     10^{-21}

 -- Macro: GSL_CONST_NUM_YOCTO

     10^{-24}


File: gsl-ref.info,  Node: Examples<36>,  Next: References and Further Reading<39>,  Prev: Prefixes,  Up: Physical Constants

46.17 Examples
==============

The following program demonstrates the use of the physical constants in
a calculation.  In this case, the goal is to calculate the range of
light-travel times from Earth to Mars.

The required data is the average distance of each planet from the Sun in
astronomical units (the eccentricities and inclinations of the orbits
will be neglected for the purposes of this calculation).  The average
radius of the orbit of Mars is 1.52 astronomical units, and for the
orbit of Earth it is 1 astronomical unit (by definition).  These values
are combined with the MKSA values of the constants for the speed of
light and the length of an astronomical unit to produce a result for the
shortest and longest light-travel times in seconds.  The figures are
converted into minutes before being displayed.

     #include <stdio.h>
     #include <gsl/gsl_const_mksa.h>

     int
     main (void)
     {
       double c  = GSL_CONST_MKSA_SPEED_OF_LIGHT;
       double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
       double minutes = GSL_CONST_MKSA_MINUTE;

       /* distance stored in meters */
       double r_earth = 1.00 * au;
       double r_mars  = 1.52 * au;

       double t_min, t_max;

       t_min = (r_mars - r_earth) / c;
       t_max = (r_mars + r_earth) / c;

       printf ("light travel time from Earth to Mars:\n");
       printf ("minimum = %.1f minutes\n", t_min / minutes);
       printf ("maximum = %.1f minutes\n", t_max / minutes);

       return 0;
     }

Here is the output from the program,

     light travel time from Earth to Mars:
     minimum = 4.3 minutes
     maximum = 21.0 minutes


File: gsl-ref.info,  Node: References and Further Reading<39>,  Prev: Examples<36>,  Up: Physical Constants

46.18 References and Further Reading
====================================

The authoritative sources for physical constants are the 2006 CODATA
recommended values, published in the article below.  Further information
on the values of physical constants is also available from the NIST
website.

   * P.J. Mohr, B.N. Taylor, D.B. Newell, “CODATA Recommended Values of
     the Fundamental Physical Constants: 2006”, Reviews of Modern
     Physics, 80(2), pp.  633–730 (2008).

   * ‘http://www.physics.nist.gov/cuu/Constants/index.html’

   * ‘http://physics.nist.gov/Pubs/SP811/appenB9.html’


File: gsl-ref.info,  Node: IEEE floating-point arithmetic,  Next: Debugging Numerical Programs,  Prev: Physical Constants,  Up: Top

47 IEEE floating-point arithmetic
*********************************

This chapter describes functions for examining the representation of
floating point numbers and controlling the floating point environment of
your program.  The functions described in this chapter are declared in
the header file ‘gsl_ieee_utils.h’.

* Menu:

* Representation of floating point numbers::
* Setting up your IEEE environment::
* References and Further Reading: References and Further Reading<40>.


File: gsl-ref.info,  Node: Representation of floating point numbers,  Next: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic

47.1 Representation of floating point numbers
=============================================

The IEEE Standard for Binary Floating-Point Arithmetic defines binary
formats for single and double precision numbers.  Each number is
composed of three parts: a `sign bit' (s), an `exponent' (E) and a
`fraction' (f).  The numerical value of the combination (s,E,f) is given
by the following formula,

     (-1)^s (1.fffff...) 2^E

The sign bit is either zero or one.  The exponent ranges from a minimum
value E_{min} to a maximum value E_{max} depending on the precision.
The exponent is converted to an unsigned number e, known as the `biased
exponent', for storage by adding a `bias' parameter,

     e = E + bias

The sequence fffff... represents the digits of the binary fraction f.
The binary digits are stored in `normalized form', by adjusting the
exponent to give a leading digit of 1.  Since the leading digit is
always 1 for normalized numbers it is assumed implicitly and does not
have to be stored.  Numbers smaller than 2^{E_{min}} are be stored in
`denormalized form' with a leading zero,

     (-1)^s (0.fffff...) 2^(E_min)

This allows gradual underflow down to 2^{E_{min} - p} for p bits of
precision.  A zero is encoded with the special exponent of 2^{E_{min}-1}
and infinities with the exponent of 2^{E_{max}+1}.

The format for single precision numbers uses 32 bits divided in the
following way:

     seeeeeeeefffffffffffffffffffffff

     s = sign bit, 1 bit
     e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
     f = fraction, 23 bits

The format for double precision numbers uses 64 bits divided in the
following way:

     seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff

     s = sign bit, 1 bit
     e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
     f = fraction, 52 bits

It is often useful to be able to investigate the behavior of a
calculation at the bit-level and the library provides functions for
printing the IEEE representations in a human-readable form.

 -- Function: void gsl_ieee_fprintf_float (FILE *stream, const float *x)
 -- Function: void gsl_ieee_fprintf_double (FILE *stream, const double
          *x)

     These functions output a formatted version of the IEEE
     floating-point number pointed to by *note x: cf2. to the stream
     *note stream: cf2.  A pointer is used to pass the number
     indirectly, to avoid any undesired promotion from ‘float’ to
     ‘double’.  The output takes one of the following forms,

     ‘NaN’

          the Not-a-Number symbol

     ‘Inf, -Inf’

          positive or negative infinity

     ‘1.fffff...*2^E, -1.fffff...*2^E’

          a normalized floating point number

     ‘0.fffff...*2^E, -0.fffff...*2^E’

          a denormalized floating point number

     ‘0, -0’

          positive or negative zero

     The output can be used directly in GNU Emacs Calc mode by preceding
     it with ‘2#’ to indicate binary.

 -- Function: void gsl_ieee_printf_float (const float *x)
 -- Function: void gsl_ieee_printf_double (const double *x)

     These functions output a formatted version of the IEEE
     floating-point number pointed to by *note x: cf4. to the stream
     ‘stdout’.

The following program demonstrates the use of the functions by printing
the single and double precision representations of the fraction 1/3.
For comparison the representation of the value promoted from single to
double precision is also printed.

     #include <stdio.h>
     #include <gsl/gsl_ieee_utils.h>

     int
     main (void)
     {
       float f = 1.0/3.0;
       double d = 1.0/3.0;

       double fd = f; /* promote from float to double */

       printf (" f="); gsl_ieee_printf_float(&f);
       printf ("\n");

       printf ("fd="); gsl_ieee_printf_double(&fd);
       printf ("\n");

       printf (" d="); gsl_ieee_printf_double(&d);
       printf ("\n");

       return 0;
     }

The binary representation of 1/3 is 0.01010101....  The output below
shows that the IEEE format normalizes this fraction to give a leading
digit of 1:

      f= 1.01010101010101010101011*2^-2
     fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
      d= 1.0101010101010101010101010101010101010101010101010101*2^-2

The output also shows that a single-precision number is promoted to
double-precision by adding zeros in the binary representation.


File: gsl-ref.info,  Node: Setting up your IEEE environment,  Next: References and Further Reading<40>,  Prev: Representation of floating point numbers,  Up: IEEE floating-point arithmetic

47.2 Setting up your IEEE environment
=====================================

The IEEE standard defines several `modes' for controlling the behavior
of floating point operations.  These modes specify the important
properties of computer arithmetic: the direction used for rounding (e.g.
whether numbers should be rounded up, down or to the nearest number),
the rounding precision and how the program should handle arithmetic
exceptions, such as division by zero.

Many of these features can now be controlled via standard functions such
as ‘fpsetround()’, which should be used whenever they are available.
Unfortunately in the past there has been no universal API for
controlling their behavior—each system has had its own low-level way of
accessing them.  To help you write portable programs GSL allows you to
specify modes in a platform-independent way using the environment
variable *note GSL_IEEE_MODE: cf6.  The library then takes care of all
the necessary machine-specific initializations for you when you call the
function *note gsl_ieee_env_setup(): cf7.

 -- Macro: GSL_IEEE_MODE

     Environment variable which specifies IEEE mode.

 -- Function: void gsl_ieee_env_setup ()

     This function reads the environment variable *note GSL_IEEE_MODE:
     cf6. and attempts to set up the corresponding specified IEEE modes.
     The environment variable should be a list of keywords, separated by
     commas, like this:

          GSL_IEEE_MODE = "keyword, keyword, ..."

     where ‘keyword’ is one of the following mode-names:

          single-precision
          double-precision
          extended-precision
          round-to-nearest
          round-down
          round-up
          round-to-zero
          mask-all
          mask-invalid
          mask-denormalized
          mask-division-by-zero
          mask-overflow
          mask-underflow
          trap-inexact
          trap-common

     If *note GSL_IEEE_MODE: cf6. is empty or undefined then the
     function returns immediately and no attempt is made to change the
     system’s IEEE mode.  When the modes from *note GSL_IEEE_MODE: cf6.
     are turned on the function prints a short message showing the new
     settings to remind you that the results of the program will be
     affected.

     If the requested modes are not supported by the platform being used
     then the function calls the error handler and returns an error code
     of ‘GSL_EUNSUP’.

     When options are specified using this method, the resulting mode is
     based on a default setting of the highest available precision
     (double precision or extended precision, depending on the platform)
     in round-to-nearest mode, with all exceptions enabled apart from
     the INEXACT exception.  The INEXACT exception is generated whenever
     rounding occurs, so it must generally be disabled in typical
     scientific calculations.  All other floating-point exceptions are
     enabled by default, including underflows and the use of
     denormalized numbers, for safety.  They can be disabled with the
     individual ‘mask-’ settings or together using ‘mask-all’.

     The following adjusted combination of modes is convenient for many
     purposes:

          GSL_IEEE_MODE="double-precision,"\
                          "mask-underflow,"\
                            "mask-denormalized"

     This choice ignores any errors relating to small numbers (either
     denormalized, or underflowing to zero) but traps overflows,
     division by zero and invalid operations.

     Note that on the x86 series of processors this function sets both
     the original x87 mode and the newer MXCSR mode, which controls SSE
     floating-point operations.  The SSE floating-point units do not
     have a precision-control bit, and always work in double-precision.
     The single-precision and extended-precision keywords have no effect
     in this case.

To demonstrate the effects of different rounding modes consider the
following program which computes e, the base of natural logarithms, by
summing a rapidly-decreasing series,

     e = 1 + 1/2! + 1/3! + 1/4! + ...
       = 2.71828182846...

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_ieee_utils.h>

     int
     main (void)
     {
       double x = 1, oldsum = 0, sum = 0;
       int i = 0;

       gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */

       do
         {
           i++;

           oldsum = sum;
           sum += x;
           x = x / i;

           printf ("i=%2d sum=%.18f error=%g\n",
                   i, sum, sum - M_E);

           if (i > 30)
              break;
         }
       while (sum != oldsum);

       return 0;
     }

Here are the results of running the program in ‘round-to-nearest’ mode.
This is the IEEE default so it isn’t really necessary to specify it
here:

     $ GSL_IEEE_MODE="round-to-nearest" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     i= 2 sum=2.000000000000000000 error=-0.718282
     ....
     i=18 sum=2.718281828459045535 error=4.44089e-16
     i=19 sum=2.718281828459045535 error=4.44089e-16

After nineteen terms the sum converges to within 4 \times 10^{-16} of
the correct value.  If we now change the rounding mode to ‘round-down’
the final result is less accurate:

     $ GSL_IEEE_MODE="round-down" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     ....
     i=19 sum=2.718281828459041094 error=-3.9968e-15

The result is about 4 \times 10^{-15} below the correct value, an order
of magnitude worse than the result obtained in the ‘round-to-nearest’
mode.

If we change to rounding mode to ‘round-up’ then the final result is
higher than the correct value (when we add each term to the sum the
final result is always rounded up, which increases the sum by at least
one tick until the added term underflows to zero).  To avoid this
problem we would need to use a safer converge criterion, such as ‘while
(fabs(sum - oldsum) > epsilon)’, with a suitably chosen value of
epsilon.

Finally we can see the effect of computing the sum using
single-precision rounding, in the default ‘round-to-nearest’ mode.  In
this case the program thinks it is still using double precision numbers
but the CPU rounds the result of each floating point operation to
single-precision accuracy.  This simulates the effect of writing the
program using single-precision ‘float’ variables instead of ‘double’
variables.  The iteration stops after about half the number of
iterations and the final result is much less accurate:

     $ GSL_IEEE_MODE="single-precision" ./a.out
     ....
     i=12 sum=2.718281984329223633 error=1.5587e-07

with an error of O(10^{-7}), which corresponds to single precision
accuracy (about 1 part in 10^7).  Continuing the iterations further does
not decrease the error because all the subsequent results are rounded to
the same value.


File: gsl-ref.info,  Node: References and Further Reading<40>,  Prev: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic

47.3 References and Further Reading
===================================

The reference for the IEEE standard is,

   * ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point
     Arithmetic.

A more pedagogical introduction to the standard can be found in the
following paper,

   * David Goldberg: What Every Computer Scientist Should Know About
     Floating-Point Arithmetic.  `ACM Computing Surveys', Vol.: 23, No.:
     1 (March 1991), pages 5–48.

   * Corrigendum: `ACM Computing Surveys', Vol.: 23, No.: 3 (September
     1991), page 413.  and see also the sections by B. A. Wichmann and
     Charles B. Dunham in Surveyor’s Forum: “What Every Computer
     Scientist Should Know About Floating-Point Arithmetic”.  `ACM
     Computing Surveys', Vol.: 24, No.: 3 (September 1992), page 319.

A detailed textbook on IEEE arithmetic and its practical use is
available from SIAM Press,

   * Michael L. Overton, `Numerical Computing with IEEE Floating Point
     Arithmetic', SIAM Press, ISBN 0898715717.


File: gsl-ref.info,  Node: Debugging Numerical Programs,  Next: Contributors to GSL,  Prev: IEEE floating-point arithmetic,  Up: Top

48 Debugging Numerical Programs
*******************************

This chapter describes some tips and tricks for debugging numerical
programs which use GSL.

* Menu:

* Using gdb::
* Examining floating point registers::
* Handling floating point exceptions::
* GCC warning options for numerical programs::
* References and Further Reading: References and Further Reading<41>.


File: gsl-ref.info,  Node: Using gdb,  Next: Examining floating point registers,  Up: Debugging Numerical Programs

48.1 Using gdb
==============

Any errors reported by the library are passed to the function
‘gsl_error()’.  By running your programs under gdb and setting a
breakpoint in this function you can automatically catch any library
errors.  You can add a breakpoint for every session by putting:

     break gsl_error

into your ‘.gdbinit’ file in the directory where your program is
started.

If the breakpoint catches an error then you can use a backtrace (‘bt’)
to see the call-tree, and the arguments which possibly caused the error.
By moving up into the calling function you can investigate the values of
variables at that point.  Here is an example from the program
‘fft/test_trap’, which contains the following line:

     status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);

The function *note gsl_fft_complex_wavetable_alloc(): 61d. takes the
length of an FFT as its first argument.  When this line is executed an
error will be generated because the length of an FFT is not allowed to
be zero.

To debug this problem we start ‘gdb’, using the file ‘.gdbinit’ to
define a breakpoint in ‘gsl_error()’:

     $ gdb test_trap

     GDB is free software and you are welcome to distribute copies
     of it under certain conditions; type "show copying" to see
     the conditions.  There is absolutely no warranty for GDB;
     type "show warranty" for details.  GDB 4.16 (i586-debian-linux),
     Copyright 1996 Free Software Foundation, Inc.

     Breakpoint 1 at 0x8050b1e: file error.c, line 14.

When we run the program this breakpoint catches the error and shows the
reason for it:

     (gdb) run
     Starting program: test_trap

     Breakpoint 1, gsl_error (reason=0x8052b0d
         "length n must be positive integer",
         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
         at error.c:14
     14        if (gsl_error_handler)

The first argument of ‘gsl_error()’ is always a string describing the
error.  Now we can look at the backtrace to see what caused the problem:

     (gdb) bt
     #0  gsl_error (reason=0x8052b0d
         "length n must be positive integer",
         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
         at error.c:14
     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
         wavetable=0xbffff778) at c_init.c:108
     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
         at test_trap.c:94
     #3  0x80488be in ___crt_dummy__ ()

We can see that the error was generated in the function *note
gsl_fft_complex_wavetable_alloc(): 61d. when it was called with an
argument of ‘n = 0’.  The original call came from line 94 in the file
‘test_trap.c’.

By moving up to the level of the original call we can find the line that
caused the error:

     (gdb) up
     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
         wavetable=0xbffff778) at c_init.c:108
     108   GSL_ERROR ("length n must be positive integer", GSL_EDOM);
     (gdb) up
     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
       at test_trap.c:94
     94    status = gsl_fft_complex_wavetable_alloc (0,
             &complex_wavetable);

Thus we have found the line that caused the problem.  From this point we
could also print out the values of other variables such as
‘complex_wavetable’.


File: gsl-ref.info,  Node: Examining floating point registers,  Next: Handling floating point exceptions,  Prev: Using gdb,  Up: Debugging Numerical Programs

48.2 Examining floating point registers
=======================================

The contents of floating point registers can be examined using the
command ‘info float’ (on supported platforms):

     (gdb) info float
          st0: 0xc4018b895aa17a945000  Valid Normal -7.838871e+308
          st1: 0x3ff9ea3f50e4d7275000  Valid Normal 0.0285946
          st2: 0x3fe790c64ce27dad4800  Valid Normal 6.7415931e-08
          st3: 0x3ffaa3ef0df6607d7800  Spec  Normal 0.0400229
          st4: 0x3c028000000000000000  Valid Normal 4.4501477e-308
          st5: 0x3ffef5412c22219d9000  Zero  Normal 0.9580257
          st6: 0x3fff8000000000000000  Valid Normal 1
          st7: 0xc4028b65a1f6d243c800  Valid Normal -1.566206e+309
        fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
        fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
         ftag: 0x3fff
          fip: 0x08048b5c
          fcs: 0x051a0023
       fopoff: 0x08086820
       fopsel: 0x002b

Individual registers can be examined using the variables ‘$reg’, where
‘reg’ is the register name:

     (gdb) p $st1
     $1 = 0.02859464454261210347719


File: gsl-ref.info,  Node: Handling floating point exceptions,  Next: GCC warning options for numerical programs,  Prev: Examining floating point registers,  Up: Debugging Numerical Programs

48.3 Handling floating point exceptions
=======================================

It is possible to stop the program whenever a ‘SIGFPE’ floating point
exception occurs.  This can be useful for finding the cause of an
unexpected infinity or ‘NaN’.  The current handler settings can be shown
with the command ‘info signal SIGFPE’:

     (gdb) info signal SIGFPE
     Signal  Stop  Print  Pass to program Description
     SIGFPE  Yes   Yes    Yes             Arithmetic exception

Unless the program uses a signal handler the default setting should be
changed so that SIGFPE is not passed to the program, as this would cause
it to exit.  The command ‘handle SIGFPE stop nopass’ prevents this:

     (gdb) handle SIGFPE stop nopass
     Signal  Stop  Print  Pass to program Description
     SIGFPE  Yes   Yes    No              Arithmetic exception

Depending on the platform it may be necessary to instruct the kernel to
generate signals for floating point exceptions.  For programs using GSL
this can be achieved using the *note GSL_IEEE_MODE: cf6. environment
variable in conjunction with the function *note gsl_ieee_env_setup():
cf7. as described in *note IEEE floating-point arithmetic: cee.:

     (gdb) set env GSL_IEEE_MODE=double-precision


File: gsl-ref.info,  Node: GCC warning options for numerical programs,  Next: References and Further Reading<41>,  Prev: Handling floating point exceptions,  Up: Debugging Numerical Programs

48.4 GCC warning options for numerical programs
===============================================

Writing reliable numerical programs in C requires great care.  The
following GCC warning options are recommended when compiling numerical
programs:

     gcc -ansi -pedantic -Werror -Wall -W
       -Wmissing-prototypes -Wstrict-prototypes
       -Wconversion -Wshadow -Wpointer-arith
       -Wcast-qual -Wcast-align
       -Wwrite-strings -Wnested-externs
       -fshort-enums -fno-common -Dinline= -g -O2

For details of each option consult the manual `Using and Porting GCC'.
The following table gives a brief explanation of what types of errors
these options catch.

‘-ansi -pedantic’

     Use ANSI C, and reject any non-ANSI extensions.  These flags help
     in writing portable programs that will compile on other systems.

‘-Werror’

     Consider warnings to be errors, so that compilation stops.  This
     prevents warnings from scrolling off the top of the screen and
     being lost.  You won’t be able to compile the program until it is
     completely warning-free.

‘-Wall’

     This turns on a set of warnings for common programming problems.
     You need ‘-Wall’, but it is not enough on its own.

‘-O2’

     Turn on optimization.  The warnings for uninitialized variables in
     ‘-Wall’ rely on the optimizer to analyze the code.  If there is no
     optimization then these warnings aren’t generated.

‘-W’

     This turns on some extra warnings not included in ‘-Wall’, such as
     missing return values and comparisons between signed and unsigned
     integers.

‘-Wmissing-prototypes -Wstrict-prototypes’

     Warn if there are any missing or inconsistent prototypes.  Without
     prototypes it is harder to detect problems with incorrect
     arguments.

‘-Wconversion’

     The main use of this option is to warn about conversions from
     signed to unsigned integers.  For example, ‘unsigned int x = -1’.
     If you need to perform such a conversion you can use an explicit
     cast.

‘-Wshadow’

     This warns whenever a local variable shadows another local
     variable.  If two variables have the same name then it is a
     potential source of confusion.

‘-Wpointer-arith -Wcast-qual -Wcast-align’

     These options warn if you try to do pointer arithmetic for types
     which don’t have a size, such as ‘void’, if you remove a ‘const’
     cast from a pointer, or if you cast a pointer to a type which has a
     different size, causing an invalid alignment.

‘-Wwrite-strings’

     This option gives string constants a ‘const’ qualifier so that it
     will be a compile-time error to attempt to overwrite them.

‘-fshort-enums’

     This option makes the type of ‘enum’ as short as possible.
     Normally this makes an ‘enum’ different from an ‘int’.
     Consequently any attempts to assign a pointer-to-int to a
     pointer-to-enum will generate a cast-alignment warning.

‘-fno-common’

     This option prevents global variables being simultaneously defined
     in different object files (you get an error at link time).  Such a
     variable should be defined in one file and referred to in other
     files with an ‘extern’ declaration.

‘-Wnested-externs’

     This warns if an ‘extern’ declaration is encountered within a
     function.

‘-Dinline=’

     The ‘inline’ keyword is not part of ANSI C. Thus if you want to use
     ‘-ansi’ with a program which uses inline functions you can use this
     preprocessor definition to remove the ‘inline’ keywords.

‘-g’

     It always makes sense to put debugging symbols in the executable so
     that you can debug it using ‘gdb’.  The only effect of debugging
     symbols is to increase the size of the file, and you can use the
     ‘strip’ command to remove them later if necessary.


File: gsl-ref.info,  Node: References and Further Reading<41>,  Prev: GCC warning options for numerical programs,  Up: Debugging Numerical Programs

48.5 References and Further Reading
===================================

The following books are essential reading for anyone writing and
debugging numerical programs with ‘gcc’ and ‘gdb’.

   * R.M. Stallman, `Using and Porting GNU CC', Free Software
     Foundation, ISBN 1882114388

   * R.M. Stallman, R.H. Pesch, `Debugging with GDB: The GNU
     Source-Level Debugger', Free Software Foundation, ISBN 1882114779

For a tutorial introduction to the GNU C Compiler and related programs,
see

   * B.J. Gough, ‘http://www.network-theory.co.uk/gcc/intro/’,’ `An
     Introduction to GCC', Network Theory Ltd, ISBN 0954161793


File: gsl-ref.info,  Node: Contributors to GSL,  Next: Autoconf Macros,  Prev: Debugging Numerical Programs,  Up: Top

49 Contributors to GSL
**********************

(See the ‘AUTHORS’ file in the distribution for up-to-date information.)

Mark Galassi

     Conceived GSL (with James Theiler) and wrote the design document.
     Wrote the simulated annealing package and the relevant chapter in
     the manual.

James Theiler

     Conceived GSL (with Mark Galassi).  Wrote the random number
     generators and the relevant chapter in this manual.

Jim Davies

     Wrote the statistical routines and the relevant chapter in this
     manual.

Brian Gough

     FFTs, numerical integration, random number generators and
     distributions, root finding, minimization and fitting, polynomial
     solvers, complex numbers, physical constants, permutations, vector
     and matrix functions, histograms, statistics, ieee-utils, revised
     CBLAS Level 2 & 3, matrix decompositions, eigensystems, cumulative
     distribution functions, testing, documentation and releases.

Reid Priedhorsky

     Wrote and documented the initial version of the root finding
     routines while at Los Alamos National Laboratory, Mathematical
     Modeling and Analysis Group.

Gerard Jungman

     Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra,
     Eigensystems, Hankel Transforms.

Patrick Alken

     Implementation of nonsymmetric and generalized eigensystems,
     B-splines, linear and nonlinear least squares, matrix
     decompositions, associated Legendre functions, running statistics,
     sparse matrices, and sparse linear algebra.

Mike Booth

     Wrote the Monte Carlo library.

Jorma Olavi Tähtinen

     Wrote the initial complex arithmetic functions.

Thomas Walter

     Wrote the initial heapsort routines and Cholesky decomposition.

Fabrice Rossi

     Multidimensional minimization.

Carlo Perassi

     Implementation of the random number generators in Knuth’s
     `Seminumerical Algorithms', 3rd Ed.

Szymon Jaroszewicz

     Wrote the routines for generating combinations.

Nicolas Darnis

     Wrote the cyclic functions and the initial functions for canonical
     permutations.

Jason H. Stover

     Wrote the major cumulative distribution functions.

Ivo Alxneit

     Wrote the routines for wavelet transforms.

Tuomo Keskitalo

     Improved the implementation of the ODE solvers and wrote the
     ode-initval2 routines.

Lowell Johnson

     Implementation of the Mathieu functions.

Rhys Ulerich

     Wrote the multiset routines.

Pavel Holoborodko

     Wrote the fixed order Gauss-Legendre quadrature routines.

Pedro Gonnet

     Wrote the CQUAD integration routines.

Thanks to Nigel Lowry for help in proofreading the manual.

The non-symmetric eigensystems routines contain code based on the LAPACK
linear algebra library.  LAPACK is distributed under the following
license:

     Copyright (c) 1992-2006 The University of Tennessee.  All rights reserved.

     Redistribution and use in source and binary forms, with or without
     modification, are permitted provided that the following conditions are
     met:

     * Redistributions of source code must retain the above copyright
       notice, this list of conditions and the following disclaimer.

     * Redistributions in binary form must reproduce the above copyright
       notice, this list of conditions and the following disclaimer listed
       in this license in the documentation and/or other materials
       provided with the distribution.

     * Neither the name of the copyright holders nor the names of its
       contributors may be used to endorse or promote products derived from
       this software without specific prior written permission.

     THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
     ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
     LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
     A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
     OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
     SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
     LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
     DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
     THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
     (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
     OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.


File: gsl-ref.info,  Node: Autoconf Macros,  Next: GSL CBLAS Library,  Prev: Contributors to GSL,  Up: Top

50 Autoconf Macros
******************

For applications using ‘autoconf’ the standard macro ‘AC_CHECK_LIB’ can
be used to link with GSL automatically from a ‘configure’ script.  The
library itself depends on the presence of a CBLAS and math library as
well, so these must also be located before linking with the main
‘libgsl’ file.  The following commands should be placed in the
‘configure.ac’ file to perform these tests:

     AC_CHECK_LIB([m],[cos])
     AC_CHECK_LIB([gslcblas],[cblas_dgemm])
     AC_CHECK_LIB([gsl],[gsl_blas_dgemm])

It is important to check for ‘libm’ and ‘libgslcblas’ before ‘libgsl’,
otherwise the tests will fail.  Assuming the libraries are found the
output during the configure stage looks like this:

     checking for cos in -lm... yes
     checking for cblas_dgemm in -lgslcblas... yes
     checking for gsl_blas_dgemm in -lgsl... yes

If the library is found then the tests will define the macros
‘HAVE_LIBGSL’, ‘HAVE_LIBGSLCBLAS’, ‘HAVE_LIBM’ and add the options
‘-lgsl -lgslcblas -lm’ to the variable ‘LIBS’.

The tests above will find any version of the library.  They are suitable
for general use, where the versions of the functions are not important.
An alternative macro is available in the file ‘gsl.m4’ to test for a
specific version of the library.  To use this macro simply add the
following line to your ‘configure.in’ file instead of the tests above:

     AX_PATH_GSL(GSL_VERSION,
                [action-if-found],
                [action-if-not-found])

The argument ‘GSL_VERSION’ should be the two or three digit
‘major.minor’ or ‘major.minor.micro’ version number of the release you
require.  A suitable choice for ‘action-if-not-found’ is:

     AC_MSG_ERROR(could not find required version of GSL)

Then you can add the variables ‘GSL_LIBS’ and ‘GSL_CFLAGS’ to your
Makefile.am files to obtain the correct compiler flags.  ‘GSL_LIBS’ is
equal to the output of the ‘gsl-config --libs’ command and ‘GSL_CFLAGS’
is equal to ‘gsl-config --cflags’ command.  For example:

     libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas

Note that the macro ‘AX_PATH_GSL’ needs to use the C compiler so it
should appear in the ‘configure.in’ file before the macro
‘AC_LANG_CPLUSPLUS’ for programs that use C++.

To test for ‘inline’ the following test should be placed in your
‘configure.in’ file:

     AC_C_INLINE

     if test "$ac_cv_c_inline" != no ; then
       AC_DEFINE(HAVE_INLINE,1)
       AC_SUBST(HAVE_INLINE)
     fi

and the macro will then be defined in the compilation flags or by
including the file ‘config.h’ before any library headers.

The following autoconf test will check for ‘extern inline’:

     dnl Check for "extern inline", using a modified version
     dnl of the test for AC_C_INLINE from acspecific.mt
     dnl
     AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
     [ac_cv_c_extern_inline=no
     AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
     extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
     double foo (double x) { return x + 1.0; };],
     [  foo(1.0)  ],
     [ac_cv_c_extern_inline="yes"])
     ])

     if test "$ac_cv_c_extern_inline" != no ; then
       AC_DEFINE(HAVE_INLINE,1)
       AC_SUBST(HAVE_INLINE)
     fi

The substitution of portability functions can be made automatically if
you use ‘autoconf’.  For example, to test whether the BSD function
‘hypot()’ is available you can include the following line in the
configure file ‘configure.in’ for your application:

     AC_CHECK_FUNCS(hypot)

and place the following macro definitions in the file ‘config.h.in’:

     /* Substitute gsl_hypot for missing system hypot */

     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

The application source files can then use the include command ‘#include
<config.h>’ to substitute *note gsl_hypot(): 1a. for each occurrence of
‘hypot()’ when ‘hypot()’ is not available.


File: gsl-ref.info,  Node: GSL CBLAS Library,  Next: GNU General Public License,  Prev: Autoconf Macros,  Up: Top

51 GSL CBLAS Library
********************

The prototypes for the low-level CBLAS functions are declared in the
file ‘gsl_cblas.h’.  For the definition of the functions consult the
documentation available from Netlib (*note see BLAS References and
Further Reading: 4e7.).

* Menu:

* Level 1: Level 1<2>.
* Level 2: Level 2<2>.
* Level 3: Level 3<2>.
* Examples: Examples<37>.


File: gsl-ref.info,  Node: Level 1<2>,  Next: Level 2<2>,  Up: GSL CBLAS Library

51.1 Level 1
============

 -- Function: float cblas_sdsdot (const int N, const float alpha, const
          float *x, const int incx, const float *y, const int incy)

 -- Function: double cblas_dsdot (const int N, const float *x, const int
          incx, const float *y, const int incy)

 -- Function: float cblas_sdot (const int N, const float *x, const int
          incx, const float *y, const int incy)

 -- Function: double cblas_ddot (const int N, const double *x, const int
          incx, const double *y, const int incy)

 -- Function: void cblas_cdotu_sub (const int N, const void *x, const
          int incx, const void *y, const int incy, void *dotu)

 -- Function: void cblas_cdotc_sub (const int N, const void *x, const
          int incx, const void *y, const int incy, void *dotc)

 -- Function: void cblas_zdotu_sub (const int N, const void *x, const
          int incx, const void *y, const int incy, void *dotu)

 -- Function: void cblas_zdotc_sub (const int N, const void *x, const
          int incx, const void *y, const int incy, void *dotc)

 -- Function: float cblas_snrm2 (const int N, const float *x, const int
          incx)

 -- Function: float cblas_sasum (const int N, const float *x, const int
          incx)

 -- Function: double cblas_dnrm2 (const int N, const double *x, const
          int incx)

 -- Function: double cblas_dasum (const int N, const double *x, const
          int incx)

 -- Function: float cblas_scnrm2 (const int N, const void *x, const int
          incx)

 -- Function: float cblas_scasum (const int N, const void *x, const int
          incx)

 -- Function: double cblas_dznrm2 (const int N, const void *x, const int
          incx)

 -- Function: double cblas_dzasum (const int N, const void *x, const int
          incx)

 -- Function: CBLAS_INDEX cblas_isamax (const int N, const float *x,
          const int incx)

 -- Function: CBLAS_INDEX cblas_idamax (const int N, const double *x,
          const int incx)

 -- Function: CBLAS_INDEX cblas_icamax (const int N, const void *x,
          const int incx)

 -- Function: CBLAS_INDEX cblas_izamax (const int N, const void *x,
          const int incx)

 -- Function: void cblas_sswap (const int N, float *x, const int incx,
          float *y, const int incy)

 -- Function: void cblas_scopy (const int N, const float *x, const int
          incx, float *y, const int incy)

 -- Function: void cblas_saxpy (const int N, const float alpha, const
          float *x, const int incx, float *y, const int incy)

 -- Function: void cblas_dswap (const int N, double *x, const int incx,
          double *y, const int incy)

 -- Function: void cblas_dcopy (const int N, const double *x, const int
          incx, double *y, const int incy)

 -- Function: void cblas_daxpy (const int N, const double alpha, const
          double *x, const int incx, double *y, const int incy)

 -- Function: void cblas_cswap (const int N, void *x, const int incx,
          void *y, const int incy)

 -- Function: void cblas_ccopy (const int N, const void *x, const int
          incx, void *y, const int incy)

 -- Function: void cblas_caxpy (const int N, const void *alpha, const
          void *x, const int incx, void *y, const int incy)

 -- Function: void cblas_zswap (const int N, void *x, const int incx,
          void *y, const int incy)

 -- Function: void cblas_zcopy (const int N, const void *x, const int
          incx, void *y, const int incy)

 -- Function: void cblas_zaxpy (const int N, const void *alpha, const
          void *x, const int incx, void *y, const int incy)

 -- Function: void cblas_srotg (float *a, float *b, float *c, float *s)

 -- Function: void cblas_srotmg (float *d1, float *d2, float *b1, const
          float b2, float *P)

 -- Function: void cblas_srot (const int N, float *x, const int incx,
          float *y, const int incy, const float c, const float s)

 -- Function: void cblas_srotm (const int N, float *x, const int incx,
          float *y, const int incy, const float *P)

 -- Function: void cblas_drotg (double *a, double *b, double *c, double
          *s)

 -- Function: void cblas_drotmg (double *d1, double *d2, double *b1,
          const double b2, double *P)

 -- Function: void cblas_drot (const int N, double *x, const int incx,
          double *y, const int incy, const double c, const double s)

 -- Function: void cblas_drotm (const int N, double *x, const int incx,
          double *y, const int incy, const double *P)

 -- Function: void cblas_sscal (const int N, const float alpha, float
          *x, const int incx)

 -- Function: void cblas_dscal (const int N, const double alpha, double
          *x, const int incx)

 -- Function: void cblas_cscal (const int N, const void *alpha, void *x,
          const int incx)

 -- Function: void cblas_zscal (const int N, const void *alpha, void *x,
          const int incx)

 -- Function: void cblas_csscal (const int N, const float alpha, void
          *x, const int incx)

 -- Function: void cblas_zdscal (const int N, const double alpha, void
          *x, const int incx)


File: gsl-ref.info,  Node: Level 2<2>,  Next: Level 3<2>,  Prev: Level 1<2>,  Up: GSL CBLAS Library

51.2 Level 2
============

 -- Function: void cblas_sgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const float
          alpha, const float *A, const int lda, const float *x, const
          int incx, const float beta, float *y, const int incy)

 -- Function: void cblas_sgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const int
          KL, const int KU, const float alpha, const float *A, const int
          lda, const float *x, const int incx, const float beta, float
          *y, const int incy)

 -- Function: void cblas_strmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float *A, const int lda,
          float *x, const int incx)

 -- Function: void cblas_stbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const float *A,
          const int lda, float *x, const int incx)

 -- Function: void cblas_stpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float *Ap, float *x, const
          int incx)

 -- Function: void cblas_strsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float *A, const int lda,
          float *x, const int incx)

 -- Function: void cblas_stbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const float *A,
          const int lda, float *x, const int incx)

 -- Function: void cblas_stpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float *Ap, float *x, const
          int incx)

 -- Function: void cblas_dgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const double
          alpha, const double *A, const int lda, const double *x, const
          int incx, const double beta, double *y, const int incy)

 -- Function: void cblas_dgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const int
          KL, const int KU, const double alpha, const double *A, const
          int lda, const double *x, const int incx, const double beta,
          double *y, const int incy)

 -- Function: void cblas_dtrmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double *A, const int lda,
          double *x, const int incx)

 -- Function: void cblas_dtbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const double *A,
          const int lda, double *x, const int incx)

 -- Function: void cblas_dtpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double *Ap, double *x,
          const int incx)

 -- Function: void cblas_dtrsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double *A, const int lda,
          double *x, const int incx)

 -- Function: void cblas_dtbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const double *A,
          const int lda, double *x, const int incx)

 -- Function: void cblas_dtpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double *Ap, double *x,
          const int incx)

 -- Function: void cblas_cgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const void
          *alpha, const void *A, const int lda, const void *x, const int
          incx, const void *beta, void *y, const int incy)

 -- Function: void cblas_cgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const int
          KL, const int KU, const void *alpha, const void *A, const int
          lda, const void *x, const int incx, const void *beta, void *y,
          const int incy)

 -- Function: void cblas_ctrmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *A, const int lda,
          void *x, const int incx)

 -- Function: void cblas_ctbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void *A,
          const int lda, void *x, const int incx)

 -- Function: void cblas_ctpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const
          int incx)

 -- Function: void cblas_ctrsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *A, const int lda,
          void *x, const int incx)

 -- Function: void cblas_ctbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void *A,
          const int lda, void *x, const int incx)

 -- Function: void cblas_ctpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const
          int incx)

 -- Function: void cblas_zgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const void
          *alpha, const void *A, const int lda, const void *x, const int
          incx, const void *beta, void *y, const int incy)

 -- Function: void cblas_zgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const int
          KL, const int KU, const void *alpha, const void *A, const int
          lda, const void *x, const int incx, const void *beta, void *y,
          const int incy)

 -- Function: void cblas_ztrmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *A, const int lda,
          void *x, const int incx)

 -- Function: void cblas_ztbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void *A,
          const int lda, void *x, const int incx)

 -- Function: void cblas_ztpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const
          int incx)

 -- Function: void cblas_ztrsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *A, const int lda,
          void *x, const int incx)

 -- Function: void cblas_ztbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void *A,
          const int lda, void *x, const int incx)

 -- Function: void cblas_ztpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void *Ap, void *x, const
          int incx)

 -- Function: void cblas_ssymv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          *A, const int lda, const float *x, const int incx, const float
          beta, float *y, const int incy)

 -- Function: void cblas_ssbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const float alpha,
          const float *A, const int lda, const float *x, const int incx,
          const float beta, float *y, const int incy)

 -- Function: void cblas_sspmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          *Ap, const float *x, const int incx, const float beta, float
          *y, const int incy)

 -- Function: void cblas_sger (const enum CBLAS_ORDER order, const int
          M, const int N, const float alpha, const float *x, const int
          incx, const float *y, const int incy, float *A, const int lda)

 -- Function: void cblas_ssyr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          *x, const int incx, float *A, const int lda)

 -- Function: void cblas_sspr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          *x, const int incx, float *Ap)

 -- Function: void cblas_ssyr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          *x, const int incx, const float *y, const int incy, float *A,
          const int lda)

 -- Function: void cblas_sspr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          *x, const int incx, const float *y, const int incy, float *A)

 -- Function: void cblas_dsymv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          *A, const int lda, const double *x, const int incx, const
          double beta, double *y, const int incy)

 -- Function: void cblas_dsbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const double alpha,
          const double *A, const int lda, const double *x, const int
          incx, const double beta, double *y, const int incy)

 -- Function: void cblas_dspmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          *Ap, const double *x, const int incx, const double beta,
          double *y, const int incy)

 -- Function: void cblas_dger (const enum CBLAS_ORDER order, const int
          M, const int N, const double alpha, const double *x, const int
          incx, const double *y, const int incy, double *A, const int
          lda)

 -- Function: void cblas_dsyr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          *x, const int incx, double *A, const int lda)

 -- Function: void cblas_dspr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          *x, const int incx, double *Ap)

 -- Function: void cblas_dsyr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          *x, const int incx, const double *y, const int incy, double
          *A, const int lda)

 -- Function: void cblas_dspr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          *x, const int incx, const double *y, const int incy, double
          *A)

 -- Function: void cblas_chemv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *A, const int lda, const void *x, const int incx, const void
          *beta, void *y, const int incy)

 -- Function: void cblas_chbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const void *alpha,
          const void *A, const int lda, const void *x, const int incx,
          const void *beta, void *y, const int incy)

 -- Function: void cblas_chpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *Ap, const void *x, const int incx, const void *beta, void *y,
          const int incy)

 -- Function: void cblas_cgeru (const enum CBLAS_ORDER order, const int
          M, const int N, const void *alpha, const void *x, const int
          incx, const void *y, const int incy, void *A, const int lda)

 -- Function: void cblas_cgerc (const enum CBLAS_ORDER order, const int
          M, const int N, const void *alpha, const void *x, const int
          incx, const void *y, const int incy, void *A, const int lda)

 -- Function: void cblas_cher (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const void
          *x, const int incx, void *A, const int lda)

 -- Function: void cblas_chpr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const void
          *x, const int incx, void *A)

 -- Function: void cblas_cher2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *x, const int incx, const void *y, const int incy, void *A,
          const int lda)

 -- Function: void cblas_chpr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *x, const int incx, const void *y, const int incy, void *Ap)

 -- Function: void cblas_zhemv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *A, const int lda, const void *x, const int incx, const void
          *beta, void *y, const int incy)

 -- Function: void cblas_zhbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const void *alpha,
          const void *A, const int lda, const void *x, const int incx,
          const void *beta, void *y, const int incy)

 -- Function: void cblas_zhpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *Ap, const void *x, const int incx, const void *beta, void *y,
          const int incy)

 -- Function: void cblas_zgeru (const enum CBLAS_ORDER order, const int
          M, const int N, const void *alpha, const void *x, const int
          incx, const void *y, const int incy, void *A, const int lda)

 -- Function: void cblas_zgerc (const enum CBLAS_ORDER order, const int
          M, const int N, const void *alpha, const void *x, const int
          incx, const void *y, const int incy, void *A, const int lda)

 -- Function: void cblas_zher (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const void
          *x, const int incx, void *A, const int lda)

 -- Function: void cblas_zhpr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const void
          *x, const int incx, void *A)

 -- Function: void cblas_zher2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *x, const int incx, const void *y, const int incy, void *A,
          const int lda)

 -- Function: void cblas_zhpr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void *alpha, const void
          *x, const int incx, const void *y, const int incy, void *Ap)


File: gsl-ref.info,  Node: Level 3<2>,  Next: Examples<37>,  Prev: Level 2<2>,  Up: GSL CBLAS Library

51.3 Level 3
============

 -- Function: void cblas_sgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const float alpha,
          const float *A, const int lda, const float *B, const int ldb,
          const float beta, float *C, const int ldc)

 -- Function: void cblas_ssymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const float alpha, const float *A, const int lda,
          const float *B, const int ldb, const float beta, float *C,
          const int ldc)

 -- Function: void cblas_ssyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int
          N, const int K, const float alpha, const float *A, const int
          lda, const float beta, float *C, const int ldc)

 -- Function: void cblas_ssyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const float alpha, const float *A, const
          int lda, const float *B, const int ldb, const float beta,
          float *C, const int ldc)

 -- Function: void cblas_strmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const float alpha, const float *A, const int
          lda, float *B, const int ldb)

 -- Function: void cblas_strsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const float alpha, const float *A, const int
          lda, float *B, const int ldb)

 -- Function: void cblas_dgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const double alpha,
          const double *A, const int lda, const double *B, const int
          ldb, const double beta, double *C, const int ldc)

 -- Function: void cblas_dsymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const double alpha, const double *A, const int
          lda, const double *B, const int ldb, const double beta, double
          *C, const int ldc)

 -- Function: void cblas_dsyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int
          N, const int K, const double alpha, const double *A, const int
          lda, const double beta, double *C, const int ldc)

 -- Function: void cblas_dsyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const double alpha, const double *A, const
          int lda, const double *B, const int ldb, const double beta,
          double *C, const int ldc)

 -- Function: void cblas_dtrmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const double alpha, const double *A, const int
          lda, double *B, const int ldb)

 -- Function: void cblas_dtrsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const double alpha, const double *A, const int
          lda, double *B, const int ldb)

 -- Function: void cblas_cgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const void *alpha,
          const void *A, const int lda, const void *B, const int ldb,
          const void *beta, void *C, const int ldc)

 -- Function: void cblas_csymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void *alpha, const void *A, const int lda,
          const void *B, const int ldb, const void *beta, void *C, const
          int ldc)

 -- Function: void cblas_csyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int
          N, const int K, const void *alpha, const void *A, const int
          lda, const void *beta, void *C, const int ldc)

 -- Function: void cblas_csyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void *alpha, const void *A, const
          int lda, const void *B, const int ldb, const void *beta, void
          *C, const int ldc)

 -- Function: void cblas_ctrmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const void *alpha, const void *A, const int
          lda, void *B, const int ldb)

 -- Function: void cblas_ctrsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const void *alpha, const void *A, const int
          lda, void *B, const int ldb)

 -- Function: void cblas_zgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const void *alpha,
          const void *A, const int lda, const void *B, const int ldb,
          const void *beta, void *C, const int ldc)

 -- Function: void cblas_zsymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void *alpha, const void *A, const int lda,
          const void *B, const int ldb, const void *beta, void *C, const
          int ldc)

 -- Function: void cblas_zsyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int
          N, const int K, const void *alpha, const void *A, const int
          lda, const void *beta, void *C, const int ldc)

 -- Function: void cblas_zsyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void *alpha, const void *A, const
          int lda, const void *B, const int ldb, const void *beta, void
          *C, const int ldc)

 -- Function: void cblas_ztrmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const void *alpha, const void *A, const int
          lda, void *B, const int ldb)

 -- Function: void cblas_ztrsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const int
          M, const int N, const void *alpha, const void *A, const int
          lda, void *B, const int ldb)

 -- Function: void cblas_chemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void *alpha, const void *A, const int lda,
          const void *B, const int ldb, const void *beta, void *C, const
          int ldc)

 -- Function: void cblas_cherk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int
          N, const int K, const float alpha, const void *A, const int
          lda, const float beta, void *C, const int ldc)

 -- Function: void cblas_cher2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void *alpha, const void *A, const
          int lda, const void *B, const int ldb, const float beta, void
          *C, const int ldc)

 -- Function: void cblas_zhemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void *alpha, const void *A, const int lda,
          const void *B, const int ldb, const void *beta, void *C, const
          int ldc)

 -- Function: void cblas_zherk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const int
          N, const int K, const double alpha, const void *A, const int
          lda, const double beta, void *C, const int ldc)

 -- Function: void cblas_zher2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void *alpha, const void *A, const
          int lda, const void *B, const int ldb, const double beta, void
          *C, const int ldc)

 -- Function: void cblas_xerbla (int p, const char *rout, const char
          *form, ...)


File: gsl-ref.info,  Node: Examples<37>,  Prev: Level 3<2>,  Up: GSL CBLAS Library

51.4 Examples
=============

The following program computes the product of two matrices using the
Level-3 BLAS function SGEMM,

     [ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
     [ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
                         [ 1031 1032 ]

The matrices are stored in row major order but could be stored in column
major order if the first argument of the call to *note cblas_sgemm():
d79. was changed to ‘CblasColMajor’.

     #include <stdio.h>
     #include <gsl/gsl_cblas.h>

     int
     main (void)
     {
       int lda = 3;

       float A[] = { 0.11, 0.12, 0.13,
                     0.21, 0.22, 0.23 };

       int ldb = 2;

       float B[] = { 1011, 1012,
                     1021, 1022,
                     1031, 1032 };

       int ldc = 2;

       float C[] = { 0.00, 0.00,
                     0.00, 0.00 };

       /* Compute C = A B */

       cblas_sgemm (CblasRowMajor,
                    CblasNoTrans, CblasNoTrans, 2, 2, 3,
                    1.0, A, lda, B, ldb, 0.0, C, ldc);

       printf ("[ %g, %g\n", C[0], C[1]);
       printf ("  %g, %g ]\n", C[2], C[3]);

       return 0;
     }

To compile the program use the following command line:

     $ gcc -Wall demo.c -lgslcblas

There is no need to link with the main library ‘-lgsl’ in this case as
the CBLAS library is an independent unit.  Here is the output from the
program,

     [ 367.76, 368.12
       674.06, 674.72 ]


File: gsl-ref.info,  Node: GNU General Public License,  Next: GNU Free Documentation License,  Prev: GSL CBLAS Library,  Up: Top

52 GNU General Public License
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                         GNU GENERAL PUBLIC LICENSE
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     to attach them to the start of each source file to most effectively
     state the exclusion of warranty; and each file should have at least
     the "copyright" line and a pointer to where the full notice is found.

         <one line to give the program's name and a brief idea of what it does.>
         Copyright (C) <year>  <name of author>

         This program is free software: you can redistribute it and/or modify
         it under the terms of the GNU General Public License as published by
         the Free Software Foundation, either version 3 of the License, or
         (at your option) any later version.

         This program is distributed in the hope that it will be useful,
         but WITHOUT ANY WARRANTY; without even the implied warranty of
         MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
         GNU General Public License for more details.

         You should have received a copy of the GNU General Public License
         along with this program.  If not, see <http://www.gnu.org/licenses/>.

     Also add information on how to contact you by electronic and paper mail.

       If the program does terminal interaction, make it output a short
     notice like this when it starts in an interactive mode:

         <program>  Copyright (C) <year>  <name of author>
         This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
         This is free software, and you are welcome to redistribute it
         under certain conditions; type `show c' for details.

     The hypothetical commands `show w' and `show c' should show the appropriate
     parts of the General Public License.  Of course, your program's commands
     might be different; for a GUI interface, you would use an "about box".

       You should also get your employer (if you work as a programmer) or school,
     if any, to sign a "copyright disclaimer" for the program, if necessary.
     For more information on this, and how to apply and follow the GNU GPL, see
     <http://www.gnu.org/licenses/>.

       The GNU General Public License does not permit incorporating your program
     into proprietary programs.  If your program is a subroutine library, you
     may consider it more useful to permit linking proprietary applications with
     the library.  If this is what you want to do, use the GNU Lesser General
     Public License instead of this License.  But first, please read
     <http://www.gnu.org/philosophy/why-not-lgpl.html>.


File: gsl-ref.info,  Node: GNU Free Documentation License,  Next: Index,  Prev: GNU General Public License,  Up: Top

53 GNU Free Documentation License
*********************************


                     GNU Free Documentation License
                      Version 1.3, 3 November 2008


      Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
          <http://fsf.org/>
      Everyone is permitted to copy and distribute verbatim copies
      of this license document, but changing it is not allowed.

     0. PREAMBLE

     The purpose of this License is to make a manual, textbook, or other
     functional and useful document "free" in the sense of freedom: to
     assure everyone the effective freedom to copy and redistribute it,
     with or without modifying it, either commercially or noncommercially.
     Secondarily, this License preserves for the author and publisher a way
     to get credit for their work, while not being considered responsible
     for modifications made by others.

     This License is a kind of "copyleft", which means that derivative
     works of the document must themselves be free in the same sense.  It
     complements the GNU General Public License, which is a copyleft
     license designed for free software.

     We have designed this License in order to use it for manuals for free
     software, because free software needs free documentation: a free
     program should come with manuals providing the same freedoms that the
     software does.  But this License is not limited to software manuals;
     it can be used for any textual work, regardless of subject matter or
     whether it is published as a printed book.  We recommend this License
     principally for works whose purpose is instruction or reference.


     1. APPLICABILITY AND DEFINITIONS

     This License applies to any manual or other work, in any medium, that
     contains a notice placed by the copyright holder saying it can be
     distributed under the terms of this License.  Such a notice grants a
     world-wide, royalty-free license, unlimited in duration, to use that
     work under the conditions stated herein.  The "Document", below,
     refers to any such manual or work.  Any member of the public is a
     licensee, and is addressed as "you".  You accept the license if you
     copy, modify or distribute the work in a way requiring permission
     under copyright law.

     A "Modified Version" of the Document means any work containing the
     Document or a portion of it, either copied verbatim, or with
     modifications and/or translated into another language.

     A "Secondary Section" is a named appendix or a front-matter section of
     the Document that deals exclusively with the relationship of the
     publishers or authors of the Document to the Document's overall
     subject (or to related matters) and contains nothing that could fall
     directly within that overall subject.  (Thus, if the Document is in
     part a textbook of mathematics, a Secondary Section may not explain
     any mathematics.)  The relationship could be a matter of historical
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     commercial, philosophical, ethical or political position regarding
     them.

     The "Invariant Sections" are certain Secondary Sections whose titles
     are designated, as being those of Invariant Sections, in the notice
     that says that the Document is released under this License.  If a
     section does not fit the above definition of Secondary then it is not
     allowed to be designated as Invariant.  The Document may contain zero
     Invariant Sections.  If the Document does not identify any Invariant
     Sections then there are none.

     The "Cover Texts" are certain short passages of text that are listed,
     as Front-Cover Texts or Back-Cover Texts, in the notice that says that
     the Document is released under this License.  A Front-Cover Text may
     be at most 5 words, and a Back-Cover Text may be at most 25 words.

     A "Transparent" copy of the Document means a machine-readable copy,
     represented in a format whose specification is available to the
     general public, that is suitable for revising the document
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     drawing editor, and that is suitable for input to text formatters or
     for automatic translation to a variety of formats suitable for input
     to text formatters.  A copy made in an otherwise Transparent file
     format whose markup, or absence of markup, has been arranged to thwart
     or discourage subsequent modification by readers is not Transparent.
     An image format is not Transparent if used for any substantial amount
     of text.  A copy that is not "Transparent" is called "Opaque".

     Examples of suitable formats for Transparent copies include plain
     ASCII without markup, Texinfo input format, LaTeX input format, SGML
     or XML using a publicly available DTD, and standard-conforming simple
     HTML, PostScript or PDF designed for human modification.  Examples of
     transparent image formats include PNG, XCF and JPG.  Opaque formats
     include proprietary formats that can be read and edited only by
     proprietary word processors, SGML or XML for which the DTD and/or
     processing tools are not generally available, and the
     machine-generated HTML, PostScript or PDF produced by some word
     processors for output purposes only.

     The "Title Page" means, for a printed book, the title page itself,
     plus such following pages as are needed to hold, legibly, the material
     this License requires to appear in the title page.  For works in
     formats which do not have any title page as such, "Title Page" means
     the text near the most prominent appearance of the work's title,
     preceding the beginning of the body of the text.

     The "publisher" means any person or entity that distributes copies of
     the Document to the public.

     A section "Entitled XYZ" means a named subunit of the Document whose
     title either is precisely XYZ or contains XYZ in parentheses following
     text that translates XYZ in another language.  (Here XYZ stands for a
     specific section name mentioned below, such as "Acknowledgements",
     "Dedications", "Endorsements", or "History".)  To "Preserve the Title"
     of such a section when you modify the Document means that it remains a
     section "Entitled XYZ" according to this definition.

     The Document may include Warranty Disclaimers next to the notice which
     states that this License applies to the Document.  These Warranty
     Disclaimers are considered to be included by reference in this
     License, but only as regards disclaiming warranties: any other
     implication that these Warranty Disclaimers may have is void and has
     no effect on the meaning of this License.

     2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either
     commercially or noncommercially, provided that this License, the
     copyright notices, and the license notice saying this License applies
     to the Document are reproduced in all copies, and that you add no
     other conditions whatsoever to those of this License.  You may not use
     technical measures to obstruct or control the reading or further
     copying of the copies you make or distribute.  However, you may accept
     compensation in exchange for copies.  If you distribute a large enough
     number of copies you must also follow the conditions in section 3.

     You may also lend copies, under the same conditions stated above, and
     you may publicly display copies.


     3. COPYING IN QUANTITY

     If you publish printed copies (or copies in media that commonly have
     printed covers) of the Document, numbering more than 100, and the
     Document's license notice requires Cover Texts, you must enclose the
     copies in covers that carry, clearly and legibly, all these Cover
     Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on
     the back cover.  Both covers must also clearly and legibly identify
     you as the publisher of these copies.  The front cover must present
     the full title with all words of the title equally prominent and
     visible.  You may add other material on the covers in addition.
     Copying with changes limited to the covers, as long as they preserve
     the title of the Document and satisfy these conditions, can be treated
     as verbatim copying in other respects.

     If the required texts for either cover are too voluminous to fit
     legibly, you should put the first ones listed (as many as fit
     reasonably) on the actual cover, and continue the rest onto adjacent
     pages.

     If you publish or distribute Opaque copies of the Document numbering
     more than 100, you must either include a machine-readable Transparent
     copy along with each Opaque copy, or state in or with each Opaque copy
     a computer-network location from which the general network-using
     public has access to download using public-standard network protocols
     a complete Transparent copy of the Document, free of added material.
     If you use the latter option, you must take reasonably prudent steps,
     when you begin distribution of Opaque copies in quantity, to ensure
     that this Transparent copy will remain thus accessible at the stated
     location until at least one year after the last time you distribute an
     Opaque copy (directly or through your agents or retailers) of that
     edition to the public.

     It is requested, but not required, that you contact the authors of the
     Document well before redistributing any large number of copies, to
     give them a chance to provide you with an updated version of the
     Document.


     4. MODIFICATIONS

     You may copy and distribute a Modified Version of the Document under
     the conditions of sections 2 and 3 above, provided that you release
     the Modified Version under precisely this License, with the Modified
     Version filling the role of the Document, thus licensing distribution
     and modification of the Modified Version to whoever possesses a copy
     of it.  In addition, you must do these things in the Modified Version:

     A. Use in the Title Page (and on the covers, if any) a title distinct
        from that of the Document, and from those of previous versions
        (which should, if there were any, be listed in the History section
        of the Document).  You may use the same title as a previous version
        if the original publisher of that version gives permission.
     B. List on the Title Page, as authors, one or more persons or entities
        responsible for authorship of the modifications in the Modified
        Version, together with at least five of the principal authors of the
        Document (all of its principal authors, if it has fewer than five),
        unless they release you from this requirement.
     C. State on the Title page the name of the publisher of the
        Modified Version, as the publisher.
     D. Preserve all the copyright notices of the Document.
     E. Add an appropriate copyright notice for your modifications
        adjacent to the other copyright notices.
     F. Include, immediately after the copyright notices, a license notice
        giving the public permission to use the Modified Version under the
        terms of this License, in the form shown in the Addendum below.
     G. Preserve in that license notice the full lists of Invariant Sections
        and required Cover Texts given in the Document's license notice.
     H. Include an unaltered copy of this License.
     I. Preserve the section Entitled "History", Preserve its Title, and add
        to it an item stating at least the title, year, new authors, and
        publisher of the Modified Version as given on the Title Page.  If
        there is no section Entitled "History" in the Document, create one
        stating the title, year, authors, and publisher of the Document as
        given on its Title Page, then add an item describing the Modified
        Version as stated in the previous sentence.
     J. Preserve the network location, if any, given in the Document for
        public access to a Transparent copy of the Document, and likewise
        the network locations given in the Document for previous versions
        it was based on.  These may be placed in the "History" section.
        You may omit a network location for a work that was published at
        least four years before the Document itself, or if the original
        publisher of the version it refers to gives permission.
     K. For any section Entitled "Acknowledgements" or "Dedications",
        Preserve the Title of the section, and preserve in the section all
        the substance and tone of each of the contributor acknowledgements
        and/or dedications given therein.
     L. Preserve all the Invariant Sections of the Document,
        unaltered in their text and in their titles.  Section numbers
        or the equivalent are not considered part of the section titles.
     M. Delete any section Entitled "Endorsements".  Such a section
        may not be included in the Modified Version.
     N. Do not retitle any existing section to be Entitled "Endorsements"
        or to conflict in title with any Invariant Section.
     O. Preserve any Warranty Disclaimers.

     If the Modified Version includes new front-matter sections or
     appendices that qualify as Secondary Sections and contain no material
     copied from the Document, you may at your option designate some or all
     of these sections as invariant.  To do this, add their titles to the
     list of Invariant Sections in the Modified Version's license notice.
     These titles must be distinct from any other section titles.

     You may add a section Entitled "Endorsements", provided it contains
     nothing but endorsements of your Modified Version by various
     parties--for example, statements of peer review or that the text has
     been approved by an organization as the authoritative definition of a
     standard.

     You may add a passage of up to five words as a Front-Cover Text, and a
     passage of up to 25 words as a Back-Cover Text, to the end of the list
     of Cover Texts in the Modified Version.  Only one passage of
     Front-Cover Text and one of Back-Cover Text may be added by (or
     through arrangements made by) any one entity.  If the Document already
     includes a cover text for the same cover, previously added by you or
     by arrangement made by the same entity you are acting on behalf of,
     you may not add another; but you may replace the old one, on explicit
     permission from the previous publisher that added the old one.

     The author(s) and publisher(s) of the Document do not by this License
     give permission to use their names for publicity for or to assert or
     imply endorsement of any Modified Version.


     5. COMBINING DOCUMENTS

     You may combine the Document with other documents released under this
     License, under the terms defined in section 4 above for modified
     versions, provided that you include in the combination all of the
     Invariant Sections of all of the original documents, unmodified, and
     list them all as Invariant Sections of your combined work in its
     license notice, and that you preserve all their Warranty Disclaimers.

     The combined work need only contain one copy of this License, and
     multiple identical Invariant Sections may be replaced with a single
     copy.  If there are multiple Invariant Sections with the same name but
     different contents, make the title of each such section unique by
     adding at the end of it, in parentheses, the name of the original
     author or publisher of that section if known, or else a unique number.
     Make the same adjustment to the section titles in the list of
     Invariant Sections in the license notice of the combined work.

     In the combination, you must combine any sections Entitled "History"
     in the various original documents, forming one section Entitled
     "History"; likewise combine any sections Entitled "Acknowledgements",
     and any sections Entitled "Dedications".  You must delete all sections
     Entitled "Endorsements".


     6. COLLECTIONS OF DOCUMENTS

     You may make a collection consisting of the Document and other
     documents released under this License, and replace the individual
     copies of this License in the various documents with a single copy
     that is included in the collection, provided that you follow the rules
     of this License for verbatim copying of each of the documents in all
     other respects.

     You may extract a single document from such a collection, and
     distribute it individually under this License, provided you insert a
     copy of this License into the extracted document, and follow this
     License in all other respects regarding verbatim copying of that
     document.


     7. AGGREGATION WITH INDEPENDENT WORKS

     A compilation of the Document or its derivatives with other separate
     and independent documents or works, in or on a volume of a storage or
     distribution medium, is called an "aggregate" if the copyright
     resulting from the compilation is not used to limit the legal rights
     of the compilation's users beyond what the individual works permit.
     When the Document is included in an aggregate, this License does not
     apply to the other works in the aggregate which are not themselves
     derivative works of the Document.

     If the Cover Text requirement of section 3 is applicable to these
     copies of the Document, then if the Document is less than one half of
     the entire aggregate, the Document's Cover Texts may be placed on
     covers that bracket the Document within the aggregate, or the
     electronic equivalent of covers if the Document is in electronic form.
     Otherwise they must appear on printed covers that bracket the whole
     aggregate.


     8. TRANSLATION

     Translation is considered a kind of modification, so you may
     distribute translations of the Document under the terms of section 4.
     Replacing Invariant Sections with translations requires special
     permission from their copyright holders, but you may include
     translations of some or all Invariant Sections in addition to the
     original versions of these Invariant Sections.  You may include a
     translation of this License, and all the license notices in the
     Document, and any Warranty Disclaimers, provided that you also include
     the original English version of this License and the original versions
     of those notices and disclaimers.  In case of a disagreement between
     the translation and the original version of this License or a notice
     or disclaimer, the original version will prevail.

     If a section in the Document is Entitled "Acknowledgements",
     "Dedications", or "History", the requirement (section 4) to Preserve
     its Title (section 1) will typically require changing the actual
     title.


     9. TERMINATION

     You may not copy, modify, sublicense, or distribute the Document
     except as expressly provided under this License.  Any attempt
     otherwise to copy, modify, sublicense, or distribute it is void, and
     will automatically terminate your rights under this License.

     However, if you cease all violation of this License, then your license
     from a particular copyright holder is reinstated (a) provisionally,
     unless and until the copyright holder explicitly and finally
     terminates your license, and (b) permanently, if the copyright holder
     fails to notify you of the violation by some reasonable means prior to
     60 days after the cessation.

     Moreover, your license from a particular copyright holder is
     reinstated permanently if the copyright holder notifies you of the
     violation by some reasonable means, this is the first time you have
     received notice of violation of this License (for any work) from that
     copyright holder, and you cure the violation prior to 30 days after
     your receipt of the notice.

     Termination of your rights under this section does not terminate the
     licenses of parties who have received copies or rights from you under
     this License.  If your rights have been terminated and not permanently
     reinstated, receipt of a copy of some or all of the same material does
     not give you any rights to use it.


     10. FUTURE REVISIONS OF THIS LICENSE

     The Free Software Foundation may publish new, revised versions of the
     GNU Free Documentation License from time to time.  Such new versions
     will be similar in spirit to the present version, but may differ in
     detail to address new problems or concerns.  See
     http://www.gnu.org/copyleft/.

     Each version of the License is given a distinguishing version number.
     If the Document specifies that a particular numbered version of this
     License "or any later version" applies to it, you have the option of
     following the terms and conditions either of that specified version or
     of any later version that has been published (not as a draft) by the
     Free Software Foundation.  If the Document does not specify a version
     number of this License, you may choose any version ever published (not
     as a draft) by the Free Software Foundation.  If the Document
     specifies that a proxy can decide which future versions of this
     License can be used, that proxy's public statement of acceptance of a
     version permanently authorizes you to choose that version for the
     Document.

     11. RELICENSING

     "Massive Multiauthor Collaboration Site" (or "MMC Site") means any
     World Wide Web server that publishes copyrightable works and also
     provides prominent facilities for anybody to edit those works.  A
     public wiki that anybody can edit is an example of such a server.  A
     "Massive Multiauthor Collaboration" (or "MMC") contained in the site
     means any set of copyrightable works thus published on the MMC site.

     "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
     license published by Creative Commons Corporation, a not-for-profit
     corporation with a principal place of business in San Francisco,
     California, as well as future copyleft versions of that license
     published by that same organization.

     "Incorporate" means to publish or republish a Document, in whole or in
     part, as part of another Document.

     An MMC is "eligible for relicensing" if it is licensed under this
     License, and if all works that were first published under this License
     somewhere other than this MMC, and subsequently incorporated in whole or
     in part into the MMC, (1) had no cover texts or invariant sections, and
     (2) were thus incorporated prior to November 1, 2008.

     The operator of an MMC Site may republish an MMC contained in the site
     under CC-BY-SA on the same site at any time before August 1, 2009,
     provided the MMC is eligible for relicensing.


     ADDENDUM: How to use this License for your documents

     To use this License in a document you have written, include a copy of
     the License in the document and put the following copyright and
     license notices just after the title page:

         Copyright (c)  YEAR  YOUR NAME.
         Permission is granted to copy, distribute and/or modify this document
         under the terms of the GNU Free Documentation License, Version 1.3
         or any later version published by the Free Software Foundation;
         with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
         A copy of the license is included in the section entitled "GNU
         Free Documentation License".

     If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
     replace the "with...Texts." line with this:

         with the Invariant Sections being LIST THEIR TITLES, with the
         Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.

     If you have Invariant Sections without Cover Texts, or some other
     combination of the three, merge those two alternatives to suit the
     situation.

     If your document contains nontrivial examples of program code, we
     recommend releasing these examples in parallel under your choice of
     free software license, such as the GNU General Public License,
     to permit their use in free software.

   * genindex


File: gsl-ref.info,  Node: Index,  Prev: GNU Free Documentation License,  Up: Top

Index
*****

[index]
* Menu:

* $, shell prompt:                       Further Information. (line  26)
* 2D histograms:                         Example programs for histograms.
                                                              (line  64)
* 2D random direction vector:            Spherical Vector Distributions.
                                                              (line  10)
* 3-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* 3D random direction vector:            Spherical Vector Distributions.
                                                              (line  33)
* 6-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* 9-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* acceleration of series:                References and Further Reading<25>.
                                                              (line  12)
* acosh:                                 Elementary Functions.
                                                              (line  36)
* Adams method:                          Stepping Functions.  (line 164)
* Adaptive step-size control, differential equations: Stepping Functions.
                                                              (line 187)
* Ai(x):                                 Airy Functions and Derivatives.
                                                              (line   6)
* Airy functions:                        Airy Functions and Derivatives.
                                                              (line   6)
* Akima splines:                         1D Interpolation Types.
                                                              (line  42)
* aliasing of arrays:                    Compatibility with C++.
                                                              (line  13)
* alternative optimized functions:       Portability functions.
                                                              (line  34)
* AMAX, Level-1 BLAS:                    Level 1.             (line  69)
* Angular Mathieu Functions:             Angular Mathieu Functions.
                                                              (line   6)
* angular reduction:                     Restriction Functions.
                                                              (line   6)
* ANSI C, use of:                        Conventions used in this manual.
                                                              (line  20)
* Apell symbol, see Pochhammer symbol:   Pochhammer Symbol.   (line   6)
* approximate comparison of floating point numbers: Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* arctangent integral:                   Arctangent Integral. (line   6)
* argument of complex number:            Properties of complex numbers.
                                                              (line   6)
* arithmetic exceptions:                 Representation of floating point numbers.
                                                              (line 135)
* asinh:                                 Elementary Functions.
                                                              (line  41)
* astronomical constants:                Fundamental Constants.
                                                              (line  55)
* ASUM, Level-1 BLAS:                    Level 1.             (line  56)
* atanh:                                 Elementary Functions.
                                                              (line  46)
* atomic physics, constants:             Astronomy and Astrophysics.
                                                              (line  30)
* autoconf, using with GSL:              Contributors to GSL. (line 145)
* AXPY, Level-1 BLAS:                    Level 1.             (line 105)
* B-spline wavelets:                     Initialization.      (line  39)
* Bader and Deuflhard, Bulirsch-Stoer method.: Stepping Functions.
                                                              (line 157)
* balancing matrices:                    Banded LDLT Decomposition.
                                                              (line  62)
* banded Cholesky Decomposition:         Banded LU Decomposition.
                                                              (line  77)
* banded general matrices:               Banded Systems.      (line  14)
* banded LDLT decomposition:             Banded Cholesky Decomposition.
                                                              (line  99)
* banded LU Decomposition:               Symmetric Banded Format.
                                                              (line  35)
* banded matrices:                       Triangular Systems.  (line  44)
* banded symmetric matrices:             General Banded Format.
                                                              (line  27)
* Basic Linear Algebra Subroutines (BLAS): References and Further Reading<7>.
                                                              (line  14)
* Basic Linear Algebra Subroutines (BLAS) <1>: Autoconf Macros.
                                                              (line 106)
* basis splines, B-splines:              References and Further Reading<34>.
                                                              (line  38)
* basis splines, derivatives:            Evaluation of B-splines.
                                                              (line  35)
* basis splines, evaluation:             Constructing the knots vector.
                                                              (line  18)
* basis splines, examples:               Working with the Greville abscissae.
                                                              (line  20)
* basis splines, Greville abscissae:     Evaluation of B-spline derivatives.
                                                              (line  37)
* basis splines, initializing:           Overview<7>.         (line  34)
* basis splines, Marsden-Schoenberg points: Evaluation of B-spline derivatives.
                                                              (line  36)
* basis splines, overview:               Basis Splines.       (line  15)
* BDF method:                            Stepping Functions.  (line 175)
* Bernoulli trial, random variates:      The Bernoulli Distribution.
                                                              (line   6)
* Bessel functions:                      Bessel Functions.    (line   6)
* Bessel Functions, Fractional Order:    Regular Bessel Function—Fractional Order.
                                                              (line   6)
* best-fit parameters, covariance:       High Level Driver.   (line  40)
* Beta distribution:                     The Beta Distribution.
                                                              (line   6)
* Beta function:                         Beta Functions.      (line   6)
* Beta function, incomplete normalized:  Incomplete Beta Function.
                                                              (line   6)
* BFGS algorithm, minimization:          Algorithms with Derivatives.
                                                              (line  45)
* Bi(x):                                 Airy Functions and Derivatives.
                                                              (line   6)
* bias, IEEE format:                     IEEE floating-point arithmetic.
                                                              (line  11)
* bicubic interpolation:                 2D Interpolation Types.
                                                              (line  16)
* bidiagonalization of real matrices:    Hessenberg-Triangular Decomposition of Real Matrices.
                                                              (line  26)
* bilinear interpolation:                2D Interpolation Types.
                                                              (line  11)
* binning data:                          References and Further Reading<18>.
                                                              (line  18)
* Binomial random variates:              The Binomial Distribution.
                                                              (line   6)
* biorthogonal wavelets:                 Initialization.      (line  39)
* bisection algorithm for finding roots: Root Bracketing Algorithms.
                                                              (line  18)
* Bivariate Gaussian distribution:       The Bivariate Gaussian Distribution.
                                                              (line   6)
* Bivariate Gaussian distribution <1>:   The Multivariate Gaussian Distribution.
                                                              (line   6)
* BLAS:                                  References and Further Reading<7>.
                                                              (line  15)
* BLAS, Low-level C interface:           Autoconf Macros.     (line 107)
* BLAS, sparse:                          References and Further Reading<36>.
                                                              (line  13)
* blocks:                                Data types.          (line  60)
* bounds checking, extension to GCC:     Vector allocation.   (line  36)
* breakpoints:                           Debugging Numerical Programs.
                                                              (line   9)
* Brent’s method for finding minima:     Minimization Algorithms.
                                                              (line  34)
* Brent’s method for finding roots:      Root Bracketing Algorithms.
                                                              (line  54)
* Broyden algorithm for multidimensional roots: Algorithms without Derivatives.
                                                              (line  57)
* BSD random number generator:           Unix random number generators.
                                                              (line  17)
* bug-gsl:                               No Warranty.         (line  11)
* bug-gsl mailing list:                  No Warranty.         (line  12)
* bugs:                                  No Warranty.         (line  12)
* Bulirsch-Stoer method:                 Stepping Functions.  (line 157)
* C extensions, compatible use of:       Conventions used in this manual.
                                                              (line  20)
* C++, compatibility:                    Support for different numeric types.
                                                              (line  67)
* C99, inline keyword:                   ANSI C Compliance.   (line  25)
* Carlson forms of Elliptic integrals:   Definition of Carlson Forms.
                                                              (line   6)
* Cash-Karp, Runge-Kutta method:         Stepping Functions.  (line 121)
* Cauchy distribution:                   The Cauchy Distribution.
                                                              (line   6)
* Cauchy principal value, by numerical quadrature: QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* CBLAS:                                 References and Further Reading<7>.
                                                              (line  15)
* CBLAS, Low-level interface:            Autoconf Macros.     (line 107)
* cblas_caxpy (C function):              Level 1<2>.          (line  90)
* cblas_ccopy (C function):              Level 1<2>.          (line  87)
* cblas_cdotc_sub (C function):          Level 1<2>.          (line  21)
* cblas_cdotu_sub (C function):          Level 1<2>.          (line  18)
* cblas_cgbmv (C function):              Level 2<2>.          (line  93)
* cblas_cgemm (C function):              Level 3<2>.          (line  76)
* cblas_cgemv (C function):              Level 2<2>.          (line  88)
* cblas_cgerc (C function):              Level 2<2>.          (line 263)
* cblas_cgeru (C function):              Level 2<2>.          (line 259)
* cblas_chbmv (C function):              Level 2<2>.          (line 249)
* cblas_chemm (C function):              Level 3<2>.          (line 146)
* cblas_chemv (C function):              Level 2<2>.          (line 244)
* cblas_cher (C function):               Level 2<2>.          (line 267)
* cblas_cher2 (C function):              Level 2<2>.          (line 275)
* cblas_cher2k (C function):             Level 3<2>.          (line 157)
* cblas_cherk (C function):              Level 3<2>.          (line 152)
* cblas_chpmv (C function):              Level 2<2>.          (line 254)
* cblas_chpr (C function):               Level 2<2>.          (line 271)
* cblas_chpr2 (C function):              Level 2<2>.          (line 280)
* cblas_cscal (C function):              Level 1<2>.          (line 131)
* cblas_csscal (C function):             Level 1<2>.          (line 137)
* cblas_cswap (C function):              Level 1<2>.          (line  84)
* cblas_csymm (C function):              Level 3<2>.          (line  82)
* cblas_csyr2k (C function):             Level 3<2>.          (line  93)
* cblas_csyrk (C function):              Level 3<2>.          (line  88)
* cblas_ctbmv (C function):              Level 2<2>.          (line 104)
* cblas_ctbsv (C function):              Level 2<2>.          (line 119)
* cblas_ctpmv (C function):              Level 2<2>.          (line 109)
* cblas_ctpsv (C function):              Level 2<2>.          (line 124)
* cblas_ctrmm (C function):              Level 3<2>.          (line  99)
* cblas_ctrmv (C function):              Level 2<2>.          (line  99)
* cblas_ctrsm (C function):              Level 3<2>.          (line 105)
* cblas_ctrsv (C function):              Level 2<2>.          (line 114)
* cblas_dasum (C function):              Level 1<2>.          (line  39)
* cblas_daxpy (C function):              Level 1<2>.          (line  81)
* cblas_dcopy (C function):              Level 1<2>.          (line  78)
* cblas_ddot (C function):               Level 1<2>.          (line  15)
* cblas_dgbmv (C function):              Level 2<2>.          (line  52)
* cblas_dgemm (C function):              Level 3<2>.          (line  41)
* cblas_dgemv (C function):              Level 2<2>.          (line  47)
* cblas_dger (C function):               Level 2<2>.          (line 221)
* cblas_dnrm2 (C function):              Level 1<2>.          (line  36)
* cblas_drot (C function):               Level 1<2>.          (line 119)
* cblas_drotg (C function):              Level 1<2>.          (line 113)
* cblas_drotm (C function):              Level 1<2>.          (line 122)
* cblas_drotmg (C function):             Level 1<2>.          (line 116)
* cblas_dsbmv (C function):              Level 2<2>.          (line 211)
* cblas_dscal (C function):              Level 1<2>.          (line 128)
* cblas_dsdot (C function):              Level 1<2>.          (line   9)
* cblas_dspmv (C function):              Level 2<2>.          (line 216)
* cblas_dspr (C function):               Level 2<2>.          (line 230)
* cblas_dspr2 (C function):              Level 2<2>.          (line 239)
* cblas_dswap (C function):              Level 1<2>.          (line  75)
* cblas_dsymm (C function):              Level 3<2>.          (line  47)
* cblas_dsymv (C function):              Level 2<2>.          (line 206)
* cblas_dsyr (C function):               Level 2<2>.          (line 226)
* cblas_dsyr2 (C function):              Level 2<2>.          (line 234)
* cblas_dsyr2k (C function):             Level 3<2>.          (line  58)
* cblas_dsyrk (C function):              Level 3<2>.          (line  53)
* cblas_dtbmv (C function):              Level 2<2>.          (line  63)
* cblas_dtbsv (C function):              Level 2<2>.          (line  78)
* cblas_dtpmv (C function):              Level 2<2>.          (line  68)
* cblas_dtpsv (C function):              Level 2<2>.          (line  83)
* cblas_dtrmm (C function):              Level 3<2>.          (line  64)
* cblas_dtrmv (C function):              Level 2<2>.          (line  58)
* cblas_dtrsm (C function):              Level 3<2>.          (line  70)
* cblas_dtrsv (C function):              Level 2<2>.          (line  73)
* cblas_dzasum (C function):             Level 1<2>.          (line  51)
* cblas_dznrm2 (C function):             Level 1<2>.          (line  48)
* cblas_icamax (C function):             Level 1<2>.          (line  60)
* cblas_idamax (C function):             Level 1<2>.          (line  57)
* cblas_isamax (C function):             Level 1<2>.          (line  54)
* cblas_izamax (C function):             Level 1<2>.          (line  63)
* cblas_sasum (C function):              Level 1<2>.          (line  33)
* cblas_saxpy (C function):              Level 1<2>.          (line  72)
* cblas_scasum (C function):             Level 1<2>.          (line  45)
* cblas_scnrm2 (C function):             Level 1<2>.          (line  42)
* cblas_scopy (C function):              Level 1<2>.          (line  69)
* cblas_sdot (C function):               Level 1<2>.          (line  12)
* cblas_sdsdot (C function):             Level 1<2>.          (line   6)
* cblas_sgbmv (C function):              Level 2<2>.          (line  11)
* cblas_sgemm (C function):              Level 3<2>.          (line   6)
* cblas_sgemv (C function):              Level 2<2>.          (line   6)
* cblas_sger (C function):               Level 2<2>.          (line 185)
* cblas_snrm2 (C function):              Level 1<2>.          (line  30)
* cblas_srot (C function):               Level 1<2>.          (line 107)
* cblas_srotg (C function):              Level 1<2>.          (line 102)
* cblas_srotm (C function):              Level 1<2>.          (line 110)
* cblas_srotmg (C function):             Level 1<2>.          (line 104)
* cblas_ssbmv (C function):              Level 2<2>.          (line 175)
* cblas_sscal (C function):              Level 1<2>.          (line 125)
* cblas_sspmv (C function):              Level 2<2>.          (line 180)
* cblas_sspr (C function):               Level 2<2>.          (line 193)
* cblas_sspr2 (C function):              Level 2<2>.          (line 202)
* cblas_sswap (C function):              Level 1<2>.          (line  66)
* cblas_ssymm (C function):              Level 3<2>.          (line  12)
* cblas_ssymv (C function):              Level 2<2>.          (line 170)
* cblas_ssyr (C function):               Level 2<2>.          (line 189)
* cblas_ssyr2 (C function):              Level 2<2>.          (line 197)
* cblas_ssyr2k (C function):             Level 3<2>.          (line  23)
* cblas_ssyrk (C function):              Level 3<2>.          (line  18)
* cblas_stbmv (C function):              Level 2<2>.          (line  22)
* cblas_stbsv (C function):              Level 2<2>.          (line  37)
* cblas_stpmv (C function):              Level 2<2>.          (line  27)
* cblas_stpsv (C function):              Level 2<2>.          (line  42)
* cblas_strmm (C function):              Level 3<2>.          (line  29)
* cblas_strmv (C function):              Level 2<2>.          (line  17)
* cblas_strsm (C function):              Level 3<2>.          (line  35)
* cblas_strsv (C function):              Level 2<2>.          (line  32)
* cblas_xerbla (C function):             Level 3<2>.          (line 180)
* cblas_zaxpy (C function):              Level 1<2>.          (line  99)
* cblas_zcopy (C function):              Level 1<2>.          (line  96)
* cblas_zdotc_sub (C function):          Level 1<2>.          (line  27)
* cblas_zdotu_sub (C function):          Level 1<2>.          (line  24)
* cblas_zdscal (C function):             Level 1<2>.          (line 140)
* cblas_zgbmv (C function):              Level 2<2>.          (line 134)
* cblas_zgemm (C function):              Level 3<2>.          (line 111)
* cblas_zgemv (C function):              Level 2<2>.          (line 129)
* cblas_zgerc (C function):              Level 2<2>.          (line 303)
* cblas_zgeru (C function):              Level 2<2>.          (line 299)
* cblas_zhbmv (C function):              Level 2<2>.          (line 289)
* cblas_zhemm (C function):              Level 3<2>.          (line 163)
* cblas_zhemv (C function):              Level 2<2>.          (line 284)
* cblas_zher (C function):               Level 2<2>.          (line 307)
* cblas_zher2 (C function):              Level 2<2>.          (line 315)
* cblas_zher2k (C function):             Level 3<2>.          (line 174)
* cblas_zherk (C function):              Level 3<2>.          (line 169)
* cblas_zhpmv (C function):              Level 2<2>.          (line 294)
* cblas_zhpr (C function):               Level 2<2>.          (line 311)
* cblas_zhpr2 (C function):              Level 2<2>.          (line 320)
* cblas_zscal (C function):              Level 1<2>.          (line 134)
* cblas_zswap (C function):              Level 1<2>.          (line  93)
* cblas_zsymm (C function):              Level 3<2>.          (line 117)
* cblas_zsyr2k (C function):             Level 3<2>.          (line 128)
* cblas_zsyrk (C function):              Level 3<2>.          (line 123)
* cblas_ztbmv (C function):              Level 2<2>.          (line 145)
* cblas_ztbsv (C function):              Level 2<2>.          (line 160)
* cblas_ztpmv (C function):              Level 2<2>.          (line 150)
* cblas_ztpsv (C function):              Level 2<2>.          (line 165)
* cblas_ztrmm (C function):              Level 3<2>.          (line 134)
* cblas_ztrmv (C function):              Level 2<2>.          (line 140)
* cblas_ztrsm (C function):              Level 3<2>.          (line 140)
* cblas_ztrsv (C function):              Level 2<2>.          (line 155)
* CDFs, cumulative distribution functions: References.        (line  14)
* ce(q,x), Mathieu function:             Angular Mathieu Functions.
                                                              (line   6)
* Chebyshev series:                      References and Further Reading<24>.
                                                              (line  15)
* checking combination for validity:     Combination properties.
                                                              (line  20)
* checking multiset for validity:        Multiset properties. (line  20)
* checking permutation for validity:     Permutation properties.
                                                              (line  15)
* Chi(x):                                Hyperbolic Integrals.
                                                              (line   6)
* Chi-squared distribution:              The Chi-squared Distribution.
                                                              (line  14)
* Cholesky decomposition:                Singular Value Decomposition.
                                                              (line  97)
* Cholesky decomposition, banded:        Banded LU Decomposition.
                                                              (line  78)
* Cholesky decomposition, modified:      Pivoted Cholesky Decomposition.
                                                              (line 111)
* Cholesky decomposition, pivoted:       Cholesky Decomposition.
                                                              (line 155)
* Cholesky decomposition, square root free: Modified Cholesky Decomposition.
                                                              (line  68)
* Ci(x):                                 Trigonometric Integrals.
                                                              (line   6)
* Clausen functions:                     Clausen Functions.   (line   6)
* Clenshaw-Curtis quadrature:            Integrands with weight functions.
                                                              (line   6)
* CMRG, combined multiple recursive random number generator: Random number generator algorithms.
                                                              (line 103)
* code reuse in applications:            Deprecated Functions.
                                                              (line  14)
* combinations:                          References and Further Reading<5>.
                                                              (line  16)
* combinatorial factor C(m:              Factorials.          (line  43)
* combinatorial optimization:            References and Further Reading<20>.
                                                              (line  21)
* comparison functions, definition:      Sorting objects.     (line  16)
* compatibility:                         Conventions used in this manual.
                                                              (line  19)
* compiling programs, include paths:     An Example Program.  (line  29)
* compiling programs, library paths:     Compiling and Linking.
                                                              (line  24)
* complementary incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line  22)
* complete Fermi-Dirac integrals:        Complete Fermi-Dirac Integrals.
                                                              (line   6)
* complete orthogonal decomposition:     QL Decomposition.    (line  36)
* complex arithmetic:                    Properties of complex numbers.
                                                              (line  28)
* complex cosine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line  12)
* Complex Gamma function:                Gamma Functions.     (line  60)
* complex hermitian matrix, eigensystem: Complex Hermitian Matrices.
                                                              (line   9)
* complex log sine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line  19)
* complex numbers:                       Approximate Comparison of Floating Point Numbers.
                                                              (line  30)
* complex sinc function, special functions: Circular Trigonometric Functions.
                                                              (line  23)
* complex sine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line   6)
* confluent hypergeometric function:     Laguerre Functions.  (line   6)
* confluent hypergeometric functions:    Hypergeometric Functions.
                                                              (line   6)
* conical functions:                     Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* Conjugate gradient algorithm, minimization: Algorithms with Derivatives.
                                                              (line  15)
* conjugate of complex number:           Complex arithmetic operators.
                                                              (line  66)
* constant matrix:                       Accessing matrix elements.
                                                              (line  48)
* constants, fundamental:                Physical Constants.  (line  22)
* constants, mathematical (defined as macros): Mathematical Functions.
                                                              (line  14)
* constants, physical:                   References and Further Reading<38>.
                                                              (line  15)
* constants, prefixes:                   Force and Energy.    (line  21)
* contacting the GSL developers:         Reporting Bugs.      (line  30)
* conventions, used in manual:           Further Information. (line  27)
* convergence, accelerating a series:    References and Further Reading<25>.
                                                              (line  11)
* conversion of units:                   References and Further Reading<38>.
                                                              (line  15)
* cooling schedule:                      Simulated Annealing algorithm.
                                                              (line  20)
* COPY, Level-1 BLAS:                    Level 1.             (line  94)
* correlation, of two datasets:          Covariance.          (line  25)
* cosine function, special functions:    Circular Trigonometric Functions.
                                                              (line  11)
* cosine of complex number:              Complex Trigonometric Functions.
                                                              (line  11)
* cost function:                         References and Further Reading<20>.
                                                              (line  20)
* Coulomb wave functions:                Coulomb Functions.   (line   6)
* coupling coefficients:                 Coupling Coefficients.
                                                              (line   6)
* covariance matrix, from linear regression: Linear regression with a constant term.
                                                              (line   9)
* covariance matrix, linear fits:        Overview<5>.         (line  18)
* covariance matrix, nonlinear fits:     High Level Driver.   (line  39)
* covariance, of two datasets:           Autocorrelation.     (line  22)
* cquad, doubly-adaptive integration:    QAWF adaptive integration for Fourier integrals.
                                                              (line  60)
* CRAY random number generator, RANF:    Other random number generators.
                                                              (line  22)
* cubic equation, solving:               Quadratic Equations. (line  44)
* cubic splines:                         1D Interpolation Types.
                                                              (line  23)
* cumulative distribution functions (CDFs): References.       (line  14)
* Cylindrical Bessel Functions:          Regular Cylindrical Bessel Functions.
                                                              (line   6)
* Daubechies wavelets:                   Initialization.      (line  24)
* Dawson function:                       Dawson Function.     (line   6)
* DAXPY, Level-1 BLAS:                   Level 1.             (line 105)
* debugging numerical programs:          Debugging Numerical Programs.
                                                              (line   9)
* Debye functions:                       Debye Functions.     (line   6)
* denormalized form, IEEE format:        Representation of floating point numbers.
                                                              (line  14)
* deprecated functions:                  Thread-safety.       (line  21)
* derivatives, calculating numerically:  References and Further Reading<23>.
                                                              (line  18)
* determinant of a matrix, by LU decomposition: LU Decomposition.
                                                              (line 111)
* Deuflhard and Bader, Bulirsch-Stoer method.: Stepping Functions.
                                                              (line 157)
* DFTs, see FFT:                         References and Further Reading<10>.
                                                              (line  27)
* diagonal, of a matrix:                 Creating row and column views.
                                                              (line  70)
* differential equations, initial value problems: References and Further Reading<21>.
                                                              (line  11)
* differentiation of functions, numeric: References and Further Reading<23>.
                                                              (line  18)
* digamma function:                      Psi Digamma Function.
                                                              (line   6)
* dilogarithm:                           Dilogarithm.         (line   6)
* direction vector, random 2D:           Spherical Vector Distributions.
                                                              (line  10)
* direction vector, random 3D:           Spherical Vector Distributions.
                                                              (line  33)
* direction vector, random N-dimensional: Spherical Vector Distributions.
                                                              (line  44)
* Dirichlet distribution:                The Dirichlet Distribution.
                                                              (line   6)
* discontinuities, in ODE systems:       Evolution.           (line  92)
* Discrete Fourier Transforms, see FFT:  References and Further Reading<10>.
                                                              (line  28)
* discrete Hankel transforms:            References and Further Reading<27>.
                                                              (line  54)
* Discrete Newton algorithm for multidimensional roots: Algorithms without Derivatives.
                                                              (line  34)
* Discrete random numbers:               General Discrete Distributions.
                                                              (line  49)
* Discrete random numbers <1>:           General Discrete Distributions.
                                                              (line  66)
* Discrete random numbers <2>:           General Discrete Distributions.
                                                              (line  72)
* Discrete random numbers <3>:           General Discrete Distributions.
                                                              (line  81)
* Discrete random numbers, preprocessing: General Discrete Distributions.
                                                              (line  49)
* divided differences, polynomials:      Polynomial Evaluation.
                                                              (line  39)
* division by zero, IEEE exceptions:     Representation of floating point numbers.
                                                              (line 135)
* Dogleg algorithm:                      Levenberg-Marquardt with Geodesic Acceleration.
                                                              (line  30)
* Dogleg algorithm, double:              Dogleg.              (line  27)
* dollar sign $, shell prompt:           Conventions used in this manual.
                                                              (line   6)
* DOT, Level-1 BLAS:                     Level 1.             (line   6)
* double Dogleg algorithm:               Dogleg.              (line  27)
* double factorial:                      Factorials.          (line  19)
* double precision, IEEE format:         Representation of floating point numbers.
                                                              (line  43)
* downloading GSL:                       GSL is Free Software.
                                                              (line  46)
* DWT initialization:                    Definitions<2>.      (line  27)
* DWT, mathematical definition:          Wavelet Transforms.  (line  11)
* DWT, one dimensional:                  Transform Functions. (line  12)
* DWT, see wavelet transforms:           References and Further Reading<26>.
                                                              (line  25)
* DWT, two dimensional:                  Wavelet transforms in one dimension.
                                                              (line  41)
* e, defined as a macro:                 Mathematical Constants.
                                                              (line   9)
* E1(x):                                 Exponential Integral.
                                                              (line   6)
* E2(x):                                 Exponential Integral.
                                                              (line   6)
* Ei(x):                                 Exponential Integral.
                                                              (line   6)
* eigenvalues and eigenvectors:          References and Further Reading<9>.
                                                              (line  64)
* elementary functions:                  Examples.            (line  39)
* elementary operations:                 Elementary Operations.
                                                              (line   6)
* elliptic functions (Jacobi):           Elliptic Functions Jacobi.
                                                              (line   6)
* elliptic integrals:                    Elliptic Integrals.  (line   6)
* energy function:                       References and Further Reading<20>.
                                                              (line  21)
* energy, units of:                      Mass and Weight.     (line  50)
* erf(x):                                Error Functions.     (line   6)
* erfc(x):                               Error Functions.     (line   6)
* Erlang distribution:                   The Gamma Distribution.
                                                              (line   6)
* error codes:                           Error Codes.         (line  13)
* error codes, reserved:                 Error Reporting.     (line  44)
* error function:                        Error Functions.     (line   6)
* error handlers:                        Error Codes.         (line  50)
* error handling:                        Code Reuse.          (line  14)
* error handling macros:                 Error Handlers.      (line  82)
* estimated standard deviation:          References and Further Reading<14>.
                                                              (line  52)
* estimated variance:                    References and Further Reading<14>.
                                                              (line  52)
* estimation, location:                  Order Statistics.    (line  22)
* estimation, scale:                     Gastwirth Estimator. (line  24)
* Eta Function:                          Eta Function.        (line   6)
* euclidean distance function, hypot:    Elementary Functions.
                                                              (line  24)
* euclidean distance function, hypot3:   Elementary Functions.
                                                              (line  30)
* Euler’s constant, defined as a macro:  Mathematical Constants.
                                                              (line   9)
* evaluation of polynomials:             Polynomials.         (line  13)
* evaluation of polynomials, in divided difference form: Polynomial Evaluation.
                                                              (line  38)
* examples, conventions used in:         Further Information. (line  27)
* exceptions, C++:                       Support for different numeric types.
                                                              (line  66)
* exceptions, floating point:            Examining floating point registers.
                                                              (line  32)
* exceptions, IEEE arithmetic:           Representation of floating point numbers.
                                                              (line 135)
* exchanging permutation elements:       Accessing permutation elements.
                                                              (line  18)
* exp:                                   Exponential Functions.
                                                              (line   6)
* expm1:                                 Elementary Functions.
                                                              (line  18)
* exponent, IEEE format:                 IEEE floating-point arithmetic.
                                                              (line  11)
* Exponential distribution:              The Exponential Distribution.
                                                              (line   6)
* exponential function:                  Exponential Functions.
                                                              (line   6)
* exponential integrals:                 Exponential Integrals.
                                                              (line   6)
* Exponential power distribution:        The Exponential Power Distribution.
                                                              (line   6)
* exponential, difference from 1 computed accurately: Elementary Functions.
                                                              (line  18)
* exponentiation of complex number:      Elementary Complex Functions.
                                                              (line  17)
* extern inline:                         ANSI C Compliance.   (line  24)
* F-distribution:                        The F-distribution.  (line  13)
* factorial:                             Factorials.          (line   6)
* factorial <1>:                         Factorials.          (line  11)
* factorization of matrices:             References and Further Reading<8>.
                                                              (line  32)
* false position algorithm for finding roots: Root Bracketing Algorithms.
                                                              (line  35)
* Fast Fourier Transforms, see FFT:      References and Further Reading<10>.
                                                              (line  28)
* Fehlberg method, differential equations: Stepping Functions.
                                                              (line 115)
* Fermi-Dirac function:                  Fermi-Dirac Function.
                                                              (line   6)
* FFT:                                   References and Further Reading<10>.
                                                              (line  28)
* FFT mathematical definition:           Fast Fourier Transforms FFTs.
                                                              (line  18)
* FFT of complex data, mixed-radix algorithm: Radix-2 FFT routines for complex data.
                                                              (line 120)
* FFT of complex data, radix-2 algorithm: Overview of complex data FFTs.
                                                              (line  73)
* FFT of real data:                      Mixed-radix FFT routines for complex data.
                                                              (line 226)
* FFT of real data, mixed-radix algorithm: Radix-2 FFT routines for real data.
                                                              (line  98)
* FFT of real data, radix-2 algorithm:   Overview of real data FFTs.
                                                              (line  50)
* FFT, complex data:                     Mathematical Definitions.
                                                              (line  54)
* finding minima:                        References and Further Reading<29>.
                                                              (line  17)
* finding roots:                         References and Further Reading<28>.
                                                              (line  15)
* finding zeros:                         References and Further Reading<28>.
                                                              (line  15)
* fits, multi-parameter linear:          Linear regression without a constant term.
                                                              (line  50)
* fitting:                               References and Further Reading<32>.
                                                              (line  24)
* fitting, using Chebyshev polynomials:  References and Further Reading<24>.
                                                              (line  15)
* Fj(x), Fermi-Dirac integral:           Complete Fermi-Dirac Integrals.
                                                              (line   6)
* Fj(x,b), incomplete Fermi-Dirac integral: Incomplete Fermi-Dirac Integrals.
                                                              (line   6)
* flat distribution:                     The Flat Uniform Distribution.
                                                              (line   6)
* Fletcher-Reeves conjugate gradient algorithm, minimization: Algorithms with Derivatives.
                                                              (line  15)
* floating point exceptions:             Examining floating point registers.
                                                              (line  31)
* floating point numbers, approximate comparison: Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* floating point registers:              Using gdb.           (line  89)
* force and energy:                      Radioactivity.       (line  18)
* Fortran range checking, equivalent in gcc: Vector allocation.
                                                              (line  35)
* Four-tap Generalized Feedback Shift Register: Random number generator algorithms.
                                                              (line 181)
* Fourier integrals, numerical:          QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* Fourier Transforms, see FFT:           References and Further Reading<10>.
                                                              (line  28)
* Fractional Order Bessel Functions:     Regular Bessel Function—Fractional Order.
                                                              (line   6)
* free software, explanation of:         Routines available in GSL.
                                                              (line  53)
* frexp:                                 Elementary Functions.
                                                              (line  56)
* functions, numerical differentiation:  References and Further Reading<23>.
                                                              (line  18)
* fundamental constants:                 Physical Constants.  (line  22)
* Gamma distribution:                    The Gamma Distribution.
                                                              (line   6)
* gamma functions:                       Gamma Functions.     (line   6)
* Gastwirth estimator:                   Trimmed Mean.        (line  25)
* Gauss-Kronrod quadrature:              Introduction<2>.     (line  57)
* Gaussian distribution:                 The Gaussian Distribution.
                                                              (line   6)
* Gaussian distribution, bivariate:      The Bivariate Gaussian Distribution.
                                                              (line   6)
* Gaussian distribution, bivariate <1>:  The Multivariate Gaussian Distribution.
                                                              (line   6)
* Gaussian Tail distribution:            The Gaussian Tail Distribution.
                                                              (line   6)
* gcc extensions, range-checking:        Vector allocation.   (line  36)
* gcc warning options:                   Handling floating point exceptions.
                                                              (line  30)
* gdb:                                   Debugging Numerical Programs.
                                                              (line   9)
* Gegenbauer functions:                  Gegenbauer Functions.
                                                              (line   6)
* GEMM, Level-3 BLAS:                    Level 3.             (line   6)
* GEMV, Level-2 BLAS:                    Level 2.             (line   6)
* general polynomial equations, solving: Cubic Equations.     (line  37)
* generalized eigensystems:              Real Generalized Nonsymmetric Eigensystems.
                                                              (line   6)
* generalized hermitian definite eigensystems: Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line   6)
* generalized symmetric eigensystems:    Real Generalized Symmetric-Definite Eigensystems.
                                                              (line   6)
* Geometric random variates:             The Geometric Distribution.
                                                              (line   6)
* Geometric random variates <1>:         The Hypergeometric Distribution.
                                                              (line   6)
* GER, Level-2 BLAS:                     Level 2.             (line 103)
* GERC, Level-2 BLAS:                    Level 2.             (line 117)
* GERU, Level-2 BLAS:                    Level 2.             (line 103)
* Givens rotation:                       Bidiagonalization.   (line  57)
* Givens Rotation, BLAS:                 Level 1.             (line 130)
* Givens Rotation, Modified, BLAS:       Level 1.             (line 152)
* gmres:                                 Types of Sparse Iterative Solvers.
                                                              (line  11)
* GNU General Public License:            Top.                 (line  12)
* golden section algorithm for finding minima: Minimization Algorithms.
                                                              (line  15)
* gsl_acosh (C function):                Elementary Functions.
                                                              (line  36)
* gsl_asinh (C function):                Elementary Functions.
                                                              (line  41)
* gsl_atanh (C function):                Elementary Functions.
                                                              (line  46)
* gsl_blas_caxpy (C function):           Level 1.             (line 105)
* gsl_blas_ccopy (C function):           Level 1.             (line  94)
* gsl_blas_cdotc (C function):           Level 1.             (line  33)
* gsl_blas_cdotu (C function):           Level 1.             (line  24)
* gsl_blas_cgemm (C function):           Level 3.             (line   6)
* gsl_blas_cgemv (C function):           Level 2.             (line   6)
* gsl_blas_cgerc (C function):           Level 2.             (line 117)
* gsl_blas_cgeru (C function):           Level 2.             (line 103)
* gsl_blas_chemm (C function):           Level 3.             (line  50)
* gsl_blas_chemv (C function):           Level 2.             (line  85)
* gsl_blas_cher (C function):            Level 2.             (line 140)
* gsl_blas_cher2 (C function):           Level 2.             (line 168)
* gsl_blas_cher2k (C function):          Level 3.             (line 187)
* gsl_blas_cherk (C function):           Level 3.             (line 144)
* gsl_blas_cscal (C function):           Level 1.             (line 117)
* gsl_blas_csscal (C function):          Level 1.             (line 117)
* gsl_blas_cswap (C function):           Level 1.             (line  83)
* gsl_blas_csymm (C function):           Level 3.             (line  27)
* gsl_blas_csyr2k (C function):          Level 3.             (line 162)
* gsl_blas_csyrk (C function):           Level 3.             (line 121)
* gsl_blas_ctrmm (C function):           Level 3.             (line  68)
* gsl_blas_ctrmv (C function):           Level 2.             (line  25)
* gsl_blas_ctrsm (C function):           Level 3.             (line  94)
* gsl_blas_ctrsv (C function):           Level 2.             (line  48)
* gsl_blas_dasum (C function):           Level 1.             (line  56)
* gsl_blas_daxpy (C function):           Level 1.             (line 105)
* gsl_blas_dcopy (C function):           Level 1.             (line  94)
* gsl_blas_ddot (C function):            Level 1.             (line  13)
* gsl_blas_dgemm (C function):           Level 3.             (line   6)
* gsl_blas_dgemv (C function):           Level 2.             (line   6)
* gsl_blas_dger (C function):            Level 2.             (line 103)
* gsl_blas_dnrm2 (C function):           Level 1.             (line  42)
* gsl_blas_drot (C function):            Level 1.             (line 144)
* gsl_blas_drotg (C function):           Level 1.             (line 130)
* gsl_blas_drotm (C function):           Level 1.             (line 161)
* gsl_blas_drotmg (C function):          Level 1.             (line 152)
* gsl_blas_dscal (C function):           Level 1.             (line 117)
* gsl_blas_dsdot (C function):           Level 1.             (line  13)
* gsl_blas_dswap (C function):           Level 1.             (line  83)
* gsl_blas_dsymm (C function):           Level 3.             (line  27)
* gsl_blas_dsymv (C function):           Level 2.             (line  70)
* gsl_blas_dsyr (C function):            Level 2.             (line 127)
* gsl_blas_dsyr2 (C function):           Level 2.             (line 154)
* gsl_blas_dsyr2k (C function):          Level 3.             (line 162)
* gsl_blas_dsyrk (C function):           Level 3.             (line 121)
* gsl_blas_dtrmm (C function):           Level 3.             (line  68)
* gsl_blas_dtrmv (C function):           Level 2.             (line  25)
* gsl_blas_dtrsm (C function):           Level 3.             (line  94)
* gsl_blas_dtrsv (C function):           Level 2.             (line  48)
* gsl_blas_dzasum (C function):          Level 1.             (line  62)
* gsl_blas_dznrm2 (C function):          Level 1.             (line  48)
* gsl_blas_icamax (C function):          Level 1.             (line  69)
* gsl_blas_idamax (C function):          Level 1.             (line  69)
* gsl_blas_isamax (C function):          Level 1.             (line  69)
* gsl_blas_izamax (C function):          Level 1.             (line  69)
* gsl_blas_sasum (C function):           Level 1.             (line  56)
* gsl_blas_saxpy (C function):           Level 1.             (line 105)
* gsl_blas_scasum (C function):          Level 1.             (line  62)
* gsl_blas_scnrm2 (C function):          Level 1.             (line  48)
* gsl_blas_scopy (C function):           Level 1.             (line  94)
* gsl_blas_sdot (C function):            Level 1.             (line  13)
* gsl_blas_sdsdot (C function):          Level 1.             (line   6)
* gsl_blas_sgemm (C function):           Level 3.             (line   6)
* gsl_blas_sgemv (C function):           Level 2.             (line   6)
* gsl_blas_sger (C function):            Level 2.             (line 103)
* gsl_blas_snrm2 (C function):           Level 1.             (line  42)
* gsl_blas_srot (C function):            Level 1.             (line 144)
* gsl_blas_srotg (C function):           Level 1.             (line 130)
* gsl_blas_srotm (C function):           Level 1.             (line 161)
* gsl_blas_srotmg (C function):          Level 1.             (line 152)
* gsl_blas_sscal (C function):           Level 1.             (line 117)
* gsl_blas_sswap (C function):           Level 1.             (line  83)
* gsl_blas_ssymm (C function):           Level 3.             (line  27)
* gsl_blas_ssymv (C function):           Level 2.             (line  70)
* gsl_blas_ssyr (C function):            Level 2.             (line 127)
* gsl_blas_ssyr2 (C function):           Level 2.             (line 154)
* gsl_blas_ssyr2k (C function):          Level 3.             (line 162)
* gsl_blas_ssyrk (C function):           Level 3.             (line 121)
* gsl_blas_strmm (C function):           Level 3.             (line  68)
* gsl_blas_strmv (C function):           Level 2.             (line  25)
* gsl_blas_strsm (C function):           Level 3.             (line  94)
* gsl_blas_strsv (C function):           Level 2.             (line  48)
* gsl_blas_zaxpy (C function):           Level 1.             (line 105)
* gsl_blas_zcopy (C function):           Level 1.             (line  94)
* gsl_blas_zdotc (C function):           Level 1.             (line  33)
* gsl_blas_zdotu (C function):           Level 1.             (line  24)
* gsl_blas_zdscal (C function):          Level 1.             (line 117)
* gsl_blas_zgemm (C function):           Level 3.             (line   6)
* gsl_blas_zgemv (C function):           Level 2.             (line   6)
* gsl_blas_zgerc (C function):           Level 2.             (line 117)
* gsl_blas_zgeru (C function):           Level 2.             (line 103)
* gsl_blas_zhemm (C function):           Level 3.             (line  50)
* gsl_blas_zhemv (C function):           Level 2.             (line  85)
* gsl_blas_zher (C function):            Level 2.             (line 140)
* gsl_blas_zher2 (C function):           Level 2.             (line 168)
* gsl_blas_zher2k (C function):          Level 3.             (line 187)
* gsl_blas_zherk (C function):           Level 3.             (line 144)
* gsl_blas_zscal (C function):           Level 1.             (line 117)
* gsl_blas_zswap (C function):           Level 1.             (line  83)
* gsl_blas_zsymm (C function):           Level 3.             (line  27)
* gsl_blas_zsyr2k (C function):          Level 3.             (line 162)
* gsl_blas_zsyrk (C function):           Level 3.             (line 121)
* gsl_blas_ztrmm (C function):           Level 3.             (line  68)
* gsl_blas_ztrmv (C function):           Level 2.             (line  25)
* gsl_blas_ztrsm (C function):           Level 3.             (line  94)
* gsl_blas_ztrsv (C function):           Level 2.             (line  48)
* gsl_block (C type):                    Blocks.              (line  10)
* gsl_block_alloc (C function):          Block allocation.    (line  14)
* gsl_block_calloc (C function):         Block allocation.    (line  26)
* gsl_block_fprintf (C function):        Reading and writing blocks.
                                                              (line  28)
* gsl_block_fread (C function):          Reading and writing blocks.
                                                              (line  17)
* gsl_block_free (C function):           Block allocation.    (line  31)
* gsl_block_fscanf (C function):         Reading and writing blocks.
                                                              (line  38)
* gsl_block_fwrite (C function):         Reading and writing blocks.
                                                              (line   9)
* gsl_bspline_alloc (C function):        Initializing the B-splines solver.
                                                              (line  11)
* gsl_bspline_deriv_eval (C function):   Evaluation of B-spline derivatives.
                                                              (line   6)
* gsl_bspline_deriv_eval_nonzero (C function): Evaluation of B-spline derivatives.
                                                              (line  21)
* gsl_bspline_eval (C function):         Evaluation of B-splines.
                                                              (line   6)
* gsl_bspline_eval_nonzero (C function): Evaluation of B-splines.
                                                              (line  18)
* gsl_bspline_free (C function):         Initializing the B-splines solver.
                                                              (line  20)
* gsl_bspline_greville_abscissa (C function): Working with the Greville abscissae.
                                                              (line  14)
* gsl_bspline_knots (C function):        Constructing the knots vector.
                                                              (line   6)
* gsl_bspline_knots_uniform (C function): Constructing the knots vector.
                                                              (line  12)
* gsl_bspline_ncoeffs (C function):      Evaluation of B-splines.
                                                              (line  31)
* gsl_bspline_workspace (C type):        Initializing the B-splines solver.
                                                              (line   6)
* GSL_C99_INLINE:                        ANSI C Compliance.   (line  25)
* GSL_C99_INLINE (C macro):              Accessing vector elements.
                                                              (line  29)
* gsl_cdf_beta_P (C function):           The Beta Distribution.
                                                              (line  22)
* gsl_cdf_beta_Pinv (C function):        The Beta Distribution.
                                                              (line  22)
* gsl_cdf_beta_Q (C function):           The Beta Distribution.
                                                              (line  22)
* gsl_cdf_beta_Qinv (C function):        The Beta Distribution.
                                                              (line  22)
* gsl_cdf_binomial_P (C function):       The Binomial Distribution.
                                                              (line  26)
* gsl_cdf_binomial_Q (C function):       The Binomial Distribution.
                                                              (line  26)
* gsl_cdf_cauchy_P (C function):         The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_cauchy_Pinv (C function):      The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_cauchy_Q (C function):         The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_cauchy_Qinv (C function):      The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_chisq_P (C function):          The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_chisq_Pinv (C function):       The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_chisq_Q (C function):          The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_chisq_Qinv (C function):       The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_exponential_P (C function):    The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exponential_Pinv (C function): The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exponential_Q (C function):    The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exponential_Qinv (C function): The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exppow_P (C function):         The Exponential Power Distribution.
                                                              (line  27)
* gsl_cdf_exppow_Q (C function):         The Exponential Power Distribution.
                                                              (line  27)
* gsl_cdf_fdist_P (C function):          The F-distribution.  (line  36)
* gsl_cdf_fdist_Pinv (C function):       The F-distribution.  (line  36)
* gsl_cdf_fdist_Q (C function):          The F-distribution.  (line  36)
* gsl_cdf_fdist_Qinv (C function):       The F-distribution.  (line  36)
* gsl_cdf_flat_P (C function):           The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_flat_Pinv (C function):        The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_flat_Q (C function):           The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_flat_Qinv (C function):        The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_gamma_P (C function):          The Gamma Distribution.
                                                              (line  36)
* gsl_cdf_gamma_Pinv (C function):       The Gamma Distribution.
                                                              (line  36)
* gsl_cdf_gamma_Q (C function):          The Gamma Distribution.
                                                              (line  36)
* gsl_cdf_gamma_Qinv (C function):       The Gamma Distribution.
                                                              (line  36)
* gsl_cdf_gaussian_P (C function):       The Gaussian Distribution.
                                                              (line  45)
* gsl_cdf_gaussian_Pinv (C function):    The Gaussian Distribution.
                                                              (line  45)
* gsl_cdf_gaussian_Q (C function):       The Gaussian Distribution.
                                                              (line  45)
* gsl_cdf_gaussian_Qinv (C function):    The Gaussian Distribution.
                                                              (line  45)
* gsl_cdf_geometric_P (C function):      The Geometric Distribution.
                                                              (line  27)
* gsl_cdf_geometric_Q (C function):      The Geometric Distribution.
                                                              (line  27)
* gsl_cdf_gumbel1_P (C function):        The Type-1 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel1_Pinv (C function):     The Type-1 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel1_Q (C function):        The Type-1 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel1_Qinv (C function):     The Type-1 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel2_P (C function):        The Type-2 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel2_Pinv (C function):     The Type-2 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel2_Q (C function):        The Type-2 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_gumbel2_Qinv (C function):     The Type-2 Gumbel Distribution.
                                                              (line  23)
* gsl_cdf_hypergeometric_P (C function): The Hypergeometric Distribution.
                                                              (line  31)
* gsl_cdf_hypergeometric_Q (C function): The Hypergeometric Distribution.
                                                              (line  31)
* gsl_cdf_laplace_P (C function):        The Laplace Distribution.
                                                              (line  22)
* gsl_cdf_laplace_Pinv (C function):     The Laplace Distribution.
                                                              (line  22)
* gsl_cdf_laplace_Q (C function):        The Laplace Distribution.
                                                              (line  22)
* gsl_cdf_laplace_Qinv (C function):     The Laplace Distribution.
                                                              (line  22)
* gsl_cdf_logistic_P (C function):       The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_logistic_Pinv (C function):    The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_logistic_Q (C function):       The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_logistic_Qinv (C function):    The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_lognormal_P (C function):      The Lognormal Distribution.
                                                              (line  24)
* gsl_cdf_lognormal_Pinv (C function):   The Lognormal Distribution.
                                                              (line  24)
* gsl_cdf_lognormal_Q (C function):      The Lognormal Distribution.
                                                              (line  24)
* gsl_cdf_lognormal_Qinv (C function):   The Lognormal Distribution.
                                                              (line  24)
* gsl_cdf_negative_binomial_P (C function): The Negative Binomial Distribution.
                                                              (line  27)
* gsl_cdf_negative_binomial_Q (C function): The Negative Binomial Distribution.
                                                              (line  27)
* gsl_cdf_pareto_P (C function):         The Pareto Distribution.
                                                              (line  23)
* gsl_cdf_pareto_Pinv (C function):      The Pareto Distribution.
                                                              (line  23)
* gsl_cdf_pareto_Q (C function):         The Pareto Distribution.
                                                              (line  23)
* gsl_cdf_pareto_Qinv (C function):      The Pareto Distribution.
                                                              (line  23)
* gsl_cdf_pascal_P (C function):         The Pascal Distribution.
                                                              (line  25)
* gsl_cdf_pascal_Q (C function):         The Pascal Distribution.
                                                              (line  25)
* gsl_cdf_poisson_P (C function):        The Poisson Distribution.
                                                              (line  23)
* gsl_cdf_poisson_Q (C function):        The Poisson Distribution.
                                                              (line  23)
* gsl_cdf_rayleigh_P (C function):       The Rayleigh Distribution.
                                                              (line  23)
* gsl_cdf_rayleigh_Pinv (C function):    The Rayleigh Distribution.
                                                              (line  23)
* gsl_cdf_rayleigh_Q (C function):       The Rayleigh Distribution.
                                                              (line  23)
* gsl_cdf_rayleigh_Qinv (C function):    The Rayleigh Distribution.
                                                              (line  23)
* gsl_cdf_tdist_P (C function):          The t-distribution.  (line  30)
* gsl_cdf_tdist_Pinv (C function):       The t-distribution.  (line  30)
* gsl_cdf_tdist_Q (C function):          The t-distribution.  (line  30)
* gsl_cdf_tdist_Qinv (C function):       The t-distribution.  (line  30)
* gsl_cdf_ugaussian_P (C function):      The Gaussian Distribution.
                                                              (line  54)
* gsl_cdf_ugaussian_Pinv (C function):   The Gaussian Distribution.
                                                              (line  54)
* gsl_cdf_ugaussian_Q (C function):      The Gaussian Distribution.
                                                              (line  54)
* gsl_cdf_ugaussian_Qinv (C function):   The Gaussian Distribution.
                                                              (line  54)
* gsl_cdf_weibull_P (C function):        The Weibull Distribution.
                                                              (line  23)
* gsl_cdf_weibull_Pinv (C function):     The Weibull Distribution.
                                                              (line  23)
* gsl_cdf_weibull_Q (C function):        The Weibull Distribution.
                                                              (line  23)
* gsl_cdf_weibull_Qinv (C function):     The Weibull Distribution.
                                                              (line  23)
* gsl_cheb_alloc (C function):           Creation and Calculation of Chebyshev Series.
                                                              (line   6)
* gsl_cheb_calc_deriv (C function):      Derivatives and Integrals.
                                                              (line  12)
* gsl_cheb_calc_integ (C function):      Derivatives and Integrals.
                                                              (line  20)
* gsl_cheb_coeffs (C function):          Auxiliary Functions. (line  13)
* gsl_cheb_eval (C function):            Chebyshev Series Evaluation.
                                                              (line   6)
* gsl_cheb_eval_err (C function):        Chebyshev Series Evaluation.
                                                              (line  11)
* gsl_cheb_eval_n (C function):          Chebyshev Series Evaluation.
                                                              (line  19)
* gsl_cheb_eval_n_err (C function):      Chebyshev Series Evaluation.
                                                              (line  26)
* gsl_cheb_free (C function):            Creation and Calculation of Chebyshev Series.
                                                              (line  13)
* gsl_cheb_init (C function):            Creation and Calculation of Chebyshev Series.
                                                              (line  18)
* gsl_cheb_order (C function):           Auxiliary Functions. (line   9)
* gsl_cheb_series (C type):              Definitions.         (line   6)
* gsl_cheb_size (C function):            Auxiliary Functions. (line  13)
* gsl_check_range (C var):               Accessing vector elements.
                                                              (line  37)
* gsl_combination (C type):              The Combination struct.
                                                              (line   6)
* gsl_combination_alloc (C function):    Combination allocation.
                                                              (line   6)
* gsl_combination_calloc (C function):   Combination allocation.
                                                              (line  17)
* gsl_combination_data (C function):     Combination properties.
                                                              (line  15)
* gsl_combination_fprintf (C function):  Reading and writing combinations.
                                                              (line  30)
* gsl_combination_fread (C function):    Reading and writing combinations.
                                                              (line  18)
* gsl_combination_free (C function):     Combination allocation.
                                                              (line  36)
* gsl_combination_fscanf (C function):   Reading and writing combinations.
                                                              (line  40)
* gsl_combination_fwrite (C function):   Reading and writing combinations.
                                                              (line   9)
* gsl_combination_get (C function):      Accessing combination elements.
                                                              (line   9)
* gsl_combination_init_first (C function): Combination allocation.
                                                              (line  25)
* gsl_combination_init_last (C function): Combination allocation.
                                                              (line  30)
* gsl_combination_k (C function):        Combination properties.
                                                              (line  10)
* gsl_combination_memcpy (C function):   Combination allocation.
                                                              (line  41)
* gsl_combination_n (C function):        Combination properties.
                                                              (line   6)
* gsl_combination_next (C function):     Combination functions.
                                                              (line   6)
* gsl_combination_prev (C function):     Combination functions.
                                                              (line  15)
* gsl_combination_valid (C function):    Combination properties.
                                                              (line  20)
* gsl_complex:                           Complex Numbers.     (line  21)
* gsl_complex_abs (C function):          Properties of complex numbers.
                                                              (line  11)
* gsl_complex_abs2 (C function):         Properties of complex numbers.
                                                              (line  16)
* gsl_complex_add (C function):          Complex arithmetic operators.
                                                              (line   6)
* gsl_complex_add_imag (C function):     Complex arithmetic operators.
                                                              (line  46)
* gsl_complex_add_real (C function):     Complex arithmetic operators.
                                                              (line  26)
* gsl_complex_arccos (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  21)
* gsl_complex_arccosh (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  12)
* gsl_complex_arccosh_real (C function): Inverse Complex Hyperbolic Functions.
                                                              (line  20)
* gsl_complex_arccos_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  27)
* gsl_complex_arccot (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  61)
* gsl_complex_arccoth (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  46)
* gsl_complex_arccsc (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  51)
* gsl_complex_arccsch (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  41)
* gsl_complex_arccsc_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  56)
* gsl_complex_arcsec (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  41)
* gsl_complex_arcsech (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  36)
* gsl_complex_arcsec_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  46)
* gsl_complex_arcsin (C function):       Inverse Complex Trigonometric Functions.
                                                              (line   6)
* gsl_complex_arcsinh (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line   6)
* gsl_complex_arcsin_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  12)
* gsl_complex_arctan (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  35)
* gsl_complex_arctanh (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  25)
* gsl_complex_arctanh_real (C function): Inverse Complex Hyperbolic Functions.
                                                              (line  31)
* gsl_complex_arg (C function):          Properties of complex numbers.
                                                              (line   6)
* gsl_complex_conjugate (C function):    Complex arithmetic operators.
                                                              (line  66)
* gsl_complex_cos (C function):          Complex Trigonometric Functions.
                                                              (line  11)
* gsl_complex_cosh (C function):         Complex Hyperbolic Functions.
                                                              (line  11)
* gsl_complex_cot (C function):          Complex Trigonometric Functions.
                                                              (line  31)
* gsl_complex_coth (C function):         Complex Hyperbolic Functions.
                                                              (line  31)
* gsl_complex_csc (C function):          Complex Trigonometric Functions.
                                                              (line  26)
* gsl_complex_csch (C function):         Complex Hyperbolic Functions.
                                                              (line  26)
* gsl_complex_div (C function):          Complex arithmetic operators.
                                                              (line  21)
* gsl_complex_div_imag (C function):     Complex arithmetic operators.
                                                              (line  61)
* gsl_complex_div_real (C function):     Complex arithmetic operators.
                                                              (line  41)
* gsl_complex_exp (C function):          Elementary Complex Functions.
                                                              (line  28)
* gsl_complex_inverse (C function):      Complex arithmetic operators.
                                                              (line  71)
* gsl_complex_log (C function):          Elementary Complex Functions.
                                                              (line  33)
* gsl_complex_log10 (C function):        Elementary Complex Functions.
                                                              (line  39)
* gsl_complex_logabs (C function):       Properties of complex numbers.
                                                              (line  21)
* gsl_complex_log_b (C function):        Elementary Complex Functions.
                                                              (line  44)
* gsl_complex_mul (C function):          Complex arithmetic operators.
                                                              (line  16)
* gsl_complex_mul_imag (C function):     Complex arithmetic operators.
                                                              (line  56)
* gsl_complex_mul_real (C function):     Complex arithmetic operators.
                                                              (line  36)
* gsl_complex_negative (C function):     Complex arithmetic operators.
                                                              (line  76)
* gsl_complex_polar (C function):        Assigning complex numbers.
                                                              (line  12)
* gsl_complex_poly_complex_eval (C function): Polynomial Evaluation.
                                                              (line  25)
* gsl_complex_pow (C function):          Elementary Complex Functions.
                                                              (line  17)
* gsl_complex_pow_real (C function):     Elementary Complex Functions.
                                                              (line  23)
* gsl_complex_rect (C function):         Assigning complex numbers.
                                                              (line   6)
* gsl_complex_sec (C function):          Complex Trigonometric Functions.
                                                              (line  21)
* gsl_complex_sech (C function):         Complex Hyperbolic Functions.
                                                              (line  21)
* gsl_complex_sin (C function):          Complex Trigonometric Functions.
                                                              (line   6)
* gsl_complex_sinh (C function):         Complex Hyperbolic Functions.
                                                              (line   6)
* gsl_complex_sqrt (C function):         Elementary Complex Functions.
                                                              (line   6)
* gsl_complex_sqrt_real (C function):    Elementary Complex Functions.
                                                              (line  12)
* gsl_complex_sub (C function):          Complex arithmetic operators.
                                                              (line  11)
* gsl_complex_sub_imag (C function):     Complex arithmetic operators.
                                                              (line  51)
* gsl_complex_sub_real (C function):     Complex arithmetic operators.
                                                              (line  31)
* gsl_complex_tan (C function):          Complex Trigonometric Functions.
                                                              (line  16)
* gsl_complex_tanh (C function):         Complex Hyperbolic Functions.
                                                              (line  16)
* GSL_CONST_MKSA_ACRE (C macro):         Volume Area and Length.
                                                              (line  14)
* GSL_CONST_MKSA_ANGSTROM (C macro):     Atomic and Nuclear Physics.
                                                              (line  47)
* GSL_CONST_MKSA_ASTRONOMICAL_UNIT (C macro): Astronomy and Astrophysics.
                                                              (line   6)
* GSL_CONST_MKSA_BAR (C macro):          Pressure.            (line   6)
* GSL_CONST_MKSA_BARN (C macro):         Atomic and Nuclear Physics.
                                                              (line  51)
* GSL_CONST_MKSA_BOHR_MAGNETON (C macro): Atomic and Nuclear Physics.
                                                              (line  55)
* GSL_CONST_MKSA_BOHR_RADIUS (C macro):  Atomic and Nuclear Physics.
                                                              (line  43)
* GSL_CONST_MKSA_BOLTZMANN (C macro):    Fundamental Constants.
                                                              (line  36)
* GSL_CONST_MKSA_BTU (C macro):          Thermal Energy and Power.
                                                              (line  10)
* GSL_CONST_MKSA_CALORIE (C macro):      Thermal Energy and Power.
                                                              (line   6)
* GSL_CONST_MKSA_CANADIAN_GALLON (C macro): Volume Area and Length.
                                                              (line  26)
* GSL_CONST_MKSA_CARAT (C macro):        Mass and Weight.     (line  30)
* GSL_CONST_MKSA_CURIE (C macro):        Radioactivity.       (line   6)
* GSL_CONST_MKSA_DAY (C macro):          Measurement of Time. (line  14)
* GSL_CONST_MKSA_DEBYE (C macro):        Atomic and Nuclear Physics.
                                                              (line  76)
* GSL_CONST_MKSA_DYNE (C macro):         Force and Energy.    (line  10)
* GSL_CONST_MKSA_ELECTRON_CHARGE (C macro): Atomic and Nuclear Physics.
                                                              (line   6)
* GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT (C macro): Atomic and Nuclear Physics.
                                                              (line  63)
* GSL_CONST_MKSA_ELECTRON_VOLT (C macro): Atomic and Nuclear Physics.
                                                              (line  10)
* GSL_CONST_MKSA_ERG (C macro):          Force and Energy.    (line  18)
* GSL_CONST_MKSA_FARADAY (C macro):      Fundamental Constants.
                                                              (line  32)
* GSL_CONST_MKSA_FATHOM (C macro):       Speed and Nautical Units.
                                                              (line  18)
* GSL_CONST_MKSA_FOOT (C macro):         Imperial Units.      (line  10)
* GSL_CONST_MKSA_FOOTCANDLE (C macro):   Light and Illumination.
                                                              (line  22)
* GSL_CONST_MKSA_FOOTLAMBERT (C macro):  Light and Illumination.
                                                              (line  30)
* GSL_CONST_MKSA_GAUSS (C macro):        Fundamental Constants.
                                                              (line  52)
* GSL_CONST_MKSA_GRAM_FORCE (C macro):   Mass and Weight.     (line  34)
* GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT (C macro): Astronomy and Astrophysics.
                                                              (line  10)
* GSL_CONST_MKSA_GRAV_ACCEL (C macro):   Astronomy and Astrophysics.
                                                              (line  22)
* GSL_CONST_MKSA_HECTARE (C macro):      Volume Area and Length.
                                                              (line  10)
* GSL_CONST_MKSA_HORSEPOWER (C macro):   Thermal Energy and Power.
                                                              (line  18)
* GSL_CONST_MKSA_HOUR (C macro):         Measurement of Time. (line  10)
* GSL_CONST_MKSA_INCH (C macro):         Imperial Units.      (line   6)
* GSL_CONST_MKSA_INCH_OF_MERCURY (C macro): Pressure.         (line  22)
* GSL_CONST_MKSA_INCH_OF_WATER (C macro): Pressure.           (line  26)
* GSL_CONST_MKSA_JOULE (C macro):        Force and Energy.    (line  14)
* GSL_CONST_MKSA_KILOMETERS_PER_HOUR (C macro): Speed and Nautical Units.
                                                              (line   6)
* GSL_CONST_MKSA_KILOPOUND_FORCE (C macro): Mass and Weight.  (line  42)
* GSL_CONST_MKSA_KNOT (C macro):         Speed and Nautical Units.
                                                              (line  22)
* GSL_CONST_MKSA_LAMBERT (C macro):      Light and Illumination.
                                                              (line  26)
* GSL_CONST_MKSA_LIGHT_YEAR (C macro):   Astronomy and Astrophysics.
                                                              (line  14)
* GSL_CONST_MKSA_LITER (C macro):        Volume Area and Length.
                                                              (line  18)
* GSL_CONST_MKSA_LUMEN (C macro):        Light and Illumination.
                                                              (line  10)
* GSL_CONST_MKSA_LUX (C macro):          Light and Illumination.
                                                              (line  14)
* GSL_CONST_MKSA_MASS_ELECTRON (C macro): Atomic and Nuclear Physics.
                                                              (line  18)
* GSL_CONST_MKSA_MASS_MUON (C macro):    Atomic and Nuclear Physics.
                                                              (line  22)
* GSL_CONST_MKSA_MASS_NEUTRON (C macro): Atomic and Nuclear Physics.
                                                              (line  30)
* GSL_CONST_MKSA_MASS_PROTON (C macro):  Atomic and Nuclear Physics.
                                                              (line  26)
* GSL_CONST_MKSA_METER_OF_MERCURY (C macro): Pressure.        (line  18)
* GSL_CONST_MKSA_METRIC_TON (C macro):   Mass and Weight.     (line  18)
* GSL_CONST_MKSA_MICRON (C macro):       Volume Area and Length.
                                                              (line   6)
* GSL_CONST_MKSA_MIL (C macro):          Imperial Units.      (line  22)
* GSL_CONST_MKSA_MILE (C macro):         Imperial Units.      (line  18)
* GSL_CONST_MKSA_MILES_PER_HOUR (C macro): Speed and Nautical Units.
                                                              (line  10)
* GSL_CONST_MKSA_MINUTE (C macro):       Measurement of Time. (line   6)
* GSL_CONST_MKSA_MOLAR_GAS (C macro):    Fundamental Constants.
                                                              (line  40)
* GSL_CONST_MKSA_NAUTICAL_MILE (C macro): Speed and Nautical Units.
                                                              (line  14)
* GSL_CONST_MKSA_NEWTON (C macro):       Force and Energy.    (line   6)
* GSL_CONST_MKSA_NUCLEAR_MAGNETON (C macro): Atomic and Nuclear Physics.
                                                              (line  59)
* GSL_CONST_MKSA_OUNCE_MASS (C macro):   Mass and Weight.     (line  10)
* GSL_CONST_MKSA_PARSEC (C macro):       Astronomy and Astrophysics.
                                                              (line  18)
* GSL_CONST_MKSA_PHOT (C macro):         Light and Illumination.
                                                              (line  18)
* GSL_CONST_MKSA_PINT (C macro):         Volume Area and Length.
                                                              (line  38)
* GSL_CONST_MKSA_PLANCKS_CONSTANT_H (C macro): Fundamental Constants.
                                                              (line  20)
* GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR (C macro): Fundamental Constants.
                                                              (line  24)
* GSL_CONST_MKSA_POINT (C macro):        Printers Units.      (line   6)
* GSL_CONST_MKSA_POISE (C macro):        Viscosity.           (line   6)
* GSL_CONST_MKSA_POUNDAL (C macro):      Mass and Weight.     (line  46)
* GSL_CONST_MKSA_POUND_FORCE (C macro):  Mass and Weight.     (line  38)
* GSL_CONST_MKSA_POUND_MASS (C macro):   Mass and Weight.     (line   6)
* GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT (C macro): Atomic and Nuclear Physics.
                                                              (line  68)
* GSL_CONST_MKSA_PSI (C macro):          Pressure.            (line  30)
* GSL_CONST_MKSA_QUART (C macro):        Volume Area and Length.
                                                              (line  34)
* GSL_CONST_MKSA_RAD (C macro):          Radioactivity.       (line  14)
* GSL_CONST_MKSA_ROENTGEN (C macro):     Radioactivity.       (line  10)
* GSL_CONST_MKSA_RYDBERG (C macro):      Atomic and Nuclear Physics.
                                                              (line  38)
* GSL_CONST_MKSA_SOLAR_MASS (C macro):   Astronomy and Astrophysics.
                                                              (line  26)
* GSL_CONST_MKSA_SPEED_OF_LIGHT (C macro): Fundamental Constants.
                                                              (line   6)
* GSL_CONST_MKSA_STANDARD_GAS_VOLUME (C macro): Fundamental Constants.
                                                              (line  44)
* GSL_CONST_MKSA_STD_ATMOSPHERE (C macro): Pressure.          (line  10)
* GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT (C macro): Fundamental Constants.
                                                              (line  48)
* GSL_CONST_MKSA_STILB (C macro):        Light and Illumination.
                                                              (line   6)
* GSL_CONST_MKSA_STOKES (C macro):       Viscosity.           (line  10)
* GSL_CONST_MKSA_TEXPOINT (C macro):     Printers Units.      (line  10)
* GSL_CONST_MKSA_THERM (C macro):        Thermal Energy and Power.
                                                              (line  14)
* GSL_CONST_MKSA_THOMSON_CROSS_SECTION (C macro): Atomic and Nuclear Physics.
                                                              (line  72)
* GSL_CONST_MKSA_TON (C macro):          Mass and Weight.     (line  14)
* GSL_CONST_MKSA_TORR (C macro):         Pressure.            (line  14)
* GSL_CONST_MKSA_TROY_OUNCE (C macro):   Mass and Weight.     (line  26)
* GSL_CONST_MKSA_UK_GALLON (C macro):    Volume Area and Length.
                                                              (line  30)
* GSL_CONST_MKSA_UK_TON (C macro):       Mass and Weight.     (line  22)
* GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS (C macro): Atomic and Nuclear Physics.
                                                              (line  14)
* GSL_CONST_MKSA_US_GALLON (C macro):    Volume Area and Length.
                                                              (line  22)
* GSL_CONST_MKSA_VACUUM_PERMEABILITY (C macro): Fundamental Constants.
                                                              (line  10)
* GSL_CONST_MKSA_VACUUM_PERMITTIVITY (C macro): Fundamental Constants.
                                                              (line  15)
* GSL_CONST_MKSA_WEEK (C macro):         Measurement of Time. (line  18)
* GSL_CONST_MKSA_YARD (C macro):         Imperial Units.      (line  14)
* GSL_CONST_NUM_ATTO (C macro):          Prefixes.            (line  60)
* GSL_CONST_NUM_AVOGADRO (C macro):      Fundamental Constants.
                                                              (line  28)
* GSL_CONST_NUM_EXA (C macro):           Prefixes.            (line  16)
* GSL_CONST_NUM_FEMTO (C macro):         Prefixes.            (line  56)
* GSL_CONST_NUM_FINE_STRUCTURE (C macro): Atomic and Nuclear Physics.
                                                              (line  34)
* GSL_CONST_NUM_GIGA (C macro):          Prefixes.            (line  28)
* GSL_CONST_NUM_KILO (C macro):          Prefixes.            (line  36)
* GSL_CONST_NUM_MEGA (C macro):          Prefixes.            (line  32)
* GSL_CONST_NUM_MICRO (C macro):         Prefixes.            (line  44)
* GSL_CONST_NUM_MILLI (C macro):         Prefixes.            (line  40)
* GSL_CONST_NUM_NANO (C macro):          Prefixes.            (line  48)
* GSL_CONST_NUM_PETA (C macro):          Prefixes.            (line  20)
* GSL_CONST_NUM_PICO (C macro):          Prefixes.            (line  52)
* GSL_CONST_NUM_TERA (C macro):          Prefixes.            (line  24)
* GSL_CONST_NUM_YOCTO (C macro):         Prefixes.            (line  68)
* GSL_CONST_NUM_YOTTA (C macro):         Prefixes.            (line   8)
* GSL_CONST_NUM_ZEPTO (C macro):         Prefixes.            (line  64)
* GSL_CONST_NUM_ZETTA (C macro):         Prefixes.            (line  12)
* gsl_deriv_backward (C function):       Functions.           (line  46)
* gsl_deriv_central (C function):        Functions.           (line   6)
* gsl_deriv_forward (C function):        Functions.           (line  25)
* gsl_dht (C type):                      Functions<2>.        (line   6)
* gsl_dht_alloc (C function):            Functions<2>.        (line  10)
* gsl_dht_apply (C function):            Functions<2>.        (line  31)
* gsl_dht_free (C function):             Functions<2>.        (line  27)
* gsl_dht_init (C function):             Functions<2>.        (line  15)
* gsl_dht_k_sample (C function):         Functions<2>.        (line  49)
* gsl_dht_new (C function):              Functions<2>.        (line  20)
* gsl_dht_x_sample (C function):         Functions<2>.        (line  42)
* GSL_EDOM (C var):                      Error Codes.         (line  13)
* gsl_eigen_gen (C function):            Real Generalized Nonsymmetric Eigensystems.
                                                              (line  89)
* gsl_eigen_genherm (C function):        Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  38)
* gsl_eigen_genhermv (C function):       Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  67)
* gsl_eigen_genhermv_alloc (C function): Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  53)
* gsl_eigen_genhermv_free (C function):  Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  61)
* gsl_eigen_genhermv_sort (C function):  Sorting Eigenvalues and Eigenvectors.
                                                              (line  58)
* gsl_eigen_genhermv_workspace (C type): Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  48)
* gsl_eigen_genherm_alloc (C function):  Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  24)
* gsl_eigen_genherm_free (C function):   Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  32)
* gsl_eigen_genherm_workspace (C type):  Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  19)
* gsl_eigen_gensymm (C function):        Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  44)
* gsl_eigen_gensymmv (C function):       Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  72)
* gsl_eigen_gensymmv_alloc (C function): Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  58)
* gsl_eigen_gensymmv_free (C function):  Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  66)
* gsl_eigen_gensymmv_sort (C function):  Sorting Eigenvalues and Eigenvectors.
                                                              (line  49)
* gsl_eigen_gensymmv_workspace (C type): Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  53)
* gsl_eigen_gensymm_alloc (C function):  Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  31)
* gsl_eigen_gensymm_free (C function):   Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  38)
* gsl_eigen_gensymm_workspace (C type):  Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  26)
* gsl_eigen_genv (C function):           Real Generalized Nonsymmetric Eigensystems.
                                                              (line 134)
* gsl_eigen_genv_alloc (C function):     Real Generalized Nonsymmetric Eigensystems.
                                                              (line 122)
* gsl_eigen_genv_free (C function):      Real Generalized Nonsymmetric Eigensystems.
                                                              (line 129)
* gsl_eigen_genv_QZ (C function):        Real Generalized Nonsymmetric Eigensystems.
                                                              (line 152)
* gsl_eigen_genv_sort (C function):      Sorting Eigenvalues and Eigenvectors.
                                                              (line  67)
* gsl_eigen_genv_workspace (C type):     Real Generalized Nonsymmetric Eigensystems.
                                                              (line 117)
* gsl_eigen_gen_alloc (C function):      Real Generalized Nonsymmetric Eigensystems.
                                                              (line  54)
* gsl_eigen_gen_free (C function):       Real Generalized Nonsymmetric Eigensystems.
                                                              (line  61)
* gsl_eigen_gen_params (C function):     Real Generalized Nonsymmetric Eigensystems.
                                                              (line  66)
* gsl_eigen_gen_QZ (C function):         Real Generalized Nonsymmetric Eigensystems.
                                                              (line 109)
* gsl_eigen_gen_workspace (C type):      Real Generalized Nonsymmetric Eigensystems.
                                                              (line  49)
* gsl_eigen_herm (C function):           Complex Hermitian Matrices.
                                                              (line  26)
* gsl_eigen_hermv (C function):          Complex Hermitian Matrices.
                                                              (line  55)
* gsl_eigen_hermv_alloc (C function):    Complex Hermitian Matrices.
                                                              (line  43)
* gsl_eigen_hermv_free (C function):     Complex Hermitian Matrices.
                                                              (line  50)
* gsl_eigen_hermv_sort (C function):     Sorting Eigenvalues and Eigenvectors.
                                                              (line  29)
* gsl_eigen_hermv_workspace (C type):    Complex Hermitian Matrices.
                                                              (line  38)
* gsl_eigen_herm_alloc (C function):     Complex Hermitian Matrices.
                                                              (line  14)
* gsl_eigen_herm_free (C function):      Complex Hermitian Matrices.
                                                              (line  21)
* gsl_eigen_herm_workspace (C type):     Complex Hermitian Matrices.
                                                              (line   9)
* gsl_eigen_nonsymm (C function):        Real Nonsymmetric Matrices.
                                                              (line  72)
* gsl_eigen_nonsymmv (C function):       Real Nonsymmetric Matrices.
                                                              (line 123)
* gsl_eigen_nonsymmv_alloc (C function): Real Nonsymmetric Matrices.
                                                              (line  99)
* gsl_eigen_nonsymmv_free (C function):  Real Nonsymmetric Matrices.
                                                              (line 106)
* gsl_eigen_nonsymmv_params (C function): Real Nonsymmetric Matrices.
                                                              (line 112)
* gsl_eigen_nonsymmv_sort (C function):  Sorting Eigenvalues and Eigenvectors.
                                                              (line  38)
* gsl_eigen_nonsymmv_workspace (C type): Real Nonsymmetric Matrices.
                                                              (line  94)
* gsl_eigen_nonsymmv_Z (C function):     Real Nonsymmetric Matrices.
                                                              (line 139)
* gsl_eigen_nonsymm_alloc (C function):  Real Nonsymmetric Matrices.
                                                              (line  22)
* gsl_eigen_nonsymm_free (C function):   Real Nonsymmetric Matrices.
                                                              (line  29)
* gsl_eigen_nonsymm_params (C function): Real Nonsymmetric Matrices.
                                                              (line  35)
* gsl_eigen_nonsymm_workspace (C type):  Real Nonsymmetric Matrices.
                                                              (line  17)
* gsl_eigen_nonsymm_Z (C function):      Real Nonsymmetric Matrices.
                                                              (line  87)
* gsl_eigen_symm (C function):           Real Symmetric Matrices.
                                                              (line  29)
* gsl_eigen_symmv (C function):          Real Symmetric Matrices.
                                                              (line  56)
* gsl_eigen_symmv_alloc (C function):    Real Symmetric Matrices.
                                                              (line  44)
* gsl_eigen_symmv_free (C function):     Real Symmetric Matrices.
                                                              (line  51)
* gsl_eigen_symmv_sort (C function):     Sorting Eigenvalues and Eigenvectors.
                                                              (line   6)
* gsl_eigen_symmv_sort.gsl_eigen_sort_t (C type): Sorting Eigenvalues and Eigenvectors.
                                                              (line  15)
* gsl_eigen_symmv_workspace (C type):    Real Symmetric Matrices.
                                                              (line  39)
* gsl_eigen_symm_alloc (C function):     Real Symmetric Matrices.
                                                              (line  17)
* gsl_eigen_symm_free (C function):      Real Symmetric Matrices.
                                                              (line  24)
* gsl_eigen_symm_workspace (C type):     Real Symmetric Matrices.
                                                              (line  12)
* GSL_EINVAL (C var):                    Error Codes.         (line  32)
* GSL_ENOMEM (C var):                    Error Codes.         (line  25)
* GSL_ERANGE (C var):                    Error Codes.         (line  19)
* GSL_ERROR (C macro):                   Using GSL error reporting in your own functions.
                                                              (line  15)
* gsl_error_handler_t (C type):          Error Handlers.      (line  23)
* GSL_ERROR_VAL (C macro):               Using GSL error reporting in your own functions.
                                                              (line  36)
* gsl_expm1 (C function):                Elementary Functions.
                                                              (line  18)
* gsl_fcmp (C function):                 Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* gsl_fft_complex_backward (C function): Mixed-radix FFT routines for complex data.
                                                              (line 121)
* gsl_fft_complex_forward (C function):  Mixed-radix FFT routines for complex data.
                                                              (line 121)
* gsl_fft_complex_inverse (C function):  Mixed-radix FFT routines for complex data.
                                                              (line 121)
* gsl_fft_complex_radix2_backward (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_radix2_dif_backward (C function): Radix-2 FFT routines for complex data.
                                                              (line  38)
* gsl_fft_complex_radix2_dif_forward (C function): Radix-2 FFT routines for complex data.
                                                              (line  38)
* gsl_fft_complex_radix2_dif_inverse (C function): Radix-2 FFT routines for complex data.
                                                              (line  38)
* gsl_fft_complex_radix2_dif_transform (C function): Radix-2 FFT routines for complex data.
                                                              (line  38)
* gsl_fft_complex_radix2_forward (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_radix2_inverse (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_radix2_transform (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_transform (C function): Mixed-radix FFT routines for complex data.
                                                              (line 121)
* gsl_fft_complex_wavetable (C type):    Mixed-radix FFT routines for complex data.
                                                              (line  80)
* gsl_fft_complex_wavetable_alloc (C function): Mixed-radix FFT routines for complex data.
                                                              (line  44)
* gsl_fft_complex_wavetable_free (C function): Mixed-radix FFT routines for complex data.
                                                              (line  65)
* gsl_fft_complex_workspace (C type):    Mixed-radix FFT routines for complex data.
                                                              (line 101)
* gsl_fft_complex_workspace_alloc (C function): Mixed-radix FFT routines for complex data.
                                                              (line 106)
* gsl_fft_complex_workspace_free (C function): Mixed-radix FFT routines for complex data.
                                                              (line 112)
* gsl_fft_halfcomplex_radix2_backward (C function): Radix-2 FFT routines for real data.
                                                              (line  57)
* gsl_fft_halfcomplex_radix2_inverse (C function): Radix-2 FFT routines for real data.
                                                              (line  57)
* gsl_fft_halfcomplex_radix2_unpack (C function): Radix-2 FFT routines for real data.
                                                              (line  68)
* gsl_fft_halfcomplex_transform (C function): Mixed-radix FFT routines for real data.
                                                              (line 133)
* gsl_fft_halfcomplex_unpack (C function): Mixed-radix FFT routines for real data.
                                                              (line 169)
* gsl_fft_halfcomplex_wavetable (C type): Mixed-radix FFT routines for real data.
                                                              (line  72)
* gsl_fft_halfcomplex_wavetable_alloc (C function): Mixed-radix FFT routines for real data.
                                                              (line  78)
* gsl_fft_halfcomplex_wavetable_free (C function): Mixed-radix FFT routines for real data.
                                                              (line 100)
* gsl_fft_real_radix2_transform (C function): Radix-2 FFT routines for real data.
                                                              (line  13)
* gsl_fft_real_transform (C function):   Mixed-radix FFT routines for real data.
                                                              (line 133)
* gsl_fft_real_unpack (C function):      Mixed-radix FFT routines for real data.
                                                              (line 153)
* gsl_fft_real_wavetable (C type):       Mixed-radix FFT routines for real data.
                                                              (line  72)
* gsl_fft_real_wavetable_alloc (C function): Mixed-radix FFT routines for real data.
                                                              (line  78)
* gsl_fft_real_wavetable_free (C function): Mixed-radix FFT routines for real data.
                                                              (line 100)
* gsl_fft_real_workspace (C type):       Mixed-radix FFT routines for real data.
                                                              (line 112)
* gsl_fft_real_workspace_alloc (C function): Mixed-radix FFT routines for real data.
                                                              (line 116)
* gsl_fft_real_workspace_free (C function): Mixed-radix FFT routines for real data.
                                                              (line 123)
* gsl_filter_end_t (C type):             Handling Endpoints<2>.
                                                              (line  12)
* gsl_filter_end_t.GSL_FILTER_END_PADVALUE (C macro): Handling Endpoints<2>.
                                                              (line  27)
* gsl_filter_end_t.GSL_FILTER_END_PADZERO (C macro): Handling Endpoints<2>.
                                                              (line  17)
* gsl_filter_end_t.GSL_FILTER_END_TRUNCATE (C macro): Handling Endpoints<2>.
                                                              (line  36)
* gsl_filter_gaussian (C function):      Gaussian Filter.     (line  50)
* gsl_filter_gaussian_alloc (C function): Gaussian Filter.    (line  37)
* gsl_filter_gaussian_free (C function): Gaussian Filter.     (line  45)
* gsl_filter_gaussian_kernel (C function): Gaussian Filter.   (line  62)
* gsl_filter_impulse (C function):       Impulse Detection Filter.
                                                              (line 115)
* gsl_filter_impulse_alloc (C function): Impulse Detection Filter.
                                                              (line 101)
* gsl_filter_impulse_free (C function):  Impulse Detection Filter.
                                                              (line 110)
* gsl_filter_median (C function):        Standard Median Filter.
                                                              (line  25)
* gsl_filter_median_alloc (C function):  Standard Median Filter.
                                                              (line  11)
* gsl_filter_median_free (C function):   Standard Median Filter.
                                                              (line  20)
* gsl_filter_rmedian (C function):       Recursive Median Filter.
                                                              (line  33)
* gsl_filter_rmedian_alloc (C function): Recursive Median Filter.
                                                              (line  19)
* gsl_filter_rmedian_free (C function):  Recursive Median Filter.
                                                              (line  28)
* gsl_filter_scale_t (C type):           Impulse Detection Filter.
                                                              (line  45)
* gsl_filter_scale_t.GSL_FILTER_SCALE_IQR (C macro): Impulse Detection Filter.
                                                              (line  61)
* gsl_filter_scale_t.GSL_FILTER_SCALE_MAD (C macro): Impulse Detection Filter.
                                                              (line  50)
* gsl_filter_scale_t.GSL_FILTER_SCALE_QN (C macro): Impulse Detection Filter.
                                                              (line  81)
* gsl_filter_scale_t.GSL_FILTER_SCALE_SN (C macro): Impulse Detection Filter.
                                                              (line  75)
* gsl_finite (C function):               Infinities and Not-a-number.
                                                              (line  30)
* gsl_fit_linear (C function):           Linear regression with a constant term.
                                                              (line   9)
* gsl_fit_linear_est (C function):       Linear regression with a constant term.
                                                              (line  47)
* gsl_fit_mul (C function):              Linear regression without a constant term.
                                                              (line  10)
* gsl_fit_mul_est (C function):          Linear regression without a constant term.
                                                              (line  43)
* gsl_fit_wlinear (C function):          Linear regression with a constant term.
                                                              (line  27)
* gsl_fit_wmul (C function):             Linear regression without a constant term.
                                                              (line  24)
* gsl_frexp (C function):                Elementary Functions.
                                                              (line  56)
* gsl_function (C type):                 Providing the function to solve.
                                                              (line  11)
* gsl_function_fdf (C type):             Providing the function to solve.
                                                              (line  54)
* gsl_heapsort (C function):             Sorting objects.     (line  16)
* gsl_heapsort.gsl_comparison_fn_t (C type): Sorting objects. (line  24)
* gsl_heapsort_index (C function):       Sorting objects.     (line  58)
* gsl_histogram (C type):                The histogram struct.
                                                              (line   8)
* gsl_histogram2d (C type):              The 2D histogram struct.
                                                              (line   8)
* gsl_histogram2d_accumulate (C function): Updating and accessing 2D histogram elements.
                                                              (line  27)
* gsl_histogram2d_add (C function):      2D Histogram Operations.
                                                              (line  12)
* gsl_histogram2d_alloc (C function):    2D Histogram allocation.
                                                              (line  14)
* gsl_histogram2d_clone (C function):    Copying 2D Histograms.
                                                              (line  14)
* gsl_histogram2d_cov (C function):      2D Histogram Statistics.
                                                              (line  60)
* gsl_histogram2d_div (C function):      2D Histogram Operations.
                                                              (line  36)
* gsl_histogram2d_equal_bins_p (C function): 2D Histogram Operations.
                                                              (line   6)
* gsl_histogram2d_find (C function):     Searching 2D histogram ranges.
                                                              (line   9)
* gsl_histogram2d_fprintf (C function):  Reading and writing 2D histograms.
                                                              (line  31)
* gsl_histogram2d_fread (C function):    Reading and writing 2D histograms.
                                                              (line  19)
* gsl_histogram2d_free (C function):     2D Histogram allocation.
                                                              (line  43)
* gsl_histogram2d_fscanf (C function):   Reading and writing 2D histograms.
                                                              (line  70)
* gsl_histogram2d_fwrite (C function):   Reading and writing 2D histograms.
                                                              (line   9)
* gsl_histogram2d_get (C function):      Updating and accessing 2D histogram elements.
                                                              (line  34)
* gsl_histogram2d_get_xrange (C function): Updating and accessing 2D histogram elements.
                                                              (line  43)
* gsl_histogram2d_get_yrange (C function): Updating and accessing 2D histogram elements.
                                                              (line  43)
* gsl_histogram2d_increment (C function): Updating and accessing 2D histogram elements.
                                                              (line  12)
* gsl_histogram2d_max_bin (C function):  2D Histogram Statistics.
                                                              (line  11)
* gsl_histogram2d_max_val (C function):  2D Histogram Statistics.
                                                              (line   6)
* gsl_histogram2d_memcpy (C function):   Copying 2D Histograms.
                                                              (line   6)
* gsl_histogram2d_min_bin (C function):  2D Histogram Statistics.
                                                              (line  24)
* gsl_histogram2d_min_val (C function):  2D Histogram Statistics.
                                                              (line  19)
* gsl_histogram2d_mul (C function):      2D Histogram Operations.
                                                              (line  28)
* gsl_histogram2d_nx (C function):       Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_ny (C function):       Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_pdf (C type):          Resampling from 2D histograms.
                                                              (line  20)
* gsl_histogram2d_pdf_alloc (C function): Resampling from 2D histograms.
                                                              (line  42)
* gsl_histogram2d_pdf_free (C function): Resampling from 2D histograms.
                                                              (line  61)
* gsl_histogram2d_pdf_init (C function): Resampling from 2D histograms.
                                                              (line  52)
* gsl_histogram2d_pdf_sample (C function): Resampling from 2D histograms.
                                                              (line  66)
* gsl_histogram2d_reset (C function):    Updating and accessing 2D histogram elements.
                                                              (line  73)
* gsl_histogram2d_scale (C function):    2D Histogram Operations.
                                                              (line  44)
* gsl_histogram2d_set_ranges (C function): 2D Histogram allocation.
                                                              (line  26)
* gsl_histogram2d_set_ranges_uniform (C function): 2D Histogram allocation.
                                                              (line  35)
* gsl_histogram2d_shift (C function):    2D Histogram Operations.
                                                              (line  52)
* gsl_histogram2d_sub (C function):      2D Histogram Operations.
                                                              (line  20)
* gsl_histogram2d_sum (C function):      2D Histogram Statistics.
                                                              (line  67)
* gsl_histogram2d_xmax (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_xmean (C function):    2D Histogram Statistics.
                                                              (line  32)
* gsl_histogram2d_xmin (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_xsigma (C function):   2D Histogram Statistics.
                                                              (line  46)
* gsl_histogram2d_ymax (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_ymean (C function):    2D Histogram Statistics.
                                                              (line  39)
* gsl_histogram2d_ymin (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_ysigma (C function):   2D Histogram Statistics.
                                                              (line  53)
* gsl_histogram_accumulate (C function): Updating and accessing histogram elements.
                                                              (line  27)
* gsl_histogram_add (C function):        Histogram Operations.
                                                              (line  12)
* gsl_histogram_alloc (C function):      Histogram allocation.
                                                              (line  14)
* gsl_histogram_bins (C function):       Updating and accessing histogram elements.
                                                              (line  58)
* gsl_histogram_clone (C function):      Copying Histograms.  (line  14)
* gsl_histogram_div (C function):        Histogram Operations.
                                                              (line  36)
* gsl_histogram_equal_bins_p (C function): Histogram Operations.
                                                              (line   6)
* gsl_histogram_find (C function):       Searching histogram ranges.
                                                              (line   9)
* gsl_histogram_fprintf (C function):    Reading and writing histograms.
                                                              (line  30)
* gsl_histogram_fread (C function):      Reading and writing histograms.
                                                              (line  19)
* gsl_histogram_free (C function):       Histogram allocation.
                                                              (line  66)
* gsl_histogram_fscanf (C function):     Reading and writing histograms.
                                                              (line  57)
* gsl_histogram_fwrite (C function):     Reading and writing histograms.
                                                              (line   9)
* gsl_histogram_get (C function):        Updating and accessing histogram elements.
                                                              (line  34)
* gsl_histogram_get_range (C function):  Updating and accessing histogram elements.
                                                              (line  43)
* gsl_histogram_increment (C function):  Updating and accessing histogram elements.
                                                              (line  11)
* gsl_histogram_max (C function):        Updating and accessing histogram elements.
                                                              (line  58)
* gsl_histogram_max_bin (C function):    Histogram Statistics.
                                                              (line  11)
* gsl_histogram_max_val (C function):    Histogram Statistics.
                                                              (line   6)
* gsl_histogram_mean (C function):       Histogram Statistics.
                                                              (line  28)
* gsl_histogram_memcpy (C function):     Copying Histograms.  (line   6)
* gsl_histogram_min (C function):        Updating and accessing histogram elements.
                                                              (line  58)
* gsl_histogram_min_bin (C function):    Histogram Statistics.
                                                              (line  22)
* gsl_histogram_min_val (C function):    Histogram Statistics.
                                                              (line  17)
* gsl_histogram_mul (C function):        Histogram Operations.
                                                              (line  28)
* gsl_histogram_pdf (C type):            The histogram probability distribution struct.
                                                              (line  18)
* gsl_histogram_pdf_alloc (C function):  The histogram probability distribution struct.
                                                              (line  36)
* gsl_histogram_pdf_free (C function):   The histogram probability distribution struct.
                                                              (line  54)
* gsl_histogram_pdf_init (C function):   The histogram probability distribution struct.
                                                              (line  45)
* gsl_histogram_pdf_sample (C function): The histogram probability distribution struct.
                                                              (line  59)
* gsl_histogram_reset (C function):      Updating and accessing histogram elements.
                                                              (line  67)
* gsl_histogram_scale (C function):      Histogram Operations.
                                                              (line  44)
* gsl_histogram_set_ranges (C function): Histogram allocation.
                                                              (line  24)
* gsl_histogram_set_ranges_uniform (C function): Histogram allocation.
                                                              (line  51)
* gsl_histogram_shift (C function):      Histogram Operations.
                                                              (line  51)
* gsl_histogram_sigma (C function):      Histogram Statistics.
                                                              (line  35)
* gsl_histogram_sub (C function):        Histogram Operations.
                                                              (line  20)
* gsl_histogram_sum (C function):        Histogram Statistics.
                                                              (line  43)
* gsl_hypot (C function):                Elementary Functions.
                                                              (line  24)
* gsl_hypot3 (C function):               Elementary Functions.
                                                              (line  30)
* gsl_ieee_env_setup (C function):       Setting up your IEEE environment.
                                                              (line  27)
* gsl_ieee_fprintf_double (C function):  Representation of floating point numbers.
                                                              (line  56)
* gsl_ieee_fprintf_float (C function):   Representation of floating point numbers.
                                                              (line  56)
* GSL_IEEE_MODE (C macro):               Setting up your IEEE environment.
                                                              (line  23)
* gsl_ieee_printf_double (C function):   Representation of floating point numbers.
                                                              (line  89)
* gsl_ieee_printf_float (C function):    Representation of floating point numbers.
                                                              (line  89)
* GSL_IMAG (C macro):                    Complex number macros.
                                                              (line   9)
* gsl_integration_cquad (C function):    CQUAD doubly-adaptive integration.
                                                              (line  36)
* gsl_integration_cquad_workspace:       CQUAD doubly-adaptive integration.
                                                              (line  20)
* gsl_integration_cquad_workspace_alloc (C function): CQUAD doubly-adaptive integration.
                                                              (line  20)
* gsl_integration_cquad_workspace_free (C function): CQUAD doubly-adaptive integration.
                                                              (line  30)
* gsl_integration_fixed (C function):    Fixed point quadratures.
                                                              (line 170)
* gsl_integration_fixed_alloc (C function): Fixed point quadratures.
                                                              (line  74)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type (C type): Fixed point quadratures.
                                                              (line  87)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_chebyshev (C var): Fixed point quadratures.
                                                              (line  99)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_chebyshev2 (C var): Fixed point quadratures.
                                                              (line 140)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_exponential (C var): Fixed point quadratures.
                                                              (line 129)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_gegenbauer (C var): Fixed point quadratures.
                                                              (line 106)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_hermite (C var): Fixed point quadratures.
                                                              (line 123)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_jacobi (C var): Fixed point quadratures.
                                                              (line 112)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_laguerre (C var): Fixed point quadratures.
                                                              (line 117)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_legendre (C var): Fixed point quadratures.
                                                              (line  92)
* gsl_integration_fixed_alloc.gsl_integration_fixed_type.gsl_integration_fixed_rational (C var): Fixed point quadratures.
                                                              (line 135)
* gsl_integration_fixed_free (C function): Fixed point quadratures.
                                                              (line 147)
* gsl_integration_fixed_n (C function):  Fixed point quadratures.
                                                              (line 153)
* gsl_integration_fixed_nodes (C function): Fixed point quadratures.
                                                              (line 158)
* gsl_integration_fixed_weights (C function): Fixed point quadratures.
                                                              (line 164)
* gsl_integration_fixed_workspace (C type): Fixed point quadratures.
                                                              (line  59)
* gsl_integration_glfixed (C function):  Gauss-Legendre integration.
                                                              (line  23)
* gsl_integration_glfixed_point (C function): Gauss-Legendre integration.
                                                              (line  29)
* gsl_integration_glfixed_table:         Gauss-Legendre integration.
                                                              (line  14)
* gsl_integration_glfixed_table <1>:     Gauss-Legendre integration.
                                                              (line  40)
* gsl_integration_glfixed_table_alloc (C function): Gauss-Legendre integration.
                                                              (line  14)
* gsl_integration_glfixed_table_free (C function): Gauss-Legendre integration.
                                                              (line  40)
* gsl_integration_qag (C function):      QAG adaptive integration.
                                                              (line  33)
* gsl_integration_qagi (C function):     QAGI adaptive integration on infinite intervals.
                                                              (line   6)
* gsl_integration_qagil (C function):    QAGI adaptive integration on infinite intervals.
                                                              (line  36)
* gsl_integration_qagiu (C function):    QAGI adaptive integration on infinite intervals.
                                                              (line  22)
* gsl_integration_qagp (C function):     QAGP adaptive integration with known singular points.
                                                              (line   6)
* gsl_integration_qags (C function):     QAGS adaptive integration with singularities.
                                                              (line  15)
* gsl_integration_qawc (C function):     QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* gsl_integration_qawf (C function):     QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* gsl_integration_qawo (C function):     QAWO adaptive integration for oscillatory functions.
                                                              (line  59)
* gsl_integration_qawo_table:            QAWO adaptive integration for oscillatory functions.
                                                              (line  11)
* gsl_integration_qawo_table_alloc (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  11)
* gsl_integration_qawo_table_alloc.GSL_INTEG_COSINE (C macro): QAWO adaptive integration for oscillatory functions.
                                                              (line  27)
* gsl_integration_qawo_table_alloc.GSL_INTEG_SINE (C macro): QAWO adaptive integration for oscillatory functions.
                                                              (line  29)
* gsl_integration_qawo_table_free (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  53)
* gsl_integration_qawo_table_set (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  40)
* gsl_integration_qawo_table_set_length (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  47)
* gsl_integration_qaws (C function):     QAWS adaptive integration for singular functions.
                                                              (line  68)
* gsl_integration_qaws_table (C type):   QAWS adaptive integration for singular functions.
                                                              (line  11)
* gsl_integration_qaws_table_alloc (C function): QAWS adaptive integration for singular functions.
                                                              (line  16)
* gsl_integration_qaws_table_free (C function): QAWS adaptive integration for singular functions.
                                                              (line  62)
* gsl_integration_qaws_table_set (C function): QAWS adaptive integration for singular functions.
                                                              (line  54)
* gsl_integration_qng (C function):      QNG non-adaptive Gauss-Kronrod integration.
                                                              (line  10)
* gsl_integration_romberg (C function):  Romberg integration. (line  36)
* gsl_integration_romberg_alloc (C function): Romberg integration.
                                                              (line  21)
* gsl_integration_romberg_free (C function): Romberg integration.
                                                              (line  30)
* gsl_integration_workspace:             QAG adaptive integration.
                                                              (line  18)
* gsl_integration_workspace (C type):    QAG adaptive integration.
                                                              (line  13)
* gsl_integration_workspace_alloc (C function): QAG adaptive integration.
                                                              (line  18)
* gsl_integration_workspace_free (C function): QAG adaptive integration.
                                                              (line  27)
* gsl_interp (C type):                   1D Interpolation Functions.
                                                              (line   9)
* gsl_interp2d (C type):                 2D Interpolation Functions.
                                                              (line  10)
* gsl_interp2d_alloc (C function):       2D Interpolation Functions.
                                                              (line  14)
* gsl_interp2d_eval (C function):        2D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp2d_eval_deriv_x (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  41)
* gsl_interp2d_eval_deriv_xx (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  77)
* gsl_interp2d_eval_deriv_xx_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  77)
* gsl_interp2d_eval_deriv_xy (C function): 2D Evaluation of Interpolating Functions.
                                                              (line 113)
* gsl_interp2d_eval_deriv_xy_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line 113)
* gsl_interp2d_eval_deriv_x_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  41)
* gsl_interp2d_eval_deriv_y (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  59)
* gsl_interp2d_eval_deriv_yy (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  95)
* gsl_interp2d_eval_deriv_yy_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  95)
* gsl_interp2d_eval_deriv_y_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  59)
* gsl_interp2d_eval_e (C function):      2D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp2d_eval_extrap (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  23)
* gsl_interp2d_eval_extrap_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  23)
* gsl_interp2d_free (C function):        2D Interpolation Functions.
                                                              (line  39)
* gsl_interp2d_get (C function):         2D Interpolation Grids.
                                                              (line  21)
* gsl_interp2d_idx (C function):         2D Interpolation Grids.
                                                              (line  27)
* gsl_interp2d_init (C function):        2D Interpolation Functions.
                                                              (line  22)
* gsl_interp2d_min_size (C function):    2D Interpolation Types.
                                                              (line  31)
* gsl_interp2d_name (C function):        2D Interpolation Types.
                                                              (line  20)
* gsl_interp2d_set (C function):         2D Interpolation Grids.
                                                              (line  15)
* gsl_interp2d_type (C type):            2D Interpolation Types.
                                                              (line   6)
* gsl_interp2d_type.gsl_interp2d_bicubic (C var): 2D Interpolation Types.
                                                              (line  16)
* gsl_interp2d_type.gsl_interp2d_bilinear (C var): 2D Interpolation Types.
                                                              (line  11)
* gsl_interp2d_type_min_size (C function): 2D Interpolation Types.
                                                              (line  31)
* gsl_interp_accel (C type):             1D Index Look-up and Acceleration.
                                                              (line   9)
* gsl_interp_accel_alloc (C function):   1D Index Look-up and Acceleration.
                                                              (line  25)
* gsl_interp_accel_find (C function):    1D Index Look-up and Acceleration.
                                                              (line  36)
* gsl_interp_accel_free (C function):    1D Index Look-up and Acceleration.
                                                              (line  52)
* gsl_interp_accel_reset (C function):   1D Index Look-up and Acceleration.
                                                              (line  46)
* gsl_interp_alloc (C function):         1D Interpolation Functions.
                                                              (line  13)
* gsl_interp_bsearch (C function):       1D Index Look-up and Acceleration.
                                                              (line  16)
* gsl_interp_eval (C function):          1D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp_eval_deriv (C function):    1D Evaluation of Interpolating Functions.
                                                              (line  20)
* gsl_interp_eval_deriv2 (C function):   1D Evaluation of Interpolating Functions.
                                                              (line  32)
* gsl_interp_eval_deriv2_e (C function): 1D Evaluation of Interpolating Functions.
                                                              (line  32)
* gsl_interp_eval_deriv_e (C function):  1D Evaluation of Interpolating Functions.
                                                              (line  20)
* gsl_interp_eval_e (C function):        1D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp_eval_integ (C function):    1D Evaluation of Interpolating Functions.
                                                              (line  44)
* gsl_interp_eval_integ_e (C function):  1D Evaluation of Interpolating Functions.
                                                              (line  44)
* gsl_interp_free (C function):          1D Interpolation Functions.
                                                              (line  31)
* gsl_interp_init (C function):          1D Interpolation Functions.
                                                              (line  19)
* gsl_interp_min_size (C function):      1D Interpolation Types.
                                                              (line  77)
* gsl_interp_name (C function):          1D Interpolation Types.
                                                              (line  66)
* gsl_interp_type (C type):              1D Interpolation Types.
                                                              (line   8)
* gsl_interp_type.gsl_interp_akima (C var): 1D Interpolation Types.
                                                              (line  42)
* gsl_interp_type.gsl_interp_akima_periodic (C var): 1D Interpolation Types.
                                                              (line  47)
* gsl_interp_type.gsl_interp_cspline (C var): 1D Interpolation Types.
                                                              (line  23)
* gsl_interp_type.gsl_interp_cspline_periodic (C var): 1D Interpolation Types.
                                                              (line  31)
* gsl_interp_type.gsl_interp_linear (C var): 1D Interpolation Types.
                                                              (line  10)
* gsl_interp_type.gsl_interp_polynomial (C var): 1D Interpolation Types.
                                                              (line  15)
* gsl_interp_type.gsl_interp_steffen (C var): 1D Interpolation Types.
                                                              (line  53)
* gsl_interp_type_min_size (C function): 1D Interpolation Types.
                                                              (line  77)
* gsl_isinf (C function):                Infinities and Not-a-number.
                                                              (line  25)
* gsl_isnan (C function):                Infinities and Not-a-number.
                                                              (line  21)
* GSL_IS_EVEN (C macro):                 Testing for Odd and Even Numbers.
                                                              (line  11)
* GSL_IS_ODD (C macro):                  Testing for Odd and Even Numbers.
                                                              (line   6)
* gsl_ldexp (C function):                Elementary Functions.
                                                              (line  51)
* gsl_linalg_balance_matrix (C function): Balancing.          (line  17)
* gsl_linalg_bidiag_decomp (C function): Bidiagonalization.   (line  15)
* gsl_linalg_bidiag_unpack (C function): Bidiagonalization.   (line  28)
* gsl_linalg_bidiag_unpack2 (C function): Bidiagonalization.  (line  40)
* gsl_linalg_bidiag_unpack_B (C function): Bidiagonalization. (line  50)
* gsl_linalg_cholesky_band_decomp (C function): Banded Cholesky Decomposition.
                                                              (line  14)
* gsl_linalg_cholesky_band_invert (C function): Banded Cholesky Decomposition.
                                                              (line  54)
* gsl_linalg_cholesky_band_rcond (C function): Banded Cholesky Decomposition.
                                                              (line  89)
* gsl_linalg_cholesky_band_scale (C function): Banded Cholesky Decomposition.
                                                              (line  71)
* gsl_linalg_cholesky_band_scale_apply (C function): Banded Cholesky Decomposition.
                                                              (line  82)
* gsl_linalg_cholesky_band_solve (C function): Banded Cholesky Decomposition.
                                                              (line  32)
* gsl_linalg_cholesky_band_solvem (C function): Banded Cholesky Decomposition.
                                                              (line  32)
* gsl_linalg_cholesky_band_svx (C function): Banded Cholesky Decomposition.
                                                              (line  42)
* gsl_linalg_cholesky_band_svxm (C function): Banded Cholesky Decomposition.
                                                              (line  42)
* gsl_linalg_cholesky_band_unpack (C function): Banded Cholesky Decomposition.
                                                              (line  63)
* gsl_linalg_cholesky_decomp (C function): Cholesky Decomposition.
                                                              (line  49)
* gsl_linalg_cholesky_decomp1 (C function): Cholesky Decomposition.
                                                              (line  29)
* gsl_linalg_cholesky_decomp2 (C function): Cholesky Decomposition.
                                                              (line  89)
* gsl_linalg_cholesky_invert (C function): Cholesky Decomposition.
                                                              (line  79)
* gsl_linalg_cholesky_rcond (C function): Cholesky Decomposition.
                                                              (line 145)
* gsl_linalg_cholesky_scale (C function): Cholesky Decomposition.
                                                              (line 127)
* gsl_linalg_cholesky_scale_apply (C function): Cholesky Decomposition.
                                                              (line 138)
* gsl_linalg_cholesky_solve (C function): Cholesky Decomposition.
                                                              (line  54)
* gsl_linalg_cholesky_solve2 (C function): Cholesky Decomposition.
                                                              (line 108)
* gsl_linalg_cholesky_svx (C function):  Cholesky Decomposition.
                                                              (line  66)
* gsl_linalg_cholesky_svx2 (C function): Cholesky Decomposition.
                                                              (line 117)
* gsl_linalg_COD_decomp (C function):    Complete Orthogonal Decomposition.
                                                              (line  49)
* gsl_linalg_COD_decomp_e (C function):  Complete Orthogonal Decomposition.
                                                              (line  49)
* gsl_linalg_COD_lssolve (C function):   Complete Orthogonal Decomposition.
                                                              (line  70)
* gsl_linalg_COD_lssolve2 (C function):  Complete Orthogonal Decomposition.
                                                              (line  86)
* gsl_linalg_COD_matZ (C function):      Complete Orthogonal Decomposition.
                                                              (line 113)
* gsl_linalg_COD_unpack (C function):    Complete Orthogonal Decomposition.
                                                              (line 103)
* gsl_linalg_complex_cholesky_decomp (C function): Cholesky Decomposition.
                                                              (line  29)
* gsl_linalg_complex_cholesky_invert (C function): Cholesky Decomposition.
                                                              (line  79)
* gsl_linalg_complex_cholesky_solve (C function): Cholesky Decomposition.
                                                              (line  54)
* gsl_linalg_complex_cholesky_svx (C function): Cholesky Decomposition.
                                                              (line  66)
* gsl_linalg_complex_householder_hm (C function): Householder Transformations.
                                                              (line  30)
* gsl_linalg_complex_householder_hv (C function): Householder Transformations.
                                                              (line  50)
* gsl_linalg_complex_householder_mh (C function): Householder Transformations.
                                                              (line  40)
* gsl_linalg_complex_householder_transform (C function): Householder Transformations.
                                                              (line  17)
* gsl_linalg_complex_LU_decomp (C function): LU Decomposition.
                                                              (line  19)
* gsl_linalg_complex_LU_det (C function): LU Decomposition.   (line 111)
* gsl_linalg_complex_LU_invert (C function): LU Decomposition.
                                                              (line  77)
* gsl_linalg_complex_LU_invx (C function): LU Decomposition.  (line  94)
* gsl_linalg_complex_LU_lndet (C function): LU Decomposition. (line 120)
* gsl_linalg_complex_LU_refine (C function): LU Decomposition.
                                                              (line  64)
* gsl_linalg_complex_LU_sgndet (C function): LU Decomposition.
                                                              (line 129)
* gsl_linalg_complex_LU_solve (C function): LU Decomposition. (line  43)
* gsl_linalg_complex_LU_svx (C function): LU Decomposition.   (line  54)
* gsl_linalg_complex_QR_decomp (C function): Level 2 Interface.
                                                              (line  10)
* gsl_linalg_complex_QR_decomp_r (C function): QR Decomposition.
                                                              (line  63)
* gsl_linalg_complex_QR_lssolve (C function): Level 2 Interface.
                                                              (line  51)
* gsl_linalg_complex_QR_lssolve_r (C function): QR Decomposition.
                                                              (line  91)
* gsl_linalg_complex_QR_QHvec (C function): Level 2 Interface.
                                                              (line  69)
* gsl_linalg_complex_QR_QHvec_r (C function): QR Decomposition.
                                                              (line 113)
* gsl_linalg_complex_QR_Qvec (C function): Level 2 Interface. (line  81)
* gsl_linalg_complex_QR_solve (C function): Level 2 Interface.
                                                              (line  30)
* gsl_linalg_complex_QR_solve_r (C function): QR Decomposition.
                                                              (line  79)
* gsl_linalg_complex_QR_svx (C function): Level 2 Interface.  (line  41)
* gsl_linalg_complex_QR_unpack_r (C function): QR Decomposition.
                                                              (line 137)
* gsl_linalg_complex_tri_invert (C function): Triangular Systems.
                                                              (line   6)
* gsl_linalg_complex_tri_LHL (C function): Triangular Systems.
                                                              (line  17)
* gsl_linalg_complex_tri_UL (C function): Triangular Systems. (line  24)
* gsl_linalg_givens (C function):        Givens Rotations.    (line  21)
* gsl_linalg_givens_gv (C function):     Givens Rotations.    (line  28)
* gsl_linalg_hermtd_decomp (C function): Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  14)
* gsl_linalg_hermtd_unpack (C function): Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  27)
* gsl_linalg_hermtd_unpack_T (C function): Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  37)
* gsl_linalg_hessenberg_decomp (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  16)
* gsl_linalg_hessenberg_set_zero (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  49)
* gsl_linalg_hessenberg_unpack (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  29)
* gsl_linalg_hessenberg_unpack_accum (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  37)
* gsl_linalg_hesstri_decomp (C function): Hessenberg-Triangular Decomposition of Real Matrices.
                                                              (line  17)
* gsl_linalg_HH_solve (C function):      Householder solver for linear systems.
                                                              (line   6)
* gsl_linalg_HH_svx (C function):        Householder solver for linear systems.
                                                              (line  14)
* gsl_linalg_householder_hm (C function): Householder Transformations.
                                                              (line  30)
* gsl_linalg_householder_hv (C function): Householder Transformations.
                                                              (line  50)
* gsl_linalg_householder_mh (C function): Householder Transformations.
                                                              (line  40)
* gsl_linalg_householder_transform (C function): Householder Transformations.
                                                              (line  17)
* gsl_linalg_ldlt_band_decomp (C function): Banded LDLT Decomposition.
                                                              (line  14)
* gsl_linalg_ldlt_band_rcond (C function): Banded LDLT Decomposition.
                                                              (line  53)
* gsl_linalg_ldlt_band_solve (C function): Banded LDLT Decomposition.
                                                              (line  26)
* gsl_linalg_ldlt_band_svx (C function): Banded LDLT Decomposition.
                                                              (line  34)
* gsl_linalg_ldlt_band_unpack (C function): Banded LDLT Decomposition.
                                                              (line  44)
* gsl_linalg_ldlt_decomp (C function):   LDLT Decomposition.  (line  22)
* gsl_linalg_ldlt_rcond (C function):    LDLT Decomposition.  (line  54)
* gsl_linalg_ldlt_solve (C function):    LDLT Decomposition.  (line  37)
* gsl_linalg_ldlt_svx (C function):      LDLT Decomposition.  (line  45)
* gsl_linalg_LQ_decomp (C function):     LQ Decomposition.    (line  27)
* gsl_linalg_LQ_lssolve (C function):    LQ Decomposition.    (line  41)
* gsl_linalg_LQ_QTvec (C function):      LQ Decomposition.    (line  60)
* gsl_linalg_LQ_unpack (C function):     LQ Decomposition.    (line  53)
* gsl_linalg_LU_band_decomp (C function): Banded LU Decomposition.
                                                              (line  28)
* gsl_linalg_LU_band_solve (C function): Banded LU Decomposition.
                                                              (line  43)
* gsl_linalg_LU_band_svx (C function):   Banded LU Decomposition.
                                                              (line  54)
* gsl_linalg_LU_band_unpack (C function): Banded LU Decomposition.
                                                              (line  66)
* gsl_linalg_LU_decomp (C function):     LU Decomposition.    (line  19)
* gsl_linalg_LU_det (C function):        LU Decomposition.    (line 111)
* gsl_linalg_LU_invert (C function):     LU Decomposition.    (line  77)
* gsl_linalg_LU_invx (C function):       LU Decomposition.    (line  94)
* gsl_linalg_LU_lndet (C function):      LU Decomposition.    (line 120)
* gsl_linalg_LU_refine (C function):     LU Decomposition.    (line  64)
* gsl_linalg_LU_sgndet (C function):     LU Decomposition.    (line 129)
* gsl_linalg_LU_solve (C function):      LU Decomposition.    (line  43)
* gsl_linalg_LU_svx (C function):        LU Decomposition.    (line  54)
* gsl_linalg_mcholesky_decomp (C function): Modified Cholesky Decomposition.
                                                              (line  25)
* gsl_linalg_mcholesky_rcond (C function): Modified Cholesky Decomposition.
                                                              (line  59)
* gsl_linalg_mcholesky_solve (C function): Modified Cholesky Decomposition.
                                                              (line  41)
* gsl_linalg_mcholesky_svx (C function): Modified Cholesky Decomposition.
                                                              (line  49)
* gsl_linalg_pcholesky_decomp (C function): Pivoted Cholesky Decomposition.
                                                              (line  20)
* gsl_linalg_pcholesky_decomp2 (C function): Pivoted Cholesky Decomposition.
                                                              (line  53)
* gsl_linalg_pcholesky_invert (C function): Pivoted Cholesky Decomposition.
                                                              (line  94)
* gsl_linalg_pcholesky_rcond (C function): Pivoted Cholesky Decomposition.
                                                              (line 101)
* gsl_linalg_pcholesky_solve (C function): Pivoted Cholesky Decomposition.
                                                              (line  35)
* gsl_linalg_pcholesky_solve2 (C function): Pivoted Cholesky Decomposition.
                                                              (line  73)
* gsl_linalg_pcholesky_svx (C function): Pivoted Cholesky Decomposition.
                                                              (line  43)
* gsl_linalg_pcholesky_svx2 (C function): Pivoted Cholesky Decomposition.
                                                              (line  83)
* gsl_linalg_QL_decomp (C function):     QL Decomposition.    (line  22)
* gsl_linalg_QL_unpack (C function):     QL Decomposition.    (line  30)
* gsl_linalg_QRPT_decomp (C function):   QR Decomposition with Column Pivoting.
                                                              (line  33)
* gsl_linalg_QRPT_decomp2 (C function):  QR Decomposition with Column Pivoting.
                                                              (line  59)
* gsl_linalg_QRPT_lssolve (C function):  QR Decomposition with Column Pivoting.
                                                              (line  85)
* gsl_linalg_QRPT_lssolve2 (C function): QR Decomposition with Column Pivoting.
                                                              (line 100)
* gsl_linalg_QRPT_QRsolve (C function):  QR Decomposition with Column Pivoting.
                                                              (line 117)
* gsl_linalg_QRPT_rank (C function):     QR Decomposition with Column Pivoting.
                                                              (line 149)
* gsl_linalg_QRPT_rcond (C function):    QR Decomposition with Column Pivoting.
                                                              (line 159)
* gsl_linalg_QRPT_Rsolve (C function):   QR Decomposition with Column Pivoting.
                                                              (line 135)
* gsl_linalg_QRPT_Rsvx (C function):     QR Decomposition with Column Pivoting.
                                                              (line 141)
* gsl_linalg_QRPT_solve (C function):    QR Decomposition with Column Pivoting.
                                                              (line  68)
* gsl_linalg_QRPT_svx (C function):      QR Decomposition with Column Pivoting.
                                                              (line  77)
* gsl_linalg_QRPT_update (C function):   QR Decomposition with Column Pivoting.
                                                              (line 125)
* gsl_linalg_QR_decomp (C function):     Level 2 Interface.   (line  10)
* gsl_linalg_QR_decomp_r (C function):   QR Decomposition.    (line  63)
* gsl_linalg_QR_lssolve (C function):    Level 2 Interface.   (line  51)
* gsl_linalg_QR_lssolve_r (C function):  QR Decomposition.    (line  91)
* gsl_linalg_QR_QRsolve (C function):    Level 2 Interface.   (line 124)
* gsl_linalg_QR_QTmat (C function):      Level 2 Interface.   (line  92)
* gsl_linalg_QR_QTmat_r (C function):    QR Decomposition.    (line 127)
* gsl_linalg_QR_QTvec (C function):      Level 2 Interface.   (line  69)
* gsl_linalg_QR_QTvec_r (C function):    QR Decomposition.    (line 113)
* gsl_linalg_QR_Qvec (C function):       Level 2 Interface.   (line  81)
* gsl_linalg_QR_rcond (C function):      QR Decomposition.    (line 151)
* gsl_linalg_QR_Rsolve (C function):     Level 2 Interface.   (line 101)
* gsl_linalg_QR_Rsvx (C function):       Level 2 Interface.   (line 108)
* gsl_linalg_QR_solve (C function):      Level 2 Interface.   (line  30)
* gsl_linalg_QR_solve_r (C function):    QR Decomposition.    (line  79)
* gsl_linalg_QR_svx (C function):        Level 2 Interface.   (line  41)
* gsl_linalg_QR_UD_decomp (C function):  Triangle on Top of Diagonal.
                                                              (line  26)
* gsl_linalg_QR_UD_lssolve (C function): Triangle on Top of Diagonal.
                                                              (line  35)
* gsl_linalg_QR_unpack (C function):     Level 2 Interface.   (line 117)
* gsl_linalg_QR_unpack_r (C function):   QR Decomposition.    (line 137)
* gsl_linalg_QR_update (C function):     Level 2 Interface.   (line 131)
* gsl_linalg_QR_UR_decomp (C function):  Triangle on Top of Rectangle.
                                                              (line  27)
* gsl_linalg_QR_UU_decomp (C function):  Triangle on Top of Triangle.
                                                              (line  25)
* gsl_linalg_QR_UU_lssolve (C function): Triangle on Top of Triangle.
                                                              (line  34)
* gsl_linalg_QR_UU_QTec (C function):    Triangle on Top of Triangle.
                                                              (line  54)
* gsl_linalg_QR_UZ_decomp (C function):  Triangle on Top of Trapezoidal.
                                                              (line  31)
* gsl_linalg_R_solve (C function):       Level 2 Interface.   (line 140)
* gsl_linalg_R_svx (C function):         Level 2 Interface.   (line 146)
* gsl_linalg_solve_cyc_tridiag (C function): Tridiagonal Systems.
                                                              (line  41)
* gsl_linalg_solve_symm_cyc_tridiag (C function): Tridiagonal Systems.
                                                              (line  56)
* gsl_linalg_solve_symm_tridiag (C function): Tridiagonal Systems.
                                                              (line  28)
* gsl_linalg_solve_tridiag (C function): Tridiagonal Systems. (line  13)
* gsl_linalg_SV_decomp (C function):     Singular Value Decomposition.
                                                              (line  38)
* gsl_linalg_SV_decomp_jacobi (C function): Singular Value Decomposition.
                                                              (line  61)
* gsl_linalg_SV_decomp_mod (C function): Singular Value Decomposition.
                                                              (line  53)
* gsl_linalg_SV_leverage (C function):   Singular Value Decomposition.
                                                              (line  87)
* gsl_linalg_SV_solve (C function):      Singular Value Decomposition.
                                                              (line  69)
* gsl_linalg_symmtd_decomp (C function): Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  13)
* gsl_linalg_symmtd_unpack (C function): Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  26)
* gsl_linalg_symmtd_unpack_T (C function): Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  36)
* gsl_linalg_tri_invert (C function):    Triangular Systems.  (line   6)
* gsl_linalg_tri_LTL (C function):       Triangular Systems.  (line  17)
* gsl_linalg_tri_rcond (C function):     Triangular Systems.  (line  33)
* gsl_linalg_tri_UL (C function):        Triangular Systems.  (line  24)
* gsl_log1p (C function):                Elementary Functions.
                                                              (line  12)
* gsl_matrix (C type):                   Matrices.            (line  11)
* gsl_matrix_add (C function):           Matrix operations.   (line   8)
* gsl_matrix_add_constant (C function):  Matrix operations.   (line  68)
* gsl_matrix_alloc (C function):         Matrix allocation.   (line  14)
* gsl_matrix_calloc (C function):        Matrix allocation.   (line  25)
* gsl_matrix_column (C function):        Creating row and column views.
                                                              (line  26)
* gsl_matrix_complex_conjtrans_memcpy (C function): Exchanging rows and columns.
                                                              (line  43)
* gsl_matrix_const_column (C function):  Creating row and column views.
                                                              (line  26)
* gsl_matrix_const_diagonal (C function): Creating row and column views.
                                                              (line  70)
* gsl_matrix_const_ptr (C function):     Accessing matrix elements.
                                                              (line  38)
* gsl_matrix_const_row (C function):     Creating row and column views.
                                                              (line  13)
* gsl_matrix_const_subcolumn (C function): Creating row and column views.
                                                              (line  54)
* gsl_matrix_const_subdiagonal (C function): Creating row and column views.
                                                              (line  84)
* gsl_matrix_const_submatrix (C function): Matrix views.      (line  21)
* gsl_matrix_const_subrow (C function):  Creating row and column views.
                                                              (line  39)
* gsl_matrix_const_superdiagonal (C function): Creating row and column views.
                                                              (line  98)
* gsl_matrix_const_view (C type):        Matrix views.        (line   6)
* gsl_matrix_const_view_array (C function): Matrix views.     (line  57)
* gsl_matrix_const_view_array_with_tda (C function): Matrix views.
                                                              (line  84)
* gsl_matrix_const_view_vector (C function): Matrix views.    (line 112)
* gsl_matrix_const_view_vector_with_tda (C function): Matrix views.
                                                              (line 139)
* gsl_matrix_diagonal (C function):      Creating row and column views.
                                                              (line  70)
* gsl_matrix_div_elements (C function):  Matrix operations.   (line  30)
* gsl_matrix_equal (C function):         Matrix properties.   (line  21)
* gsl_matrix_fprintf (C function):       Reading and writing matrices.
                                                              (line  28)
* gsl_matrix_fread (C function):         Reading and writing matrices.
                                                              (line  17)
* gsl_matrix_free (C function):          Matrix allocation.   (line  32)
* gsl_matrix_fscanf (C function):        Reading and writing matrices.
                                                              (line  38)
* gsl_matrix_fwrite (C function):        Reading and writing matrices.
                                                              (line   9)
* gsl_matrix_get (C function):           Accessing matrix elements.
                                                              (line  20)
* gsl_matrix_get_col (C function):       Copying rows and columns.
                                                              (line  20)
* gsl_matrix_get_row (C function):       Copying rows and columns.
                                                              (line  13)
* gsl_matrix_isneg (C function):         Matrix properties.   (line  10)
* gsl_matrix_isnonneg (C function):      Matrix properties.   (line  10)
* gsl_matrix_isnull (C function):        Matrix properties.   (line  10)
* gsl_matrix_ispos (C function):         Matrix properties.   (line  10)
* gsl_matrix_max (C function):           Finding maximum and minimum elements of matrices.
                                                              (line   8)
* gsl_matrix_max_index (C function):     Finding maximum and minimum elements of matrices.
                                                              (line  23)
* gsl_matrix_memcpy (C function):        Copying matrices.    (line   6)
* gsl_matrix_min (C function):           Finding maximum and minimum elements of matrices.
                                                              (line  12)
* gsl_matrix_minmax (C function):        Finding maximum and minimum elements of matrices.
                                                              (line  16)
* gsl_matrix_minmax_index (C function):  Finding maximum and minimum elements of matrices.
                                                              (line  39)
* gsl_matrix_min_index (C function):     Finding maximum and minimum elements of matrices.
                                                              (line  31)
* gsl_matrix_mul_elements (C function):  Matrix operations.   (line  22)
* gsl_matrix_norm1 (C function):         Matrix properties.   (line  27)
* gsl_matrix_ptr (C function):           Accessing matrix elements.
                                                              (line  38)
* gsl_matrix_row (C function):           Creating row and column views.
                                                              (line  13)
* gsl_matrix_scale (C function):         Matrix operations.   (line  38)
* gsl_matrix_scale_columns (C function): Matrix operations.   (line  44)
* gsl_matrix_scale_rows (C function):    Matrix operations.   (line  56)
* gsl_matrix_set (C function):           Accessing matrix elements.
                                                              (line  29)
* gsl_matrix_set_all (C function):       Initializing matrix elements.
                                                              (line   6)
* gsl_matrix_set_col (C function):       Copying rows and columns.
                                                              (line  34)
* gsl_matrix_set_identity (C function):  Initializing matrix elements.
                                                              (line  16)
* gsl_matrix_set_row (C function):       Copying rows and columns.
                                                              (line  27)
* gsl_matrix_set_zero (C function):      Initializing matrix elements.
                                                              (line  11)
* gsl_matrix_sub (C function):           Matrix operations.   (line  15)
* gsl_matrix_subcolumn (C function):     Creating row and column views.
                                                              (line  54)
* gsl_matrix_subdiagonal (C function):   Creating row and column views.
                                                              (line  84)
* gsl_matrix_submatrix (C function):     Matrix views.        (line  21)
* gsl_matrix_subrow (C function):        Creating row and column views.
                                                              (line  39)
* gsl_matrix_superdiagonal (C function): Creating row and column views.
                                                              (line  98)
* gsl_matrix_swap (C function):          Copying matrices.    (line  13)
* gsl_matrix_swap_columns (C function):  Exchanging rows and columns.
                                                              (line  15)
* gsl_matrix_swap_rowcol (C function):   Exchanging rows and columns.
                                                              (line  21)
* gsl_matrix_swap_rows (C function):     Exchanging rows and columns.
                                                              (line   9)
* gsl_matrix_transpose (C function):     Exchanging rows and columns.
                                                              (line  37)
* gsl_matrix_transpose_memcpy (C function): Exchanging rows and columns.
                                                              (line  28)
* gsl_matrix_view (C type):              Matrix views.        (line   6)
* gsl_matrix_view_array (C function):    Matrix views.        (line  57)
* gsl_matrix_view_array_with_tda (C function): Matrix views.  (line  84)
* gsl_matrix_view_vector (C function):   Matrix views.        (line 112)
* gsl_matrix_view_vector_with_tda (C function): Matrix views. (line 139)
* GSL_MAX (C macro):                     Maximum and Minimum functions.
                                                              (line  10)
* GSL_MAX_DBL (C function):              Maximum and Minimum functions.
                                                              (line  20)
* GSL_MAX_INT (C function):              Maximum and Minimum functions.
                                                              (line  38)
* GSL_MAX_LDBL (C function):             Maximum and Minimum functions.
                                                              (line  46)
* GSL_MIN (C macro):                     Maximum and Minimum functions.
                                                              (line  15)
* GSL_MIN_DBL (C function):              Maximum and Minimum functions.
                                                              (line  29)
* gsl_min_fminimizer (C type):           Initializing the Minimizer.
                                                              (line   6)
* gsl_min_fminimizer_alloc (C function): Initializing the Minimizer.
                                                              (line  10)
* gsl_min_fminimizer_free (C function):  Initializing the Minimizer.
                                                              (line  46)
* gsl_min_fminimizer_f_lower (C function): Iteration<2>.      (line  46)
* gsl_min_fminimizer_f_minimum (C function): Iteration<2>.    (line  46)
* gsl_min_fminimizer_f_upper (C function): Iteration<2>.      (line  46)
* gsl_min_fminimizer_iterate (C function): Iteration<2>.      (line  12)
* gsl_min_fminimizer_name (C function):  Initializing the Minimizer.
                                                              (line  51)
* gsl_min_fminimizer_set (C function):   Initializing the Minimizer.
                                                              (line  24)
* gsl_min_fminimizer_set_with_values (C function): Initializing the Minimizer.
                                                              (line  36)
* gsl_min_fminimizer_type (C type):      Minimization Algorithms.
                                                              (line  13)
* gsl_min_fminimizer_type.gsl_min_fminimizer_brent (C var): Minimization Algorithms.
                                                              (line  34)
* gsl_min_fminimizer_type.gsl_min_fminimizer_goldensection (C var): Minimization Algorithms.
                                                              (line  15)
* gsl_min_fminimizer_type.gsl_min_fminimizer_quad_golden (C var): Minimization Algorithms.
                                                              (line  52)
* gsl_min_fminimizer_x_lower (C function): Iteration<2>.      (line  38)
* gsl_min_fminimizer_x_minimum (C function): Iteration<2>.    (line  32)
* gsl_min_fminimizer_x_upper (C function): Iteration<2>.      (line  38)
* GSL_MIN_INT (C function):              Maximum and Minimum functions.
                                                              (line  38)
* GSL_MIN_LDBL (C function):             Maximum and Minimum functions.
                                                              (line  46)
* gsl_min_test_interval (C function):    Stopping Parameters. (line  18)
* gsl_mode_t (C type):                   Modes.               (line  14)
* gsl_mode_t.GSL_PREC_APPROX (C macro):  Modes.               (line  26)
* gsl_mode_t.GSL_PREC_DOUBLE (C macro):  Modes.               (line  16)
* gsl_mode_t.GSL_PREC_SINGLE (C macro):  Modes.               (line  21)
* gsl_monte_function (C type):           Interface.           (line  27)
* gsl_monte_miser_alloc (C function):    MISER.               (line  48)
* gsl_monte_miser_free (C function):     MISER.               (line  77)
* gsl_monte_miser_init (C function):     MISER.               (line  55)
* gsl_monte_miser_integrate (C function): MISER.              (line  61)
* gsl_monte_miser_params (C type):       MISER.               (line 104)
* gsl_monte_miser_params.alpha (C var):  MISER.               (line 133)
* gsl_monte_miser_params.dither (C var): MISER.               (line 151)
* gsl_monte_miser_params.estimate_frac (C var): MISER.        (line 106)
* gsl_monte_miser_params.min_calls (C var): MISER.            (line 113)
* gsl_monte_miser_params.min_calls_per_bisection (C var): MISER.
                                                              (line 123)
* gsl_monte_miser_params_get (C function): MISER.             (line  85)
* gsl_monte_miser_params_set (C function): MISER.             (line  91)
* gsl_monte_miser_state (C type):        MISER.               (line  44)
* gsl_monte_plain_alloc (C function):    PLAIN Monte Carlo.   (line  33)
* gsl_monte_plain_free (C function):     PLAIN Monte Carlo.   (line  61)
* gsl_monte_plain_init (C function):     PLAIN Monte Carlo.   (line  39)
* gsl_monte_plain_integrate (C function): PLAIN Monte Carlo.  (line  45)
* gsl_monte_plain_state (C type):        PLAIN Monte Carlo.   (line  29)
* gsl_monte_vegas_alloc (C function):    VEGAS.               (line  54)
* gsl_monte_vegas_chisq (C function):    VEGAS.               (line 124)
* gsl_monte_vegas_free (C function):     VEGAS.               (line  86)
* gsl_monte_vegas_init (C function):     VEGAS.               (line  61)
* gsl_monte_vegas_integrate (C function): VEGAS.              (line  67)
* gsl_monte_vegas_params (C type):       VEGAS.               (line 163)
* gsl_monte_vegas_params.alpha (C var):  VEGAS.               (line 165)
* gsl_monte_vegas_params.iterations (C var): VEGAS.           (line 172)
* gsl_monte_vegas_params.mode (C var):   VEGAS.               (line 193)
* gsl_monte_vegas_params.ostream (C var): VEGAS.              (line 203)
* gsl_monte_vegas_params.stage (C var):  VEGAS.               (line 177)
* gsl_monte_vegas_params.verbose (C var): VEGAS.              (line 203)
* gsl_monte_vegas_params_get (C function): VEGAS.             (line 144)
* gsl_monte_vegas_params_set (C function): VEGAS.             (line 150)
* gsl_monte_vegas_runval (C function):   VEGAS.               (line 134)
* gsl_monte_vegas_state (C type):        VEGAS.               (line  50)
* gsl_movstat_accum (C type):            Accumulators.        (line  11)
* gsl_movstat_accum.delete (C member):   Accumulators.        (line  45)
* gsl_movstat_accum.get (C member):      Accumulators.        (line  50)
* gsl_movstat_accum.init (C member):     Accumulators.        (line  32)
* gsl_movstat_accum.insert (C member):   Accumulators.        (line  37)
* gsl_movstat_accum.size (C member):     Accumulators.        (line  27)
* gsl_movstat_accum_max (C var):         Accumulators.        (line  60)
* gsl_movstat_accum_mean (C var):        Accumulators.        (line  67)
* gsl_movstat_accum_median (C var):      Accumulators.        (line  74)
* gsl_movstat_accum_min (C var):         Accumulators.        (line  60)
* gsl_movstat_accum_minmax (C var):      Accumulators.        (line  60)
* gsl_movstat_accum_Qn (C var):          Accumulators.        (line  79)
* gsl_movstat_accum_qqr (C var):         Accumulators.        (line  89)
* gsl_movstat_accum_sd (C var):          Accumulators.        (line  67)
* gsl_movstat_accum_Sn (C var):          Accumulators.        (line  79)
* gsl_movstat_accum_sum (C var):         Accumulators.        (line  85)
* gsl_movstat_accum_variance (C var):    Accumulators.        (line  67)
* gsl_movstat_alloc (C function):        Allocation for Moving Window Statistics.
                                                              (line  10)
* gsl_movstat_alloc2 (C function):       Allocation for Moving Window Statistics.
                                                              (line  18)
* gsl_movstat_apply (C function):        User-defined Moving Statistics.
                                                              (line  41)
* gsl_movstat_end_t (C type):            Handling Endpoints.  (line  12)
* gsl_movstat_end_t.GSL_MOVSTAT_END_PADVALUE (C macro): Handling Endpoints.
                                                              (line  27)
* gsl_movstat_end_t.GSL_MOVSTAT_END_PADZERO (C macro): Handling Endpoints.
                                                              (line  17)
* gsl_movstat_end_t.GSL_MOVSTAT_END_TRUNCATE (C macro): Handling Endpoints.
                                                              (line  36)
* gsl_movstat_fill (C function):         User-defined Moving Statistics.
                                                              (line  52)
* gsl_movstat_free (C function):         Allocation for Moving Window Statistics.
                                                              (line  26)
* gsl_movstat_function (C type):         User-defined Moving Statistics.
                                                              (line  16)
* gsl_movstat_function.function (C member): User-defined Moving Statistics.
                                                              (line  30)
* gsl_movstat_function.params (C member): User-defined Moving Statistics.
                                                              (line  37)
* gsl_movstat_mad (C function):          Moving MAD.          (line  15)
* gsl_movstat_mad0 (C function):         Moving MAD.          (line  15)
* gsl_movstat_max (C function):          Moving Minimum and Maximum.
                                                              (line  21)
* gsl_movstat_mean (C function):         Moving Mean.         (line  16)
* gsl_movstat_median (C function):       Moving Median.       (line  11)
* gsl_movstat_min (C function):          Moving Minimum and Maximum.
                                                              (line  12)
* gsl_movstat_minmax (C function):       Moving Minimum and Maximum.
                                                              (line  30)
* gsl_movstat_Qn (C function):           Moving Q_n.          (line  10)
* gsl_movstat_qqr (C function):          Moving QQR.          (line  19)
* gsl_movstat_sd (C function):           Moving Variance and Standard Deviation.
                                                              (line  23)
* gsl_movstat_Sn (C function):           Moving S_n.          (line  11)
* gsl_movstat_sum (C function):          Moving Sum.          (line  11)
* gsl_movstat_variance (C function):     Moving Variance and Standard Deviation.
                                                              (line  14)
* gsl_movstat_workspace (C type):        Allocation for Moving Window Statistics.
                                                              (line   6)
* gsl_multifit_linear (C function):      Multi-parameter regression.
                                                              (line  76)
* gsl_multifit_linear_alloc (C function): Multi-parameter regression.
                                                              (line  48)
* gsl_multifit_linear_applyW (C function): Regularized regression.
                                                              (line 234)
* gsl_multifit_linear_bsvd (C function): Multi-parameter regression.
                                                              (line  68)
* gsl_multifit_linear_est (C function):  Multi-parameter regression.
                                                              (line 157)
* gsl_multifit_linear_free (C function): Multi-parameter regression.
                                                              (line  56)
* gsl_multifit_linear_gcv (C function):  Regularized regression.
                                                              (line 365)
* gsl_multifit_linear_gcv_calc (C function): Regularized regression.
                                                              (line 358)
* gsl_multifit_linear_gcv_curve (C function): Regularized regression.
                                                              (line 336)
* gsl_multifit_linear_gcv_init (C function): Regularized regression.
                                                              (line 324)
* gsl_multifit_linear_gcv_min (C function): Regularized regression.
                                                              (line 346)
* gsl_multifit_linear_genform1 (C function): Regularized regression.
                                                              (line 201)
* gsl_multifit_linear_genform2 (C function): Regularized regression.
                                                              (line 213)
* gsl_multifit_linear_lcorner (C function): Regularized regression.
                                                              (line 284)
* gsl_multifit_linear_lcorner2 (C function): Regularized regression.
                                                              (line 302)
* gsl_multifit_linear_lcurvature (C function): Regularized regression.
                                                              (line 264)
* gsl_multifit_linear_lcurve (C function): Regularized regression.
                                                              (line 246)
* gsl_multifit_linear_Lk (C function):   Regularized regression.
                                                              (line 378)
* gsl_multifit_linear_Lsobolev (C function): Regularized regression.
                                                              (line 386)
* gsl_multifit_linear_L_decomp (C function): Regularized regression.
                                                              (line 143)
* gsl_multifit_linear_rank (C function): Multi-parameter regression.
                                                              (line 174)
* gsl_multifit_linear_rcond (C function): Regularized regression.
                                                              (line 403)
* gsl_multifit_linear_residuals (C function): Multi-parameter regression.
                                                              (line 167)
* gsl_multifit_linear_solve (C function): Regularized regression.
                                                              (line 182)
* gsl_multifit_linear_stdform1 (C function): Regularized regression.
                                                              (line 117)
* gsl_multifit_linear_stdform2 (C function): Regularized regression.
                                                              (line 157)
* gsl_multifit_linear_svd (C function):  Multi-parameter regression.
                                                              (line  61)
* gsl_multifit_linear_tsvd (C function): Multi-parameter regression.
                                                              (line  99)
* gsl_multifit_linear_wgenform2 (C function): Regularized regression.
                                                              (line 213)
* gsl_multifit_linear_workspace (C type): Multi-parameter regression.
                                                              (line  43)
* gsl_multifit_linear_wstdform1 (C function): Regularized regression.
                                                              (line 117)
* gsl_multifit_linear_wstdform2 (C function): Regularized regression.
                                                              (line 157)
* gsl_multifit_nlinear_alloc (C function): Initializing the Solver<3>.
                                                              (line  18)
* gsl_multifit_nlinear_avratio (C function): Iteration<5>.    (line 109)
* gsl_multifit_nlinear_covar (C function): Covariance matrix of best fit parameters.
                                                              (line   6)
* gsl_multifit_nlinear_default_parameters (C function): Initializing the Solver<3>.
                                                              (line  53)
* gsl_multifit_nlinear_driver (C function): High Level Driver.
                                                              (line   9)
* gsl_multifit_nlinear_fdf (C type):     Providing the Function to be Minimized.
                                                              (line  10)
* gsl_multifit_nlinear_fdtype (C type):  Tunable Parameters.  (line 211)
* gsl_multifit_nlinear_fdtype.GSL_MULTIFIT_NLINEAR_CTRDIFF (C macro): Tunable Parameters.
                                                              (line 236)
* gsl_multifit_nlinear_fdtype.GSL_MULTIFIT_NLINEAR_FWDIFF (C macro): Tunable Parameters.
                                                              (line 218)
* gsl_multifit_nlinear_free (C function): Initializing the Solver<3>.
                                                              (line  82)
* gsl_multifit_nlinear_init (C function): Initializing the Solver<3>.
                                                              (line  63)
* gsl_multifit_nlinear_iterate (C function): Iteration<5>.    (line  10)
* gsl_multifit_nlinear_jac (C function): Iteration<5>.        (line  61)
* gsl_multifit_nlinear_name (C function): Initializing the Solver<3>.
                                                              (line  90)
* gsl_multifit_nlinear_niter (C function): Iteration<5>.      (line  68)
* gsl_multifit_nlinear_parameters (C type): Tunable Parameters.
                                                              (line  11)
* gsl_multifit_nlinear_position (C function): Iteration<5>.   (line  44)
* gsl_multifit_nlinear_rcond (C function): Iteration<5>.      (line  77)
* gsl_multifit_nlinear_residual (C function): Iteration<5>.   (line  52)
* gsl_multifit_nlinear_scale (C type):   Tunable Parameters.  (line  97)
* gsl_multifit_nlinear_solver (C type):  Tunable Parameters.  (line 147)
* gsl_multifit_nlinear_test (C function): Testing for Convergence.
                                                              (line  19)
* gsl_multifit_nlinear_trs (C type):     Tunable Parameters.  (line  49)
* gsl_multifit_nlinear_trs_name (C function): Initializing the Solver<3>.
                                                              (line 102)
* gsl_multifit_nlinear_type (C type):    Initializing the Solver<3>.
                                                              (line   6)
* gsl_multifit_nlinear_type.gsl_multifit_nlinear_trust (C var): Initializing the Solver<3>.
                                                              (line  12)
* gsl_multifit_nlinear_winit (C function): Initializing the Solver<3>.
                                                              (line  63)
* gsl_multifit_robust (C function):      Robust linear regression.
                                                              (line 217)
* gsl_multifit_robust_alloc (C function): Robust linear regression.
                                                              (line  81)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type (C type): Robust linear regression.
                                                              (line  90)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_bisquare (C var): Robust linear regression.
                                                              (line  99)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_cauchy (C var): Robust linear regression.
                                                              (line 111)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_default (C var): Robust linear regression.
                                                              (line  92)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_fair (C var): Robust linear regression.
                                                              (line 125)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_huber (C var): Robust linear regression.
                                                              (line 136)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_ols (C var): Robust linear regression.
                                                              (line 151)
* gsl_multifit_robust_alloc.gsl_multifit_robust_type.gsl_multifit_robust_welsch (C var): Robust linear regression.
                                                              (line 163)
* gsl_multifit_robust_est (C function):  Robust linear regression.
                                                              (line 239)
* gsl_multifit_robust_free (C function): Robust linear regression.
                                                              (line 174)
* gsl_multifit_robust_maxiter (C function): Robust linear regression.
                                                              (line 195)
* gsl_multifit_robust_name (C function): Robust linear regression.
                                                              (line 180)
* gsl_multifit_robust_residuals (C function): Robust linear regression.
                                                              (line 249)
* gsl_multifit_robust_statistics (C function): Robust linear regression.
                                                              (line 261)
* gsl_multifit_robust_statistics.gsl_multifit_robust_stats (C type): Robust linear regression.
                                                              (line 272)
* gsl_multifit_robust_tune (C function): Robust linear regression.
                                                              (line 186)
* gsl_multifit_robust_weights (C function): Robust linear regression.
                                                              (line 203)
* gsl_multifit_robust_workspace (C type): Robust linear regression.
                                                              (line  77)
* gsl_multifit_wlinear (C function):     Multi-parameter regression.
                                                              (line 120)
* gsl_multifit_wlinear_tsvd (C function): Multi-parameter regression.
                                                              (line 136)
* gsl_multilarge_linear_accumulate (C function): Large Dense Linear Least Squares Routines.
                                                              (line 117)
* gsl_multilarge_linear_alloc (C function): Large Dense Linear Least Squares Routines.
                                                              (line  11)
* gsl_multilarge_linear_alloc.gsl_multilarge_linear_type (C type): Large Dense Linear Least Squares Routines.
                                                              (line  19)
* gsl_multilarge_linear_alloc.gsl_multilarge_linear_type.gsl_multilarge_linear_normal (C var): Large Dense Linear Least Squares Routines.
                                                              (line  25)
* gsl_multilarge_linear_alloc.gsl_multilarge_linear_type.gsl_multilarge_linear_tsqr (C var): Large Dense Linear Least Squares Routines.
                                                              (line  34)
* gsl_multilarge_linear_free (C function): Large Dense Linear Least Squares Routines.
                                                              (line  44)
* gsl_multilarge_linear_genform1 (C function): Large Dense Linear Least Squares Routines.
                                                              (line 138)
* gsl_multilarge_linear_genform2 (C function): Large Dense Linear Least Squares Routines.
                                                              (line 150)
* gsl_multilarge_linear_lcurve (C function): Large Dense Linear Least Squares Routines.
                                                              (line 160)
* gsl_multilarge_linear_L_decomp (C function): Large Dense Linear Least Squares Routines.
                                                              (line  85)
* gsl_multilarge_linear_matrix_ptr (C function): Large Dense Linear Least Squares Routines.
                                                              (line 178)
* gsl_multilarge_linear_name (C function): Large Dense Linear Least Squares Routines.
                                                              (line  50)
* gsl_multilarge_linear_rcond (C function): Large Dense Linear Least Squares Routines.
                                                              (line 194)
* gsl_multilarge_linear_reset (C function): Large Dense Linear Least Squares Routines.
                                                              (line  56)
* gsl_multilarge_linear_rhs_ptr (C function): Large Dense Linear Least Squares Routines.
                                                              (line 186)
* gsl_multilarge_linear_solve (C function): Large Dense Linear Least Squares Routines.
                                                              (line 127)
* gsl_multilarge_linear_stdform1 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  62)
* gsl_multilarge_linear_stdform2 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  95)
* gsl_multilarge_linear_workspace (C type): Large Dense Linear Least Squares Routines.
                                                              (line   6)
* gsl_multilarge_linear_wstdform1 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  62)
* gsl_multilarge_linear_wstdform2 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  95)
* gsl_multilarge_nlinear_alloc (C function): Initializing the Solver<3>.
                                                              (line  18)
* gsl_multilarge_nlinear_avratio (C function): Iteration<5>.  (line 109)
* gsl_multilarge_nlinear_covar (C function): Covariance matrix of best fit parameters.
                                                              (line   6)
* gsl_multilarge_nlinear_default_parameters (C function): Initializing the Solver<3>.
                                                              (line  53)
* gsl_multilarge_nlinear_driver (C function): High Level Driver.
                                                              (line   9)
* gsl_multilarge_nlinear_fdf (C type):   Providing the Function to be Minimized.
                                                              (line  85)
* gsl_multilarge_nlinear_free (C function): Initializing the Solver<3>.
                                                              (line  82)
* gsl_multilarge_nlinear_init (C function): Initializing the Solver<3>.
                                                              (line  63)
* gsl_multilarge_nlinear_iterate (C function): Iteration<5>.  (line  10)
* gsl_multilarge_nlinear_name (C function): Initializing the Solver<3>.
                                                              (line  90)
* gsl_multilarge_nlinear_niter (C function): Iteration<5>.    (line  68)
* gsl_multilarge_nlinear_parameters (C type): Tunable Parameters.
                                                              (line  30)
* gsl_multilarge_nlinear_position (C function): Iteration<5>. (line  44)
* gsl_multilarge_nlinear_rcond (C function): Iteration<5>.    (line  77)
* gsl_multilarge_nlinear_residual (C function): Iteration<5>. (line  52)
* gsl_multilarge_nlinear_scale (C type): Tunable Parameters.  (line  97)
* gsl_multilarge_nlinear_scale.gsl_multifit_nlinear_scale_levenberg (C var): Tunable Parameters.
                                                              (line 123)
* gsl_multilarge_nlinear_scale.gsl_multifit_nlinear_scale_marquardt (C var): Tunable Parameters.
                                                              (line 136)
* gsl_multilarge_nlinear_scale.gsl_multifit_nlinear_scale_more (C var): Tunable Parameters.
                                                              (line 103)
* gsl_multilarge_nlinear_scale.gsl_multilarge_nlinear_scale_levenberg (C var): Tunable Parameters.
                                                              (line 123)
* gsl_multilarge_nlinear_scale.gsl_multilarge_nlinear_scale_marquardt (C var): Tunable Parameters.
                                                              (line 136)
* gsl_multilarge_nlinear_scale.gsl_multilarge_nlinear_scale_more (C var): Tunable Parameters.
                                                              (line 103)
* gsl_multilarge_nlinear_solver (C type): Tunable Parameters. (line 147)
* gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_cholesky (C var): Tunable Parameters.
                                                              (line 167)
* gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_mcholesky (C var): Tunable Parameters.
                                                              (line 186)
* gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_qr (C var): Tunable Parameters.
                                                              (line 158)
* gsl_multilarge_nlinear_solver.gsl_multifit_nlinear_solver_svd (C var): Tunable Parameters.
                                                              (line 203)
* gsl_multilarge_nlinear_solver.gsl_multilarge_nlinear_solver_cholesky (C var): Tunable Parameters.
                                                              (line 167)
* gsl_multilarge_nlinear_solver.gsl_multilarge_nlinear_solver_mcholesky (C var): Tunable Parameters.
                                                              (line 186)
* gsl_multilarge_nlinear_test (C function): Testing for Convergence.
                                                              (line  19)
* gsl_multilarge_nlinear_trs (C type):   Tunable Parameters.  (line  49)
* gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_ddogleg (C var): Tunable Parameters.
                                                              (line  77)
* gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_dogleg (C var): Tunable Parameters.
                                                              (line  70)
* gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_lm (C var): Tunable Parameters.
                                                              (line  55)
* gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_lmaccel (C var): Tunable Parameters.
                                                              (line  62)
* gsl_multilarge_nlinear_trs.gsl_multifit_nlinear_trs_subspace2D (C var): Tunable Parameters.
                                                              (line  84)
* gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_cgst (C var): Tunable Parameters.
                                                              (line  91)
* gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_ddogleg (C var): Tunable Parameters.
                                                              (line  77)
* gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_dogleg (C var): Tunable Parameters.
                                                              (line  70)
* gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_lm (C var): Tunable Parameters.
                                                              (line  55)
* gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_lmaccel (C var): Tunable Parameters.
                                                              (line  62)
* gsl_multilarge_nlinear_trs.gsl_multilarge_nlinear_trs_subspace2D (C var): Tunable Parameters.
                                                              (line  84)
* gsl_multilarge_nlinear_trs_name (C function): Initializing the Solver<3>.
                                                              (line 102)
* gsl_multimin_fdfminimizer (C type):    Initializing the Multidimensional Minimizer.
                                                              (line  10)
* gsl_multimin_fdfminimizer_alloc (C function): Initializing the Multidimensional Minimizer.
                                                              (line  18)
* gsl_multimin_fdfminimizer_dx (C function): Iteration<4>.    (line  27)
* gsl_multimin_fdfminimizer_free (C function): Initializing the Multidimensional Minimizer.
                                                              (line  58)
* gsl_multimin_fdfminimizer_gradient (C function): Iteration<4>.
                                                              (line  27)
* gsl_multimin_fdfminimizer_iterate (C function): Iteration<4>.
                                                              (line  11)
* gsl_multimin_fdfminimizer_minimum (C function): Iteration<4>.
                                                              (line  27)
* gsl_multimin_fdfminimizer_name (C function): Initializing the Multidimensional Minimizer.
                                                              (line  66)
* gsl_multimin_fdfminimizer_restart (C function): Iteration<4>.
                                                              (line  47)
* gsl_multimin_fdfminimizer_set (C function): Initializing the Multidimensional Minimizer.
                                                              (line  31)
* gsl_multimin_fdfminimizer_type (C type): Algorithms with Derivatives.
                                                              (line  11)
* gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_conjugate_fr (C var): Algorithms with Derivatives.
                                                              (line  15)
* gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_conjugate_pr (C var): Algorithms with Derivatives.
                                                              (line  35)
* gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_steepest_descent (C var): Algorithms with Derivatives.
                                                              (line  69)
* gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_vector_bfgs (C var): Algorithms with Derivatives.
                                                              (line  45)
* gsl_multimin_fdfminimizer_type.gsl_multimin_fdfminimizer_vector_bfgs2 (C var): Algorithms with Derivatives.
                                                              (line  45)
* gsl_multimin_fdfminimizer_x (C function): Iteration<4>.     (line  27)
* gsl_multimin_fminimizer (C type):      Initializing the Multidimensional Minimizer.
                                                              (line  14)
* gsl_multimin_fminimizer_alloc (C function): Initializing the Multidimensional Minimizer.
                                                              (line  18)
* gsl_multimin_fminimizer_free (C function): Initializing the Multidimensional Minimizer.
                                                              (line  58)
* gsl_multimin_fminimizer_iterate (C function): Iteration<4>. (line  11)
* gsl_multimin_fminimizer_minimum (C function): Iteration<4>. (line  27)
* gsl_multimin_fminimizer_name (C function): Initializing the Multidimensional Minimizer.
                                                              (line  66)
* gsl_multimin_fminimizer_set (C function): Initializing the Multidimensional Minimizer.
                                                              (line  31)
* gsl_multimin_fminimizer_size (C function): Iteration<4>.    (line  27)
* gsl_multimin_fminimizer_type (C type): Algorithms without Derivatives<2>.
                                                              (line   9)
* gsl_multimin_fminimizer_type.gsl_multimin_fminimizer_nmsimplex (C var): Algorithms without Derivatives<2>.
                                                              (line  14)
* gsl_multimin_fminimizer_type.gsl_multimin_fminimizer_nmsimplex2 (C var): Algorithms without Derivatives<2>.
                                                              (line  14)
* gsl_multimin_fminimizer_type.gsl_multimin_fminimizer_nmsimplex2rand (C var): Algorithms without Derivatives<2>.
                                                              (line  62)
* gsl_multimin_fminimizer_x (C function): Iteration<4>.       (line  27)
* gsl_multimin_function (C type):        Providing a function to minimize.
                                                              (line  53)
* gsl_multimin_function_fdf (C type):    Providing a function to minimize.
                                                              (line  13)
* gsl_multimin_test_gradient (C function): Stopping Criteria. (line  18)
* gsl_multimin_test_size (C function):   Stopping Criteria.   (line  33)
* gsl_multiroot_fdfsolver (C type):      Initializing the Solver<2>.
                                                              (line  16)
* gsl_multiroot_fdfsolver_alloc (C function): Initializing the Solver<2>.
                                                              (line  37)
* gsl_multiroot_fdfsolver_dx (C function): Iteration<3>.      (line  51)
* gsl_multiroot_fdfsolver_f (C function): Iteration<3>.       (line  43)
* gsl_multiroot_fdfsolver_free (C function): Initializing the Solver<2>.
                                                              (line  65)
* gsl_multiroot_fdfsolver_iterate (C function): Iteration<3>. (line  12)
* gsl_multiroot_fdfsolver_name (C function): Initializing the Solver<2>.
                                                              (line  72)
* gsl_multiroot_fdfsolver_root (C function): Iteration<3>.    (line  35)
* gsl_multiroot_fdfsolver_set (C function): Initializing the Solver<2>.
                                                              (line  54)
* gsl_multiroot_fdfsolver_type (C type): Algorithms using Derivatives.
                                                              (line  13)
* gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_gnewton (C var): Algorithms using Derivatives.
                                                              (line 105)
* gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_hybridj (C var): Algorithms using Derivatives.
                                                              (line  76)
* gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_hybridsj (C var): Algorithms using Derivatives.
                                                              (line  18)
* gsl_multiroot_fdfsolver_type.gsl_multiroot_fdfsolver_newton (C var): Algorithms using Derivatives.
                                                              (line  84)
* gsl_multiroot_fsolver (C type):        Initializing the Solver<2>.
                                                              (line  11)
* gsl_multiroot_fsolver_alloc (C function): Initializing the Solver<2>.
                                                              (line  21)
* gsl_multiroot_fsolver_dx (C function): Iteration<3>.        (line  51)
* gsl_multiroot_fsolver_f (C function):  Iteration<3>.        (line  43)
* gsl_multiroot_fsolver_free (C function): Initializing the Solver<2>.
                                                              (line  65)
* gsl_multiroot_fsolver_iterate (C function): Iteration<3>.   (line  12)
* gsl_multiroot_fsolver_name (C function): Initializing the Solver<2>.
                                                              (line  72)
* gsl_multiroot_fsolver_root (C function): Iteration<3>.      (line  35)
* gsl_multiroot_fsolver_set (C function): Initializing the Solver<2>.
                                                              (line  54)
* gsl_multiroot_fsolver_type (C type):   Algorithms without Derivatives.
                                                              (line  13)
* gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_broyden (C var): Algorithms without Derivatives.
                                                              (line  57)
* gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_dnewton (C var): Algorithms without Derivatives.
                                                              (line  34)
* gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_hybrid (C var): Algorithms without Derivatives.
                                                              (line  28)
* gsl_multiroot_fsolver_type.gsl_multiroot_fsolver_hybrids (C var): Algorithms without Derivatives.
                                                              (line  18)
* gsl_multiroot_function (C type):       Providing the function to solve<2>.
                                                              (line  10)
* gsl_multiroot_function_fdf (C type):   Providing the function to solve<2>.
                                                              (line  62)
* gsl_multiroot_test_delta (C function): Search Stopping Parameters<2>.
                                                              (line  20)
* gsl_multiroot_test_residual (C function): Search Stopping Parameters<2>.
                                                              (line  34)
* gsl_multiset (C type):                 The Multiset struct. (line   6)
* gsl_multiset_alloc (C function):       Multiset allocation. (line   6)
* gsl_multiset_calloc (C function):      Multiset allocation. (line  17)
* gsl_multiset_data (C function):        Multiset properties. (line  15)
* gsl_multiset_fprintf (C function):     Reading and writing multisets.
                                                              (line  29)
* gsl_multiset_fread (C function):       Reading and writing multisets.
                                                              (line  18)
* gsl_multiset_free (C function):        Multiset allocation. (line  37)
* gsl_multiset_fscanf (C function):      Reading and writing multisets.
                                                              (line  39)
* gsl_multiset_fwrite (C function):      Reading and writing multisets.
                                                              (line   9)
* gsl_multiset_get (C function):         Accessing multiset elements.
                                                              (line   8)
* gsl_multiset_init_first (C function):  Multiset allocation. (line  26)
* gsl_multiset_init_last (C function):   Multiset allocation. (line  31)
* gsl_multiset_k (C function):           Multiset properties. (line  10)
* gsl_multiset_memcpy (C function):      Multiset allocation. (line  42)
* gsl_multiset_n (C function):           Multiset properties. (line   6)
* gsl_multiset_next (C function):        Multiset functions.  (line   6)
* gsl_multiset_prev (C function):        Multiset functions.  (line  15)
* gsl_multiset_valid (C function):       Multiset properties. (line  20)
* GSL_NAN (C macro):                     Infinities and Not-a-number.
                                                              (line  16)
* GSL_NEGINF (C macro):                  Infinities and Not-a-number.
                                                              (line  11)
* gsl_ntuple (C type):                   The ntuple struct.   (line   6)
* gsl_ntuple_bookdata (C function):      Writing ntuples.     (line  11)
* gsl_ntuple_close (C function):         Closing an ntuple file.
                                                              (line   6)
* gsl_ntuple_create (C function):        Creating ntuples.    (line   6)
* gsl_ntuple_open (C function):          Opening an existing ntuple file.
                                                              (line   6)
* gsl_ntuple_project (C function):       Histogramming ntuple values.
                                                              (line  41)
* gsl_ntuple_read (C function):          Reading ntuples.     (line   6)
* gsl_ntuple_select_fn (C type):         Histogramming ntuple values.
                                                              (line  13)
* gsl_ntuple_value_fn (C type):          Histogramming ntuple values.
                                                              (line  27)
* gsl_ntuple_write (C function):         Writing ntuples.     (line   6)
* gsl_odeiv2_control (C type):           Adaptive Step-size Control.
                                                              (line  10)
* gsl_odeiv2_control_alloc (C function): Adaptive Step-size Control.
                                                              (line  88)
* gsl_odeiv2_control_errlevel (C function): Adaptive Step-size Control.
                                                              (line 137)
* gsl_odeiv2_control_free (C function):  Adaptive Step-size Control.
                                                              (line 105)
* gsl_odeiv2_control_hadjust (C function): Adaptive Step-size Control.
                                                              (line 110)
* gsl_odeiv2_control_init (C function):  Adaptive Step-size Control.
                                                              (line  97)
* gsl_odeiv2_control_name (C function):  Adaptive Step-size Control.
                                                              (line 127)
* gsl_odeiv2_control_scaled_new (C function): Adaptive Step-size Control.
                                                              (line  72)
* gsl_odeiv2_control_set_driver (C function): Adaptive Step-size Control.
                                                              (line 146)
* gsl_odeiv2_control_standard_new (C function): Adaptive Step-size Control.
                                                              (line  18)
* gsl_odeiv2_control_type (C type):      Adaptive Step-size Control.
                                                              (line  14)
* gsl_odeiv2_control_yp_new (C function): Adaptive Step-size Control.
                                                              (line  63)
* gsl_odeiv2_control_y_new (C function): Adaptive Step-size Control.
                                                              (line  54)
* gsl_odeiv2_driver_alloc_scaled_new (C function): Driver.    (line   9)
* gsl_odeiv2_driver_alloc_standard_new (C function): Driver.  (line   9)
* gsl_odeiv2_driver_alloc_yp_new (C function): Driver.        (line   9)
* gsl_odeiv2_driver_alloc_y_new (C function): Driver.         (line   9)
* gsl_odeiv2_driver_apply (C function):  Driver.              (line  52)
* gsl_odeiv2_driver_apply_fixed_step (C function): Driver.    (line  71)
* gsl_odeiv2_driver_free (C function):   Driver.              (line  93)
* gsl_odeiv2_driver_reset (C function):  Driver.              (line  82)
* gsl_odeiv2_driver_reset_hstart (C function): Driver.        (line  86)
* gsl_odeiv2_driver_set_hmax (C function): Driver.            (line  39)
* gsl_odeiv2_driver_set_hmin (C function): Driver.            (line  33)
* gsl_odeiv2_driver_set_nmax (C function): Driver.            (line  45)
* gsl_odeiv2_evolve (C type):            Evolution.           (line  10)
* gsl_odeiv2_evolve_alloc (C function):  Evolution.           (line  15)
* gsl_odeiv2_evolve_apply (C function):  Evolution.           (line  21)
* gsl_odeiv2_evolve_apply_fixed_step (C function): Evolution. (line  63)
* gsl_odeiv2_evolve_free (C function):   Evolution.           (line  81)
* gsl_odeiv2_evolve_reset (C function):  Evolution.           (line  75)
* gsl_odeiv2_evolve_set_driver (C function): Evolution.       (line  86)
* gsl_odeiv2_step (C type):              Stepping Functions.  (line  10)
* gsl_odeiv2_step_alloc (C function):    Stepping Functions.  (line  14)
* gsl_odeiv2_step_apply (C function):    Stepping Functions.  (line  61)
* gsl_odeiv2_step_free (C function):     Stepping Functions.  (line  29)
* gsl_odeiv2_step_name (C function):     Stepping Functions.  (line  34)
* gsl_odeiv2_step_order (C function):    Stepping Functions.  (line  44)
* gsl_odeiv2_step_reset (C function):    Stepping Functions.  (line  23)
* gsl_odeiv2_step_set_driver (C function): Stepping Functions.
                                                              (line  51)
* gsl_odeiv2_step_type (C type):         Stepping Functions.  (line 102)
* gsl_odeiv2_step_type.gsl_odeiv2_step_bsimp (C var): Stepping Functions.
                                                              (line 157)
* gsl_odeiv2_step_type.gsl_odeiv2_step_msadams (C var): Stepping Functions.
                                                              (line 164)
* gsl_odeiv2_step_type.gsl_odeiv2_step_msbdf (C var): Stepping Functions.
                                                              (line 175)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rk1imp (C var): Stepping Functions.
                                                              (line 131)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rk2 (C var): Stepping Functions.
                                                              (line 104)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rk2imp (C var): Stepping Functions.
                                                              (line 140)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rk4 (C var): Stepping Functions.
                                                              (line 108)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rk4imp (C var): Stepping Functions.
                                                              (line 149)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rk8pd (C var): Stepping Functions.
                                                              (line 126)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rkck (C var): Stepping Functions.
                                                              (line 121)
* gsl_odeiv2_step_type.gsl_odeiv2_step_rkf45 (C var): Stepping Functions.
                                                              (line 115)
* gsl_odeiv2_system (C type):            Defining the ODE System.
                                                              (line  18)
* gsl_permutation (C type):              The Permutation struct.
                                                              (line   6)
* gsl_permutation_alloc (C function):    Permutation allocation.
                                                              (line   6)
* gsl_permutation_calloc (C function):   Permutation allocation.
                                                              (line  16)
* gsl_permutation_canonical_cycles (C function): Permutations in cyclic form.
                                                              (line  73)
* gsl_permutation_canonical_to_linear (C function): Permutations in cyclic form.
                                                              (line  51)
* gsl_permutation_data (C function):     Permutation properties.
                                                              (line  10)
* gsl_permutation_fprintf (C function):  Reading and writing permutations.
                                                              (line  29)
* gsl_permutation_fread (C function):    Reading and writing permutations.
                                                              (line  18)
* gsl_permutation_free (C function):     Permutation allocation.
                                                              (line  29)
* gsl_permutation_fscanf (C function):   Reading and writing permutations.
                                                              (line  39)
* gsl_permutation_fwrite (C function):   Reading and writing permutations.
                                                              (line   9)
* gsl_permutation_get (C function):      Accessing permutation elements.
                                                              (line   9)
* gsl_permutation_init (C function):     Permutation allocation.
                                                              (line  24)
* gsl_permutation_inverse (C function):  Permutation functions.
                                                              (line  11)
* gsl_permutation_inversions (C function): Permutations in cyclic form.
                                                              (line  58)
* gsl_permutation_linear_cycles (C function): Permutations in cyclic form.
                                                              (line  67)
* gsl_permutation_linear_to_canonical (C function): Permutations in cyclic form.
                                                              (line  45)
* gsl_permutation_memcpy (C function):   Permutation allocation.
                                                              (line  34)
* gsl_permutation_mul (C function):      Applying Permutations.
                                                              (line  56)
* gsl_permutation_next (C function):     Permutation functions.
                                                              (line  17)
* gsl_permutation_prev (C function):     Permutation functions.
                                                              (line  26)
* gsl_permutation_reverse (C function):  Permutation functions.
                                                              (line   6)
* gsl_permutation_size (C function):     Permutation properties.
                                                              (line   6)
* gsl_permutation_swap (C function):     Accessing permutation elements.
                                                              (line  18)
* gsl_permutation_valid (C function):    Permutation properties.
                                                              (line  15)
* gsl_permute (C function):              Applying Permutations.
                                                              (line   9)
* gsl_permute_inverse (C function):      Applying Permutations.
                                                              (line  16)
* gsl_permute_matrix (C function):       Applying Permutations.
                                                              (line  45)
* gsl_permute_vector (C function):       Applying Permutations.
                                                              (line  23)
* gsl_permute_vector_inverse (C function): Applying Permutations.
                                                              (line  33)
* gsl_poly_complex_eval (C function):    Polynomial Evaluation.
                                                              (line  19)
* gsl_poly_complex_solve (C function):   General Polynomial Equations.
                                                              (line  34)
* gsl_poly_complex_solve_cubic (C function): Cubic Equations. (line  25)
* gsl_poly_complex_solve_quadratic (C function): Quadratic Equations.
                                                              (line  31)
* gsl_poly_complex_workspace (C type):   General Polynomial Equations.
                                                              (line  11)
* gsl_poly_complex_workspace_alloc (C function): General Polynomial Equations.
                                                              (line  16)
* gsl_poly_complex_workspace_free (C function): General Polynomial Equations.
                                                              (line  28)
* gsl_poly_dd_eval (C function):         Divided Difference Representation of Polynomials.
                                                              (line  42)
* gsl_poly_dd_hermite_init (C function): Divided Difference Representation of Polynomials.
                                                              (line  62)
* gsl_poly_dd_init (C function):         Divided Difference Representation of Polynomials.
                                                              (line  32)
* gsl_poly_dd_taylor (C function):       Divided Difference Representation of Polynomials.
                                                              (line  50)
* gsl_poly_eval (C function):            Polynomial Evaluation.
                                                              (line  13)
* gsl_poly_eval_derivs (C function):     Polynomial Evaluation.
                                                              (line  31)
* gsl_poly_solve_cubic (C function):     Cubic Equations.     (line   6)
* gsl_poly_solve_quadratic (C function): Quadratic Equations. (line   6)
* GSL_POSINF (C macro):                  Infinities and Not-a-number.
                                                              (line   6)
* gsl_pow_2 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_3 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_4 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_5 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_6 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_7 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_8 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_9 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_int (C function):              Small integer powers.
                                                              (line  11)
* gsl_pow_uint (C function):             Small integer powers.
                                                              (line  11)
* gsl_qrng (C type):                     Quasi-random number generator initialization.
                                                              (line   6)
* gsl_qrng_alloc (C function):           Quasi-random number generator initialization.
                                                              (line  10)
* gsl_qrng_clone (C function):           Saving and restoring quasi-random number generator state.
                                                              (line  13)
* gsl_qrng_free (C function):            Quasi-random number generator initialization.
                                                              (line  19)
* gsl_qrng_get (C function):             Sampling from a quasi-random number generator.
                                                              (line   6)
* gsl_qrng_init (C function):            Quasi-random number generator initialization.
                                                              (line  24)
* gsl_qrng_memcpy (C function):          Saving and restoring quasi-random number generator state.
                                                              (line   6)
* gsl_qrng_name (C function):            Auxiliary quasi-random number generator functions.
                                                              (line   6)
* gsl_qrng_size (C function):            Auxiliary quasi-random number generator functions.
                                                              (line  10)
* gsl_qrng_state (C function):           Auxiliary quasi-random number generator functions.
                                                              (line  10)
* gsl_qrng_type (C type):                Quasi-random number generator algorithms.
                                                              (line   8)
* gsl_qrng_type.gsl_qrng_halton (C var): Quasi-random number generator algorithms.
                                                              (line  22)
* gsl_qrng_type.gsl_qrng_niederreiter_2 (C var): Quasi-random number generator algorithms.
                                                              (line  10)
* gsl_qrng_type.gsl_qrng_reversehalton (C var): Quasi-random number generator algorithms.
                                                              (line  22)
* gsl_qrng_type.gsl_qrng_sobol (C var):  Quasi-random number generator algorithms.
                                                              (line  16)
* GSL_RANGE_CHECK_OFF (C macro):         Accessing vector elements.
                                                              (line  17)
* gsl_ran_bernoulli (C function):        The Bernoulli Distribution.
                                                              (line   6)
* gsl_ran_bernoulli_pdf (C function):    The Bernoulli Distribution.
                                                              (line  16)
* gsl_ran_beta (C function):             The Beta Distribution.
                                                              (line   6)
* gsl_ran_beta_pdf (C function):         The Beta Distribution.
                                                              (line  15)
* gsl_ran_binomial (C function):         The Binomial Distribution.
                                                              (line   6)
* gsl_ran_binomial_pdf (C function):     The Binomial Distribution.
                                                              (line  18)
* gsl_ran_bivariate_gaussian (C function): The Bivariate Gaussian Distribution.
                                                              (line   6)
* gsl_ran_bivariate_gaussian_pdf (C function): The Bivariate Gaussian Distribution.
                                                              (line  20)
* gsl_ran_cauchy (C function):           The Cauchy Distribution.
                                                              (line   6)
* gsl_ran_cauchy_pdf (C function):       The Cauchy Distribution.
                                                              (line  17)
* gsl_ran_chisq (C function):            The Chi-squared Distribution.
                                                              (line  14)
* gsl_ran_chisq_pdf (C function):        The Chi-squared Distribution.
                                                              (line  24)
* gsl_ran_choose (C function):           Shuffling and Sampling.
                                                              (line  34)
* gsl_ran_dirichlet (C function):        The Dirichlet Distribution.
                                                              (line   6)
* gsl_ran_dirichlet_lnpdf (C function):  The Dirichlet Distribution.
                                                              (line  33)
* gsl_ran_dirichlet_pdf (C function):    The Dirichlet Distribution.
                                                              (line  26)
* gsl_ran_dir_2d (C function):           Spherical Vector Distributions.
                                                              (line  10)
* gsl_ran_dir_2d_trig_method (C function): Spherical Vector Distributions.
                                                              (line  10)
* gsl_ran_dir_3d (C function):           Spherical Vector Distributions.
                                                              (line  33)
* gsl_ran_dir_nd (C function):           Spherical Vector Distributions.
                                                              (line  44)
* gsl_ran_discrete (C function):         General Discrete Distributions.
                                                              (line  66)
* gsl_ran_discrete_free (C function):    General Discrete Distributions.
                                                              (line  81)
* gsl_ran_discrete_pdf (C function):     General Discrete Distributions.
                                                              (line  72)
* gsl_ran_discrete_preproc (C function): General Discrete Distributions.
                                                              (line  54)
* gsl_ran_discrete_t (C type):           General Discrete Distributions.
                                                              (line  49)
* gsl_ran_exponential (C function):      The Exponential Distribution.
                                                              (line   6)
* gsl_ran_exponential_pdf (C function):  The Exponential Distribution.
                                                              (line  15)
* gsl_ran_exppow (C function):           The Exponential Power Distribution.
                                                              (line   6)
* gsl_ran_exppow_pdf (C function):       The Exponential Power Distribution.
                                                              (line  19)
* gsl_ran_fdist (C function):            The F-distribution.  (line  13)
* gsl_ran_fdist_pdf (C function):        The F-distribution.  (line  28)
* gsl_ran_flat (C function):             The Flat Uniform Distribution.
                                                              (line   6)
* gsl_ran_flat_pdf (C function):         The Flat Uniform Distribution.
                                                              (line  16)
* gsl_ran_gamma (C function):            The Gamma Distribution.
                                                              (line   6)
* gsl_ran_gamma_knuth (C function):      The Gamma Distribution.
                                                              (line  23)
* gsl_ran_gamma_pdf (C function):        The Gamma Distribution.
                                                              (line  29)
* gsl_ran_gaussian (C function):         The Gaussian Distribution.
                                                              (line   6)
* gsl_ran_gaussian_pdf (C function):     The Gaussian Distribution.
                                                              (line  20)
* gsl_ran_gaussian_ratio_method (C function): The Gaussian Distribution.
                                                              (line  27)
* gsl_ran_gaussian_tail (C function):    The Gaussian Tail Distribution.
                                                              (line   6)
* gsl_ran_gaussian_tail_pdf (C function): The Gaussian Tail Distribution.
                                                              (line  25)
* gsl_ran_gaussian_ziggurat (C function): The Gaussian Distribution.
                                                              (line  27)
* gsl_ran_geometric (C function):        The Geometric Distribution.
                                                              (line   6)
* gsl_ran_geometric_pdf (C function):    The Geometric Distribution.
                                                              (line  20)
* gsl_ran_gumbel1 (C function):          The Type-1 Gumbel Distribution.
                                                              (line   6)
* gsl_ran_gumbel1_pdf (C function):      The Type-1 Gumbel Distribution.
                                                              (line  16)
* gsl_ran_gumbel2 (C function):          The Type-2 Gumbel Distribution.
                                                              (line   6)
* gsl_ran_gumbel2_pdf (C function):      The Type-2 Gumbel Distribution.
                                                              (line  16)
* gsl_ran_hypergeometric (C function):   The Hypergeometric Distribution.
                                                              (line   6)
* gsl_ran_hypergeometric_pdf (C function): The Hypergeometric Distribution.
                                                              (line  23)
* gsl_ran_landau (C function):           The Landau Distribution.
                                                              (line   6)
* gsl_ran_landau_pdf (C function):       The Landau Distribution.
                                                              (line  19)
* gsl_ran_laplace (C function):          The Laplace Distribution.
                                                              (line   6)
* gsl_ran_laplace_pdf (C function):      The Laplace Distribution.
                                                              (line  15)
* gsl_ran_levy (C function):             The Levy alpha-Stable Distributions.
                                                              (line   6)
* gsl_ran_levy_skew (C function):        The Levy skew alpha-Stable Distribution.
                                                              (line   6)
* gsl_ran_logarithmic (C function):      The Logarithmic Distribution.
                                                              (line   6)
* gsl_ran_logarithmic_pdf (C function):  The Logarithmic Distribution.
                                                              (line  17)
* gsl_ran_logistic (C function):         The Logistic Distribution.
                                                              (line   6)
* gsl_ran_logistic_pdf (C function):     The Logistic Distribution.
                                                              (line  15)
* gsl_ran_lognormal (C function):        The Lognormal Distribution.
                                                              (line   6)
* gsl_ran_lognormal_pdf (C function):    The Lognormal Distribution.
                                                              (line  16)
* gsl_ran_multinomial (C function):      The Multinomial Distribution.
                                                              (line   6)
* gsl_ran_multinomial_lnpdf (C function): The Multinomial Distribution.
                                                              (line  36)
* gsl_ran_multinomial_pdf (C function):  The Multinomial Distribution.
                                                              (line  29)
* gsl_ran_multivariate_gaussian (C function): The Multivariate Gaussian Distribution.
                                                              (line   6)
* gsl_ran_multivariate_gaussian_log_pdf (C function): The Multivariate Gaussian Distribution.
                                                              (line  21)
* gsl_ran_multivariate_gaussian_mean (C function): The Multivariate Gaussian Distribution.
                                                              (line  34)
* gsl_ran_multivariate_gaussian_pdf (C function): The Multivariate Gaussian Distribution.
                                                              (line  21)
* gsl_ran_multivariate_gaussian_vcov (C function): The Multivariate Gaussian Distribution.
                                                              (line  47)
* gsl_ran_negative_binomial (C function): The Negative Binomial Distribution.
                                                              (line   6)
* gsl_ran_negative_binomial_pdf (C function): The Negative Binomial Distribution.
                                                              (line  19)
* gsl_ran_pareto (C function):           The Pareto Distribution.
                                                              (line   6)
* gsl_ran_pareto_pdf (C function):       The Pareto Distribution.
                                                              (line  16)
* gsl_ran_pascal (C function):           The Pascal Distribution.
                                                              (line   6)
* gsl_ran_pascal_pdf (C function):       The Pascal Distribution.
                                                              (line  17)
* gsl_ran_poisson (C function):          The Poisson Distribution.
                                                              (line   6)
* gsl_ran_poisson_pdf (C function):      The Poisson Distribution.
                                                              (line  16)
* gsl_ran_rayleigh (C function):         The Rayleigh Distribution.
                                                              (line   6)
* gsl_ran_rayleigh_pdf (C function):     The Rayleigh Distribution.
                                                              (line  16)
* gsl_ran_rayleigh_tail (C function):    The Rayleigh Tail Distribution.
                                                              (line   6)
* gsl_ran_rayleigh_tail_pdf (C function): The Rayleigh Tail Distribution.
                                                              (line  17)
* gsl_ran_sample (C function):           Shuffling and Sampling.
                                                              (line  63)
* gsl_ran_shuffle (C function):          Shuffling and Sampling.
                                                              (line  13)
* gsl_ran_tdist (C function):            The t-distribution.  (line  14)
* gsl_ran_tdist_pdf (C function):        The t-distribution.  (line  23)
* gsl_ran_ugaussian (C function):        The Gaussian Distribution.
                                                              (line  37)
* gsl_ran_ugaussian_pdf (C function):    The Gaussian Distribution.
                                                              (line  37)
* gsl_ran_ugaussian_ratio_method (C function): The Gaussian Distribution.
                                                              (line  37)
* gsl_ran_ugaussian_tail (C function):   The Gaussian Tail Distribution.
                                                              (line  34)
* gsl_ran_ugaussian_tail_pdf (C function): The Gaussian Tail Distribution.
                                                              (line  34)
* gsl_ran_weibull (C function):          The Weibull Distribution.
                                                              (line   6)
* gsl_ran_weibull_pdf (C function):      The Weibull Distribution.
                                                              (line  16)
* gsl_ran_wishart (C function):          The Wishart Distribution.
                                                              (line   6)
* gsl_ran_wishart_log_pdf (C function):  The Wishart Distribution.
                                                              (line  21)
* gsl_ran_wishart_pdf (C function):      The Wishart Distribution.
                                                              (line  21)
* GSL_REAL (C macro):                    Complex number macros.
                                                              (line   9)
* gsl_rng (C type):                      The Random Number Generator Interface.
                                                              (line  18)
* gsl_rng_alloc (C function):            Random number generator initialization.
                                                              (line   6)
* gsl_rng_borosh13 (C var):              Other random number generators.
                                                              (line 193)
* gsl_rng_clone (C function):            Copying random number generator state.
                                                              (line  18)
* gsl_rng_cmrg (C var):                  Random number generator algorithms.
                                                              (line 103)
* gsl_rng_coveyou (C var):               Other random number generators.
                                                              (line 222)
* gsl_rng_default (C var):               Random number environment variables.
                                                              (line  23)
* gsl_rng_default_seed (C var):          Random number environment variables.
                                                              (line  31)
* gsl_rng_env_setup (C function):        Random number environment variables.
                                                              (line  40)
* gsl_rng_fishman18 (C var):             Other random number generators.
                                                              (line 193)
* gsl_rng_fishman20 (C var):             Other random number generators.
                                                              (line 193)
* gsl_rng_fishman2x (C var):             Other random number generators.
                                                              (line 211)
* gsl_rng_fread (C function):            Reading and writing random number generator state.
                                                              (line  18)
* gsl_rng_free (C function):             Random number generator initialization.
                                                              (line  48)
* gsl_rng_fwrite (C function):           Reading and writing random number generator state.
                                                              (line   9)
* gsl_rng_get (C function):              Sampling from a random number generator.
                                                              (line  12)
* gsl_rng_gfsr4 (C var):                 Random number generator algorithms.
                                                              (line 181)
* gsl_rng_knuthran (C var):              Other random number generators.
                                                              (line 183)
* gsl_rng_knuthran2 (C var):             Other random number generators.
                                                              (line 173)
* gsl_rng_knuthran2002 (C var):          Other random number generators.
                                                              (line 183)
* gsl_rng_lecuyer21 (C var):             Other random number generators.
                                                              (line 193)
* gsl_rng_max (C function):              Auxiliary random number generator functions.
                                                              (line  21)
* gsl_rng_memcpy (C function):           Copying random number generator state.
                                                              (line  11)
* gsl_rng_min (C function):              Auxiliary random number generator functions.
                                                              (line  26)
* gsl_rng_minstd (C var):                Other random number generators.
                                                              (line 123)
* gsl_rng_mrg (C var):                   Random number generator algorithms.
                                                              (line 126)
* gsl_rng_mt19937 (C var):               Random number generator algorithms.
                                                              (line  19)
* gsl_rng_name (C function):             Auxiliary random number generator functions.
                                                              (line  10)
* gsl_rng_r250 (C var):                  Other random number generators.
                                                              (line  62)
* gsl_rng_rand (C var):                  Unix random number generators.
                                                              (line  17)
* gsl_rng_rand48 (C var):                Unix random number generators.
                                                              (line  59)
* gsl_rng_random_bsd (C var):            Unix random number generators.
                                                              (line  27)
* gsl_rng_random_glibc2 (C var):         Unix random number generators.
                                                              (line  27)
* gsl_rng_random_libc5 (C var):          Unix random number generators.
                                                              (line  27)
* gsl_rng_randu (C var):                 Other random number generators.
                                                              (line 113)
* gsl_rng_ranf (C var):                  Other random number generators.
                                                              (line  22)
* gsl_rng_ranlux (C var):                Random number generator algorithms.
                                                              (line  74)
* gsl_rng_ranlux389 (C var):             Random number generator algorithms.
                                                              (line  74)
* gsl_rng_ranlxd1 (C var):               Random number generator algorithms.
                                                              (line  67)
* gsl_rng_ranlxd2 (C var):               Random number generator algorithms.
                                                              (line  67)
* gsl_rng_ranlxs0 (C var):               Random number generator algorithms.
                                                              (line  47)
* gsl_rng_ranlxs1 (C var):               Random number generator algorithms.
                                                              (line  47)
* gsl_rng_ranlxs2 (C var):               Random number generator algorithms.
                                                              (line  47)
* gsl_rng_ranmar (C var):                Other random number generators.
                                                              (line  55)
* GSL_RNG_SEED (C macro):                Random number environment variables.
                                                              (line  18)
* gsl_rng_set (C function):              Random number generator initialization.
                                                              (line  26)
* gsl_rng_size (C function):             Auxiliary random number generator functions.
                                                              (line  33)
* gsl_rng_slatec (C var):                Other random number generators.
                                                              (line 153)
* gsl_rng_state (C function):            Auxiliary random number generator functions.
                                                              (line  33)
* gsl_rng_taus (C var):                  Random number generator algorithms.
                                                              (line 144)
* gsl_rng_taus2 (C var):                 Random number generator algorithms.
                                                              (line 144)
* gsl_rng_transputer (C var):            Other random number generators.
                                                              (line 103)
* gsl_rng_tt800 (C var):                 Other random number generators.
                                                              (line  79)
* GSL_RNG_TYPE (C macro):                Random number environment variables.
                                                              (line  12)
* gsl_rng_type (C type):                 The Random Number Generator Interface.
                                                              (line  18)
* gsl_rng_types_setup (C function):      Auxiliary random number generator functions.
                                                              (line  45)
* gsl_rng_uni (C var):                   Other random number generators.
                                                              (line 145)
* gsl_rng_uni32 (C var):                 Other random number generators.
                                                              (line 145)
* gsl_rng_uniform (C function):          Sampling from a random number generator.
                                                              (line  20)
* gsl_rng_uniform_int (C function):      Sampling from a random number generator.
                                                              (line  40)
* gsl_rng_uniform_pos (C function):      Sampling from a random number generator.
                                                              (line  31)
* gsl_rng_vax (C var):                   Other random number generators.
                                                              (line  93)
* gsl_rng_waterman14 (C var):            Other random number generators.
                                                              (line 193)
* gsl_rng_zuf (C var):                   Other random number generators.
                                                              (line 158)
* gsl_root_fdfsolver (C type):           Initializing the Solver.
                                                              (line  11)
* gsl_root_fdfsolver_alloc (C function): Initializing the Solver.
                                                              (line  30)
* gsl_root_fdfsolver_free (C function):  Initializing the Solver.
                                                              (line  58)
* gsl_root_fdfsolver_iterate (C function): Iteration.         (line  12)
* gsl_root_fdfsolver_name (C function):  Initializing the Solver.
                                                              (line  64)
* gsl_root_fdfsolver_root (C function):  Iteration.           (line  35)
* gsl_root_fdfsolver_set (C function):   Initializing the Solver.
                                                              (line  51)
* gsl_root_fdfsolver_type (C type):      Root Finding Algorithms using Derivatives.
                                                              (line  15)
* gsl_root_fdfsolver_type.gsl_root_fdfsolver_newton (C var): Root Finding Algorithms using Derivatives.
                                                              (line  17)
* gsl_root_fdfsolver_type.gsl_root_fdfsolver_secant (C var): Root Finding Algorithms using Derivatives.
                                                              (line  32)
* gsl_root_fdfsolver_type.gsl_root_fdfsolver_steffenson (C var): Root Finding Algorithms using Derivatives.
                                                              (line  64)
* gsl_root_fsolver (C type):             Initializing the Solver.
                                                              (line   6)
* gsl_root_fsolver_alloc (C function):   Initializing the Solver.
                                                              (line  16)
* gsl_root_fsolver_free (C function):    Initializing the Solver.
                                                              (line  58)
* gsl_root_fsolver_iterate (C function): Iteration.           (line  12)
* gsl_root_fsolver_name (C function):    Initializing the Solver.
                                                              (line  64)
* gsl_root_fsolver_root (C function):    Iteration.           (line  35)
* gsl_root_fsolver_set (C function):     Initializing the Solver.
                                                              (line  44)
* gsl_root_fsolver_type (C type):        Root Bracketing Algorithms.
                                                              (line  16)
* gsl_root_fsolver_type.gsl_root_fsolver_bisection (C var): Root Bracketing Algorithms.
                                                              (line  18)
* gsl_root_fsolver_type.gsl_root_fsolver_brent (C var): Root Bracketing Algorithms.
                                                              (line  54)
* gsl_root_fsolver_type.gsl_root_fsolver_falsepos (C var): Root Bracketing Algorithms.
                                                              (line  35)
* gsl_root_fsolver_x_lower (C function): Iteration.           (line  42)
* gsl_root_fsolver_x_upper (C function): Iteration.           (line  42)
* gsl_root_test_delta (C function):      Search Stopping Parameters.
                                                              (line  43)
* gsl_root_test_interval (C function):   Search Stopping Parameters.
                                                              (line  19)
* gsl_root_test_residual (C function):   Search Stopping Parameters.
                                                              (line  55)
* gsl_rstat_add (C function):            Adding Data to the Accumulator.
                                                              (line   6)
* gsl_rstat_alloc (C function):          Initializing the Accumulator.
                                                              (line  12)
* gsl_rstat_free (C function):           Initializing the Accumulator.
                                                              (line  17)
* gsl_rstat_kurtosis (C function):       Current Statistics.  (line  55)
* gsl_rstat_max (C function):            Current Statistics.  (line  10)
* gsl_rstat_mean (C function):           Current Statistics.  (line  14)
* gsl_rstat_median (C function):         Current Statistics.  (line  62)
* gsl_rstat_min (C function):            Current Statistics.  (line   6)
* gsl_rstat_n (C function):              Adding Data to the Accumulator.
                                                              (line  12)
* gsl_rstat_quantile_add (C function):   Quantiles.           (line  38)
* gsl_rstat_quantile_alloc (C function): Quantiles.           (line  18)
* gsl_rstat_quantile_free (C function):  Quantiles.           (line  26)
* gsl_rstat_quantile_get (C function):   Quantiles.           (line  44)
* gsl_rstat_quantile_reset (C function): Quantiles.           (line  32)
* gsl_rstat_quantile_workspace (C type): Quantiles.           (line  13)
* gsl_rstat_reset (C function):          Initializing the Accumulator.
                                                              (line  22)
* gsl_rstat_rms (C function):            Current Statistics.  (line  41)
* gsl_rstat_sd (C function):             Current Statistics.  (line  28)
* gsl_rstat_sd_mean (C function):        Current Statistics.  (line  34)
* gsl_rstat_skew (C function):           Current Statistics.  (line  48)
* gsl_rstat_variance (C function):       Current Statistics.  (line  21)
* gsl_rstat_workspace (C type):          Initializing the Accumulator.
                                                              (line   6)
* GSL_SET_COMPLEX (C macro):             Complex number macros.
                                                              (line  27)
* gsl_set_error_handler (C function):    Error Handlers.      (line  43)
* gsl_set_error_handler_off (C function): Error Handlers.     (line  69)
* gsl_sf_airy_Ai (C function):           Airy Functions.      (line   6)
* gsl_sf_airy_Ai_deriv (C function):     Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_Ai_deriv_e (C function):   Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_Ai_deriv_scaled (C function): Derivatives of Airy Functions.
                                                              (line  20)
* gsl_sf_airy_Ai_deriv_scaled_e (C function): Derivatives of Airy Functions.
                                                              (line  20)
* gsl_sf_airy_Ai_e (C function):         Airy Functions.      (line   6)
* gsl_sf_airy_Ai_scaled (C function):    Airy Functions.      (line  20)
* gsl_sf_airy_Ai_scaled_e (C function):  Airy Functions.      (line  20)
* gsl_sf_airy_Bi (C function):           Airy Functions.      (line  13)
* gsl_sf_airy_Bi_deriv (C function):     Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_Bi_deriv_e (C function):   Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_Bi_deriv_scaled (C function): Derivatives of Airy Functions.
                                                              (line  29)
* gsl_sf_airy_Bi_deriv_scaled_e (C function): Derivatives of Airy Functions.
                                                              (line  29)
* gsl_sf_airy_Bi_e (C function):         Airy Functions.      (line  13)
* gsl_sf_airy_Bi_scaled (C function):    Airy Functions.      (line  28)
* gsl_sf_airy_Bi_scaled_e (C function):  Airy Functions.      (line  28)
* gsl_sf_airy_zero_Ai (C function):      Zeros of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Ai_deriv (C function): Zeros of Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Ai_deriv_e (C function): Zeros of Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Ai_e (C function):    Zeros of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Bi (C function):      Zeros of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_zero_Bi_deriv (C function): Zeros of Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_zero_Bi_deriv_e (C function): Zeros of Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_zero_Bi_e (C function):    Zeros of Airy Functions.
                                                              (line  13)
* gsl_sf_angle_restrict_pos (C function): Restriction Functions.
                                                              (line  16)
* gsl_sf_angle_restrict_pos_e (C function): Restriction Functions.
                                                              (line  16)
* gsl_sf_angle_restrict_symm (C function): Restriction Functions.
                                                              (line   6)
* gsl_sf_angle_restrict_symm_e (C function): Restriction Functions.
                                                              (line   6)
* gsl_sf_atanint (C function):           Arctangent Integral. (line   6)
* gsl_sf_atanint_e (C function):         Arctangent Integral. (line   6)
* gsl_sf_bessel_I0 (C function):         Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_I0_e (C function):       Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_I0_scaled (C function):  Regular Modified Cylindrical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_i0_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_I0_scaled_e (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_i0_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_I1 (C function):         Regular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_I1_e (C function):       Regular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_I1_scaled (C function):  Regular Modified Cylindrical Bessel Functions.
                                                              (line  43)
* gsl_sf_bessel_i1_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_I1_scaled_e (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  43)
* gsl_sf_bessel_i1_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_i2_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_i2_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_il_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_il_scaled_array (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  38)
* gsl_sf_bessel_il_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_In (C function):         Regular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Inu (C function):        Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Inu_e (C function):      Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Inu_scaled (C function): Regular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_Inu_scaled_e (C function): Regular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_In_array (C function):   Regular Modified Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_In_e (C function):       Regular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_In_scaled (C function):  Regular Modified Cylindrical Bessel Functions.
                                                              (line  50)
* gsl_sf_bessel_In_scaled_array (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  57)
* gsl_sf_bessel_In_scaled_e (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  50)
* gsl_sf_bessel_J0 (C function):         Regular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_j0 (C function):         Regular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_J0_e (C function):       Regular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_j0_e (C function):       Regular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_J1 (C function):         Regular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_j1 (C function):         Regular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_J1_e (C function):       Regular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_j1_e (C function):       Regular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_j2 (C function):         Regular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_j2_e (C function):       Regular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_jl (C function):         Regular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_jl_array (C function):   Regular Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_jl_e (C function):       Regular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_jl_steed_array (C function): Regular Spherical Bessel Functions.
                                                              (line  41)
* gsl_sf_bessel_Jn (C function):         Regular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Jnu (C function):        Regular Bessel Function—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Jnu_e (C function):      Regular Bessel Function—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Jn_array (C function):   Regular Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_Jn_e (C function):       Regular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_K0 (C function):         Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_K0_e (C function):       Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_K0_scaled (C function):  Irregular Modified Cylindrical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_k0_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_K0_scaled_e (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_k0_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_K1 (C function):         Irregular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_K1_e (C function):       Irregular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_K1_scaled (C function):  Irregular Modified Cylindrical Bessel Functions.
                                                              (line  43)
* gsl_sf_bessel_k1_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_K1_scaled_e (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  43)
* gsl_sf_bessel_k1_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_k2_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_k2_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_kl_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_kl_scaled_array (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  38)
* gsl_sf_bessel_kl_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_Kn (C function):         Irregular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Knu (C function):        Irregular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Knu_e (C function):      Irregular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Knu_scaled (C function): Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  21)
* gsl_sf_bessel_Knu_scaled_e (C function): Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  21)
* gsl_sf_bessel_Kn_array (C function):   Irregular Modified Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_Kn_e (C function):       Irregular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Kn_scaled (C function):  Irregular Modified Cylindrical Bessel Functions.
                                                              (line  50)
* gsl_sf_bessel_Kn_scaled_array (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  57)
* gsl_sf_bessel_Kn_scaled_e (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  50)
* gsl_sf_bessel_lnKnu (C function):      Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_lnKnu_e (C function):    Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_sequence_Jnu_e (C function): Regular Bessel Function—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_Y0 (C function):         Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_y0 (C function):         Irregular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_Y0_e (C function):       Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_y0_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_Y1 (C function):         Irregular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_y1 (C function):         Irregular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_Y1_e (C function):       Irregular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_y1_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_y2 (C function):         Irregular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_y2_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_yl (C function):         Irregular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_yl_array (C function):   Irregular Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_yl_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_Yn (C function):         Irregular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Ynu (C function):        Irregular Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Ynu_e (C function):      Irregular Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Yn_array (C function):   Irregular Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_Yn_e (C function):       Irregular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_zero_J0 (C function):    Zeros of Regular Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_zero_J0_e (C function):  Zeros of Regular Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_zero_J1 (C function):    Zeros of Regular Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_zero_J1_e (C function):  Zeros of Regular Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_zero_Jnu (C function):   Zeros of Regular Bessel Functions.
                                                              (line  20)
* gsl_sf_bessel_zero_Jnu_e (C function): Zeros of Regular Bessel Functions.
                                                              (line  20)
* gsl_sf_beta (C function):              Beta Functions.      (line   6)
* gsl_sf_beta_e (C function):            Beta Functions.      (line   6)
* gsl_sf_beta_inc (C function):          Incomplete Beta Function.
                                                              (line   6)
* gsl_sf_beta_inc_e (C function):        Incomplete Beta Function.
                                                              (line   6)
* gsl_sf_Chi (C function):               Hyperbolic Integrals.
                                                              (line  13)
* gsl_sf_Chi_e (C function):             Hyperbolic Integrals.
                                                              (line  13)
* gsl_sf_choose (C function):            Factorials.          (line  43)
* gsl_sf_choose_e (C function):          Factorials.          (line  43)
* gsl_sf_Ci (C function):                Trigonometric Integrals.
                                                              (line  13)
* gsl_sf_Ci_e (C function):              Trigonometric Integrals.
                                                              (line  13)
* gsl_sf_clausen (C function):           Clausen Functions.   (line  14)
* gsl_sf_clausen_e (C function):         Clausen Functions.   (line  14)
* gsl_sf_complex_cos_e (C function):     Trigonometric Functions for Complex Arguments.
                                                              (line  12)
* gsl_sf_complex_dilog_e (C function):   Complex Argument.    (line   6)
* gsl_sf_complex_logsin_e (C function):  Trigonometric Functions for Complex Arguments.
                                                              (line  19)
* gsl_sf_complex_log_e (C function):     Logarithm and Related Functions.
                                                              (line  22)
* gsl_sf_complex_sin_e (C function):     Trigonometric Functions for Complex Arguments.
                                                              (line   6)
* gsl_sf_conicalP_0 (C function):        Conical Functions.   (line  24)
* gsl_sf_conicalP_0_e (C function):      Conical Functions.   (line  24)
* gsl_sf_conicalP_1 (C function):        Conical Functions.   (line  31)
* gsl_sf_conicalP_1_e (C function):      Conical Functions.   (line  31)
* gsl_sf_conicalP_cyl_reg (C function):  Conical Functions.   (line  46)
* gsl_sf_conicalP_cyl_reg_e (C function): Conical Functions.  (line  46)
* gsl_sf_conicalP_half (C function):     Conical Functions.   (line  10)
* gsl_sf_conicalP_half_e (C function):   Conical Functions.   (line  10)
* gsl_sf_conicalP_mhalf (C function):    Conical Functions.   (line  17)
* gsl_sf_conicalP_mhalf_e (C function):  Conical Functions.   (line  17)
* gsl_sf_conicalP_sph_reg (C function):  Conical Functions.   (line  38)
* gsl_sf_conicalP_sph_reg_e (C function): Conical Functions.  (line  38)
* gsl_sf_cos (C function):               Circular Trigonometric Functions.
                                                              (line  11)
* gsl_sf_cos_e (C function):             Circular Trigonometric Functions.
                                                              (line  11)
* gsl_sf_cos_err_e (C function):         Trigonometric Functions With Error Estimates.
                                                              (line  14)
* gsl_sf_coulomb_CL_array (C function):  Coulomb Wave Function Normalization Constant.
                                                              (line  15)
* gsl_sf_coulomb_CL_e (C function):      Coulomb Wave Function Normalization Constant.
                                                              (line   9)
* gsl_sf_coulomb_wave_FGp_array (C function): Coulomb Wave Functions.
                                                              (line  52)
* gsl_sf_coulomb_wave_FG_array (C function): Coulomb Wave Functions.
                                                              (line  42)
* gsl_sf_coulomb_wave_FG_e (C function): Coulomb Wave Functions.
                                                              (line  20)
* gsl_sf_coulomb_wave_F_array (C function): Coulomb Wave Functions.
                                                              (line  34)
* gsl_sf_coulomb_wave_sphF_array (C function): Coulomb Wave Functions.
                                                              (line  64)
* gsl_sf_coupling_3j (C function):       3-j Symbols.         (line   6)
* gsl_sf_coupling_3j_e (C function):     3-j Symbols.         (line   6)
* gsl_sf_coupling_6j (C function):       6-j Symbols.         (line   6)
* gsl_sf_coupling_6j_e (C function):     6-j Symbols.         (line   6)
* gsl_sf_coupling_9j (C function):       9-j Symbols.         (line   6)
* gsl_sf_coupling_9j_e (C function):     9-j Symbols.         (line   6)
* gsl_sf_dawson (C function):            Dawson Function.     (line  14)
* gsl_sf_dawson_e (C function):          Dawson Function.     (line  14)
* gsl_sf_debye_1 (C function):           Debye Functions.     (line  13)
* gsl_sf_debye_1_e (C function):         Debye Functions.     (line  13)
* gsl_sf_debye_2 (C function):           Debye Functions.     (line  18)
* gsl_sf_debye_2_e (C function):         Debye Functions.     (line  18)
* gsl_sf_debye_3 (C function):           Debye Functions.     (line  23)
* gsl_sf_debye_3_e (C function):         Debye Functions.     (line  23)
* gsl_sf_debye_4 (C function):           Debye Functions.     (line  28)
* gsl_sf_debye_4_e (C function):         Debye Functions.     (line  28)
* gsl_sf_debye_5 (C function):           Debye Functions.     (line  33)
* gsl_sf_debye_5_e (C function):         Debye Functions.     (line  33)
* gsl_sf_debye_6 (C function):           Debye Functions.     (line  38)
* gsl_sf_debye_6_e (C function):         Debye Functions.     (line  38)
* gsl_sf_dilog (C function):             Real Argument.       (line   6)
* gsl_sf_dilog_e (C function):           Real Argument.       (line   6)
* gsl_sf_doublefact (C function):        Factorials.          (line  19)
* gsl_sf_doublefact_e (C function):      Factorials.          (line  19)
* gsl_sf_ellint_D (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  37)
* gsl_sf_ellint_D_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  37)
* gsl_sf_ellint_E (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  16)
* gsl_sf_ellint_Ecomp (C function):      Legendre Form of Complete Elliptic Integrals.
                                                              (line  15)
* gsl_sf_ellint_Ecomp_e (C function):    Legendre Form of Complete Elliptic Integrals.
                                                              (line  15)
* gsl_sf_ellint_E_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  16)
* gsl_sf_ellint_F (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_F_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_Kcomp (C function):      Legendre Form of Complete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_Kcomp_e (C function):    Legendre Form of Complete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_P (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  26)
* gsl_sf_ellint_Pcomp (C function):      Legendre Form of Complete Elliptic Integrals.
                                                              (line  24)
* gsl_sf_ellint_Pcomp_e (C function):    Legendre Form of Complete Elliptic Integrals.
                                                              (line  24)
* gsl_sf_ellint_P_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  26)
* gsl_sf_ellint_RC (C function):         Carlson Forms.       (line   6)
* gsl_sf_ellint_RC_e (C function):       Carlson Forms.       (line   6)
* gsl_sf_ellint_RD (C function):         Carlson Forms.       (line  14)
* gsl_sf_ellint_RD_e (C function):       Carlson Forms.       (line  14)
* gsl_sf_ellint_RF (C function):         Carlson Forms.       (line  22)
* gsl_sf_ellint_RF_e (C function):       Carlson Forms.       (line  22)
* gsl_sf_ellint_RJ (C function):         Carlson Forms.       (line  30)
* gsl_sf_ellint_RJ_e (C function):       Carlson Forms.       (line  30)
* gsl_sf_elljac_e (C function):          Elliptic Functions Jacobi.
                                                              (line  10)
* gsl_sf_erf (C function):               Error Function.      (line   6)
* gsl_sf_erfc (C function):              Complementary Error Function.
                                                              (line   6)
* gsl_sf_erfc_e (C function):            Complementary Error Function.
                                                              (line   6)
* gsl_sf_erf_e (C function):             Error Function.      (line   6)
* gsl_sf_erf_Q (C function):             Probability functions.
                                                              (line  15)
* gsl_sf_erf_Q_e (C function):           Probability functions.
                                                              (line  15)
* gsl_sf_erf_Z (C function):             Probability functions.
                                                              (line   9)
* gsl_sf_erf_Z_e (C function):           Probability functions.
                                                              (line   9)
* gsl_sf_eta (C function):               Eta Function.        (line  16)
* gsl_sf_eta_e (C function):             Eta Function.        (line  16)
* gsl_sf_eta_int (C function):           Eta Function.        (line  10)
* gsl_sf_eta_int_e (C function):         Eta Function.        (line  10)
* gsl_sf_exp (C function):               Exponential Function.
                                                              (line   6)
* gsl_sf_expint_3 (C function):          Ei_3 x.              (line   6)
* gsl_sf_expint_3_e (C function):        Ei_3 x.              (line   6)
* gsl_sf_expint_E1 (C function):         Exponential Integral.
                                                              (line   6)
* gsl_sf_expint_E1_e (C function):       Exponential Integral.
                                                              (line   6)
* gsl_sf_expint_E2 (C function):         Exponential Integral.
                                                              (line  13)
* gsl_sf_expint_E2_e (C function):       Exponential Integral.
                                                              (line  13)
* gsl_sf_expint_Ei (C function):         Ei x.                (line   6)
* gsl_sf_expint_Ei_e (C function):       Ei x.                (line   6)
* gsl_sf_expint_En (C function):         Exponential Integral.
                                                              (line  21)
* gsl_sf_expint_En_e (C function):       Exponential Integral.
                                                              (line  21)
* gsl_sf_expm1 (C function):             Relative Exponential Functions.
                                                              (line   6)
* gsl_sf_expm1_e (C function):           Relative Exponential Functions.
                                                              (line   6)
* gsl_sf_exprel (C function):            Relative Exponential Functions.
                                                              (line  12)
* gsl_sf_exprel_2 (C function):          Relative Exponential Functions.
                                                              (line  20)
* gsl_sf_exprel_2_e (C function):        Relative Exponential Functions.
                                                              (line  20)
* gsl_sf_exprel_e (C function):          Relative Exponential Functions.
                                                              (line  12)
* gsl_sf_exprel_n (C function):          Relative Exponential Functions.
                                                              (line  28)
* gsl_sf_exprel_n_e (C function):        Relative Exponential Functions.
                                                              (line  28)
* gsl_sf_exp_e (C function):             Exponential Function.
                                                              (line   6)
* gsl_sf_exp_e10_e (C function):         Exponential Function.
                                                              (line  12)
* gsl_sf_exp_err_e (C function):         Exponentiation With Error Estimate.
                                                              (line   6)
* gsl_sf_exp_err_e10_e (C function):     Exponentiation With Error Estimate.
                                                              (line  12)
* gsl_sf_exp_mult (C function):          Exponential Function.
                                                              (line  19)
* gsl_sf_exp_mult_e (C function):        Exponential Function.
                                                              (line  19)
* gsl_sf_exp_mult_e10_e (C function):    Exponential Function.
                                                              (line  26)
* gsl_sf_exp_mult_err_e (C function):    Exponentiation With Error Estimate.
                                                              (line  19)
* gsl_sf_exp_mult_err_e10_e (C function): Exponentiation With Error Estimate.
                                                              (line  26)
* gsl_sf_fact (C function):              Factorials.          (line  11)
* gsl_sf_fact_e (C function):            Factorials.          (line  11)
* gsl_sf_fermi_dirac_0 (C function):     Complete Fermi-Dirac Integrals.
                                                              (line  21)
* gsl_sf_fermi_dirac_0_e (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  21)
* gsl_sf_fermi_dirac_1 (C function):     Complete Fermi-Dirac Integrals.
                                                              (line  28)
* gsl_sf_fermi_dirac_1_e (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  28)
* gsl_sf_fermi_dirac_2 (C function):     Complete Fermi-Dirac Integrals.
                                                              (line  35)
* gsl_sf_fermi_dirac_2_e (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  35)
* gsl_sf_fermi_dirac_3half (C function): Complete Fermi-Dirac Integrals.
                                                              (line  64)
* gsl_sf_fermi_dirac_3half_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  64)
* gsl_sf_fermi_dirac_half (C function):  Complete Fermi-Dirac Integrals.
                                                              (line  57)
* gsl_sf_fermi_dirac_half_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  57)
* gsl_sf_fermi_dirac_inc_0 (C function): Incomplete Fermi-Dirac Integrals.
                                                              (line  10)
* gsl_sf_fermi_dirac_inc_0_e (C function): Incomplete Fermi-Dirac Integrals.
                                                              (line  10)
* gsl_sf_fermi_dirac_int (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  42)
* gsl_sf_fermi_dirac_int_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  42)
* gsl_sf_fermi_dirac_m1 (C function):    Complete Fermi-Dirac Integrals.
                                                              (line  13)
* gsl_sf_fermi_dirac_m1_e (C function):  Complete Fermi-Dirac Integrals.
                                                              (line  13)
* gsl_sf_fermi_dirac_mhalf (C function): Complete Fermi-Dirac Integrals.
                                                              (line  50)
* gsl_sf_fermi_dirac_mhalf_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  50)
* gsl_sf_gamma (C function):             Gamma Functions.     (line  14)
* gsl_sf_gammainv (C function):          Gamma Functions.     (line  54)
* gsl_sf_gammainv_e (C function):        Gamma Functions.     (line  54)
* gsl_sf_gammastar (C function):         Gamma Functions.     (line  43)
* gsl_sf_gammastar_e (C function):       Gamma Functions.     (line  43)
* gsl_sf_gamma_e (C function):           Gamma Functions.     (line  14)
* gsl_sf_gamma_inc (C function):         Incomplete Gamma Functions.
                                                              (line   6)
* gsl_sf_gamma_inc_e (C function):       Incomplete Gamma Functions.
                                                              (line   6)
* gsl_sf_gamma_inc_P (C function):       Incomplete Gamma Functions.
                                                              (line  22)
* gsl_sf_gamma_inc_P_e (C function):     Incomplete Gamma Functions.
                                                              (line  22)
* gsl_sf_gamma_inc_Q (C function):       Incomplete Gamma Functions.
                                                              (line  14)
* gsl_sf_gamma_inc_Q_e (C function):     Incomplete Gamma Functions.
                                                              (line  14)
* gsl_sf_gegenpoly_1 (C function):       Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_1_e (C function):     Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_2 (C function):       Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_2_e (C function):     Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_3 (C function):       Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_3_e (C function):     Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_array (C function):   Gegenbauer Functions.
                                                              (line  32)
* gsl_sf_gegenpoly_n (C function):       Gegenbauer Functions.
                                                              (line  24)
* gsl_sf_gegenpoly_n_e (C function):     Gegenbauer Functions.
                                                              (line  24)
* gsl_sf_hazard (C function):            Probability functions.
                                                              (line  29)
* gsl_sf_hazard_e (C function):          Probability functions.
                                                              (line  29)
* gsl_sf_hermite (C function):           Hermite Polynomials. (line  23)
* gsl_sf_hermite_array (C function):     Hermite Polynomials. (line  32)
* gsl_sf_hermite_array_deriv (C function): Derivatives of Hermite Polynomials.
                                                              (line  15)
* gsl_sf_hermite_deriv (C function):     Derivatives of Hermite Polynomials.
                                                              (line   6)
* gsl_sf_hermite_deriv_array (C function): Derivatives of Hermite Polynomials.
                                                              (line  24)
* gsl_sf_hermite_deriv_e (C function):   Derivatives of Hermite Polynomials.
                                                              (line   6)
* gsl_sf_hermite_e (C function):         Hermite Polynomials. (line  23)
* gsl_sf_hermite_func (C function):      Hermite Functions.   (line  28)
* gsl_sf_hermite_func_array (C function): Hermite Functions.  (line  46)
* gsl_sf_hermite_func_der (C function):  Derivatives of Hermite Functions.
                                                              (line   6)
* gsl_sf_hermite_func_der_e (C function): Derivatives of Hermite Functions.
                                                              (line   6)
* gsl_sf_hermite_func_e (C function):    Hermite Functions.   (line  28)
* gsl_sf_hermite_func_fast (C function): Hermite Functions.   (line  36)
* gsl_sf_hermite_func_fast_e (C function): Hermite Functions. (line  36)
* gsl_sf_hermite_func_series (C function): Hermite Functions. (line  54)
* gsl_sf_hermite_func_series_e (C function): Hermite Functions.
                                                              (line  54)
* gsl_sf_hermite_func_zero (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  26)
* gsl_sf_hermite_func_zero_e (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  26)
* gsl_sf_hermite_prob (C function):      Hermite Polynomials. (line  48)
* gsl_sf_hermite_prob_array (C function): Hermite Polynomials.
                                                              (line  57)
* gsl_sf_hermite_prob_array_deriv (C function): Derivatives of Hermite Polynomials.
                                                              (line  42)
* gsl_sf_hermite_prob_deriv (C function): Derivatives of Hermite Polynomials.
                                                              (line  33)
* gsl_sf_hermite_prob_deriv_array (C function): Derivatives of Hermite Polynomials.
                                                              (line  51)
* gsl_sf_hermite_prob_deriv_e (C function): Derivatives of Hermite Polynomials.
                                                              (line  33)
* gsl_sf_hermite_prob_e (C function):    Hermite Polynomials. (line  48)
* gsl_sf_hermite_prob_series (C function): Hermite Polynomials.
                                                              (line  64)
* gsl_sf_hermite_prob_series_e (C function): Hermite Polynomials.
                                                              (line  64)
* gsl_sf_hermite_prob_zero (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  19)
* gsl_sf_hermite_prob_zero_e (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  19)
* gsl_sf_hermite_series (C function):    Hermite Polynomials. (line  39)
* gsl_sf_hermite_series_e (C function):  Hermite Polynomials. (line  39)
* gsl_sf_hermite_zero (C function):      Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  12)
* gsl_sf_hermite_zero_e (C function):    Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  12)
* gsl_sf_hydrogenicR (C function):       Normalized Hydrogenic Bound States.
                                                              (line  13)
* gsl_sf_hydrogenicR_1 (C function):     Normalized Hydrogenic Bound States.
                                                              (line   6)
* gsl_sf_hydrogenicR_1_e (C function):   Normalized Hydrogenic Bound States.
                                                              (line   6)
* gsl_sf_hydrogenicR_e (C function):     Normalized Hydrogenic Bound States.
                                                              (line  13)
* gsl_sf_hyperg_0F1 (C function):        Hypergeometric Functions.
                                                              (line  10)
* gsl_sf_hyperg_0F1_e (C function):      Hypergeometric Functions.
                                                              (line  10)
* gsl_sf_hyperg_1F1 (C function):        Hypergeometric Functions.
                                                              (line  28)
* gsl_sf_hyperg_1F1_e (C function):      Hypergeometric Functions.
                                                              (line  28)
* gsl_sf_hyperg_1F1_int (C function):    Hypergeometric Functions.
                                                              (line  18)
* gsl_sf_hyperg_1F1_int_e (C function):  Hypergeometric Functions.
                                                              (line  18)
* gsl_sf_hyperg_2F0 (C function):        Hypergeometric Functions.
                                                              (line 116)
* gsl_sf_hyperg_2F0_e (C function):      Hypergeometric Functions.
                                                              (line 116)
* gsl_sf_hyperg_2F1 (C function):        Hypergeometric Functions.
                                                              (line  67)
* gsl_sf_hyperg_2F1_conj (C function):   Hypergeometric Functions.
                                                              (line  81)
* gsl_sf_hyperg_2F1_conj_e (C function): Hypergeometric Functions.
                                                              (line  81)
* gsl_sf_hyperg_2F1_conj_renorm (C function): Hypergeometric Functions.
                                                              (line 104)
* gsl_sf_hyperg_2F1_conj_renorm_e (C function): Hypergeometric Functions.
                                                              (line 104)
* gsl_sf_hyperg_2F1_e (C function):      Hypergeometric Functions.
                                                              (line  67)
* gsl_sf_hyperg_2F1_renorm (C function): Hypergeometric Functions.
                                                              (line  92)
* gsl_sf_hyperg_2F1_renorm_e (C function): Hypergeometric Functions.
                                                              (line  92)
* gsl_sf_hyperg_U (C function):          Hypergeometric Functions.
                                                              (line  53)
* gsl_sf_hyperg_U_e (C function):        Hypergeometric Functions.
                                                              (line  53)
* gsl_sf_hyperg_U_e10_e (C function):    Hypergeometric Functions.
                                                              (line  60)
* gsl_sf_hyperg_U_int (C function):      Hypergeometric Functions.
                                                              (line  38)
* gsl_sf_hyperg_U_int_e (C function):    Hypergeometric Functions.
                                                              (line  38)
* gsl_sf_hyperg_U_int_e10_e (C function): Hypergeometric Functions.
                                                              (line  45)
* gsl_sf_hypot (C function):             Circular Trigonometric Functions.
                                                              (line  16)
* gsl_sf_hypot_e (C function):           Circular Trigonometric Functions.
                                                              (line  16)
* gsl_sf_hzeta (C function):             Hurwitz Zeta Function.
                                                              (line  10)
* gsl_sf_hzeta_e (C function):           Hurwitz Zeta Function.
                                                              (line  10)
* gsl_sf_laguerre_1 (C function):        Laguerre Functions.  (line  20)
* gsl_sf_laguerre_1_e (C function):      Laguerre Functions.  (line  20)
* gsl_sf_laguerre_2 (C function):        Laguerre Functions.  (line  20)
* gsl_sf_laguerre_2_e (C function):      Laguerre Functions.  (line  20)
* gsl_sf_laguerre_3 (C function):        Laguerre Functions.  (line  20)
* gsl_sf_laguerre_3_e (C function):      Laguerre Functions.  (line  20)
* gsl_sf_laguerre_n (C function):        Laguerre Functions.  (line  33)
* gsl_sf_laguerre_n_e (C function):      Laguerre Functions.  (line  33)
* gsl_sf_lambert_W0 (C function):        Lambert W Functions. (line  13)
* gsl_sf_lambert_W0_e (C function):      Lambert W Functions. (line  13)
* gsl_sf_lambert_Wm1 (C function):       Lambert W Functions. (line  19)
* gsl_sf_lambert_Wm1_e (C function):     Lambert W Functions. (line  19)
* gsl_sf_legendre_array (C function):    Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  84)
* gsl_sf_legendre_array_e (C function):  Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  84)
* gsl_sf_legendre_array_index (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 193)
* gsl_sf_legendre_array_n (C function):  Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 184)
* gsl_sf_legendre_array_size (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 239)
* gsl_sf_legendre_deriv2_alt_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 157)
* gsl_sf_legendre_deriv2_alt_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 157)
* gsl_sf_legendre_deriv2_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 136)
* gsl_sf_legendre_deriv2_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 136)
* gsl_sf_legendre_deriv_alt_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 118)
* gsl_sf_legendre_deriv_alt_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 118)
* gsl_sf_legendre_deriv_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 101)
* gsl_sf_legendre_deriv_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 101)
* gsl_sf_legendre_H3d (C function):      Radial Functions for Hyperbolic Space.
                                                              (line  35)
* gsl_sf_legendre_H3d_0 (C function):    Radial Functions for Hyperbolic Space.
                                                              (line  11)
* gsl_sf_legendre_H3d_0_e (C function):  Radial Functions for Hyperbolic Space.
                                                              (line  11)
* gsl_sf_legendre_H3d_1 (C function):    Radial Functions for Hyperbolic Space.
                                                              (line  23)
* gsl_sf_legendre_H3d_1_e (C function):  Radial Functions for Hyperbolic Space.
                                                              (line  23)
* gsl_sf_legendre_H3d_array (C function): Radial Functions for Hyperbolic Space.
                                                              (line  45)
* gsl_sf_legendre_H3d_e (C function):    Radial Functions for Hyperbolic Space.
                                                              (line  35)
* gsl_sf_legendre_nlm (C function):      Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 178)
* gsl_sf_legendre_P1 (C function):       Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P1_e (C function):     Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P2 (C function):       Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P2_e (C function):     Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P3 (C function):       Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P3_e (C function):     Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_Pl (C function):       Legendre Polynomials.
                                                              (line  16)
* gsl_sf_legendre_Plm (C function):      Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 204)
* gsl_sf_legendre_Plm_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 221)
* gsl_sf_legendre_Plm_deriv_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 221)
* gsl_sf_legendre_Plm_e (C function):    Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 204)
* gsl_sf_legendre_Pl_array (C function): Legendre Polynomials.
                                                              (line  24)
* gsl_sf_legendre_Pl_deriv_array (C function): Legendre Polynomials.
                                                              (line  24)
* gsl_sf_legendre_Pl_e (C function):     Legendre Polynomials.
                                                              (line  16)
* gsl_sf_legendre_Q0 (C function):       Legendre Polynomials.
                                                              (line  32)
* gsl_sf_legendre_Q0_e (C function):     Legendre Polynomials.
                                                              (line  32)
* gsl_sf_legendre_Q1 (C function):       Legendre Polynomials.
                                                              (line  38)
* gsl_sf_legendre_Q1_e (C function):     Legendre Polynomials.
                                                              (line  38)
* gsl_sf_legendre_Ql (C function):       Legendre Polynomials.
                                                              (line  44)
* gsl_sf_legendre_Ql_e (C function):     Legendre Polynomials.
                                                              (line  44)
* gsl_sf_legendre_sphPlm (C function):   Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 211)
* gsl_sf_legendre_sphPlm_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 230)
* gsl_sf_legendre_sphPlm_deriv_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 230)
* gsl_sf_legendre_sphPlm_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 211)
* gsl_sf_legendre_t (C type):            Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  66)
* gsl_sf_lnbeta (C function):            Beta Functions.      (line  14)
* gsl_sf_lnbeta_e (C function):          Beta Functions.      (line  14)
* gsl_sf_lnchoose (C function):          Factorials.          (line  50)
* gsl_sf_lnchoose_e (C function):        Factorials.          (line  50)
* gsl_sf_lncosh (C function):            Hyperbolic Trigonometric Functions.
                                                              (line  11)
* gsl_sf_lncosh_e (C function):          Hyperbolic Trigonometric Functions.
                                                              (line  11)
* gsl_sf_lndoublefact (C function):      Factorials.          (line  36)
* gsl_sf_lndoublefact_e (C function):    Factorials.          (line  36)
* gsl_sf_lnfact (C function):            Factorials.          (line  27)
* gsl_sf_lnfact_e (C function):          Factorials.          (line  27)
* gsl_sf_lngamma (C function):           Gamma Functions.     (line  23)
* gsl_sf_lngamma_complex_e (C function): Gamma Functions.     (line  60)
* gsl_sf_lngamma_e (C function):         Gamma Functions.     (line  23)
* gsl_sf_lngamma_sgn_e (C function):     Gamma Functions.     (line  32)
* gsl_sf_lnpoch (C function):            Pochhammer Symbol.   (line  16)
* gsl_sf_lnpoch_e (C function):          Pochhammer Symbol.   (line  16)
* gsl_sf_lnpoch_sgn_e (C function):      Pochhammer Symbol.   (line  23)
* gsl_sf_lnsinh (C function):            Hyperbolic Trigonometric Functions.
                                                              (line   6)
* gsl_sf_lnsinh_e (C function):          Hyperbolic Trigonometric Functions.
                                                              (line   6)
* gsl_sf_log (C function):               Logarithm and Related Functions.
                                                              (line  10)
* gsl_sf_log_1plusx (C function):        Logarithm and Related Functions.
                                                              (line  30)
* gsl_sf_log_1plusx_e (C function):      Logarithm and Related Functions.
                                                              (line  30)
* gsl_sf_log_1plusx_mx (C function):     Logarithm and Related Functions.
                                                              (line  36)
* gsl_sf_log_1plusx_mx_e (C function):   Logarithm and Related Functions.
                                                              (line  36)
* gsl_sf_log_abs (C function):           Logarithm and Related Functions.
                                                              (line  16)
* gsl_sf_log_abs_e (C function):         Logarithm and Related Functions.
                                                              (line  16)
* gsl_sf_log_e (C function):             Logarithm and Related Functions.
                                                              (line  10)
* gsl_sf_log_erfc (C function):          Log Complementary Error Function.
                                                              (line   6)
* gsl_sf_log_erfc_e (C function):        Log Complementary Error Function.
                                                              (line   6)
* gsl_sf_mathieu_a (C function):         Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_alloc (C function):     Mathieu Function Workspace.
                                                              (line  14)
* gsl_sf_mathieu_a_array (C function):   Mathieu Function Characteristic Values.
                                                              (line  16)
* gsl_sf_mathieu_a_e (C function):       Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_b (C function):         Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_b_array (C function):   Mathieu Function Characteristic Values.
                                                              (line  16)
* gsl_sf_mathieu_b_e (C function):       Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_ce (C function):        Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_ce_array (C function):  Angular Mathieu Functions.
                                                              (line  16)
* gsl_sf_mathieu_ce_e (C function):      Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_free (C function):      Mathieu Function Workspace.
                                                              (line  22)
* gsl_sf_mathieu_Mc (C function):        Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_Mc_array (C function):  Radial Mathieu Functions.
                                                              (line  22)
* gsl_sf_mathieu_Mc_e (C function):      Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_Ms (C function):        Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_Ms_array (C function):  Radial Mathieu Functions.
                                                              (line  22)
* gsl_sf_mathieu_Ms_e (C function):      Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_se (C function):        Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_se_array (C function):  Angular Mathieu Functions.
                                                              (line  16)
* gsl_sf_mathieu_se_e (C function):      Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_workspace (C type):     Mathieu Function Workspace.
                                                              (line  10)
* gsl_sf_multiply (C function):          Elementary Operations.
                                                              (line  10)
* gsl_sf_multiply_e (C function):        Elementary Operations.
                                                              (line  10)
* gsl_sf_multiply_err_e (C function):    Elementary Operations.
                                                              (line  17)
* gsl_sf_poch (C function):              Pochhammer Symbol.   (line   6)
* gsl_sf_pochrel (C function):           Pochhammer Symbol.   (line  31)
* gsl_sf_pochrel_e (C function):         Pochhammer Symbol.   (line  31)
* gsl_sf_poch_e (C function):            Pochhammer Symbol.   (line   6)
* gsl_sf_polar_to_rect (C function):     Conversion Functions.
                                                              (line   6)
* gsl_sf_pow_int (C function):           Power Function.      (line  10)
* gsl_sf_pow_int_e (C function):         Power Function.      (line  10)
* gsl_sf_psi (C function):               Digamma Function.    (line  13)
* gsl_sf_psi_1 (C function):             Trigamma Function.   (line  12)
* gsl_sf_psi_1piy (C function):          Digamma Function.    (line  19)
* gsl_sf_psi_1piy_e (C function):        Digamma Function.    (line  19)
* gsl_sf_psi_1_e (C function):           Trigamma Function.   (line  12)
* gsl_sf_psi_1_int (C function):         Trigamma Function.   (line   6)
* gsl_sf_psi_1_int_e (C function):       Trigamma Function.   (line   6)
* gsl_sf_psi_e (C function):             Digamma Function.    (line  13)
* gsl_sf_psi_int (C function):           Digamma Function.    (line   6)
* gsl_sf_psi_int_e (C function):         Digamma Function.    (line   6)
* gsl_sf_psi_n (C function):             Polygamma Function.  (line   6)
* gsl_sf_psi_n_e (C function):           Polygamma Function.  (line   6)
* gsl_sf_rect_to_polar (C function):     Conversion Functions.
                                                              (line  13)
* gsl_sf_result (C type):                The gsl_sf_result struct.
                                                              (line  13)
* gsl_sf_result_e10 (C type):            The gsl_sf_result struct.
                                                              (line  31)
* gsl_sf_Shi (C function):               Hyperbolic Integrals.
                                                              (line   6)
* gsl_sf_Shi_e (C function):             Hyperbolic Integrals.
                                                              (line   6)
* gsl_sf_Si (C function):                Trigonometric Integrals.
                                                              (line   6)
* gsl_sf_sin (C function):               Circular Trigonometric Functions.
                                                              (line   6)
* gsl_sf_sinc (C function):              Circular Trigonometric Functions.
                                                              (line  23)
* gsl_sf_sinc_e (C function):            Circular Trigonometric Functions.
                                                              (line  23)
* gsl_sf_sin_e (C function):             Circular Trigonometric Functions.
                                                              (line   6)
* gsl_sf_sin_err_e (C function):         Trigonometric Functions With Error Estimates.
                                                              (line   6)
* gsl_sf_Si_e (C function):              Trigonometric Integrals.
                                                              (line   6)
* gsl_sf_synchrotron_1 (C function):     Synchrotron Functions.
                                                              (line   9)
* gsl_sf_synchrotron_1_e (C function):   Synchrotron Functions.
                                                              (line   9)
* gsl_sf_synchrotron_2 (C function):     Synchrotron Functions.
                                                              (line  16)
* gsl_sf_synchrotron_2_e (C function):   Synchrotron Functions.
                                                              (line  16)
* gsl_sf_taylorcoeff (C function):       Factorials.          (line  57)
* gsl_sf_taylorcoeff_e (C function):     Factorials.          (line  57)
* gsl_sf_transport_2 (C function):       Transport Functions. (line  13)
* gsl_sf_transport_2_e (C function):     Transport Functions. (line  13)
* gsl_sf_transport_3 (C function):       Transport Functions. (line  18)
* gsl_sf_transport_3_e (C function):     Transport Functions. (line  18)
* gsl_sf_transport_4 (C function):       Transport Functions. (line  23)
* gsl_sf_transport_4_e (C function):     Transport Functions. (line  23)
* gsl_sf_transport_5 (C function):       Transport Functions. (line  28)
* gsl_sf_transport_5_e (C function):     Transport Functions. (line  28)
* gsl_sf_zeta (C function):              Riemann Zeta Function.
                                                              (line  16)
* gsl_sf_zetam1 (C function):            Riemann Zeta Function Minus One.
                                                              (line  16)
* gsl_sf_zetam1_e (C function):          Riemann Zeta Function Minus One.
                                                              (line  16)
* gsl_sf_zetam1_int (C function):        Riemann Zeta Function Minus One.
                                                              (line  10)
* gsl_sf_zetam1_int_e (C function):      Riemann Zeta Function Minus One.
                                                              (line  10)
* gsl_sf_zeta_e (C function):            Riemann Zeta Function.
                                                              (line  16)
* gsl_sf_zeta_int (C function):          Riemann Zeta Function.
                                                              (line  10)
* gsl_sf_zeta_int_e (C function):        Riemann Zeta Function.
                                                              (line  10)
* GSL_SIGN (C macro):                    Testing the Sign of Numbers.
                                                              (line   6)
* gsl_siman_copy_construct_t (C type):   Simulated Annealing functions.
                                                              (line  93)
* gsl_siman_copy_t (C type):             Simulated Annealing functions.
                                                              (line  86)
* gsl_siman_destroy_t (C type):          Simulated Annealing functions.
                                                              (line 100)
* gsl_siman_Efunc_t (C type):            Simulated Annealing functions.
                                                              (line  56)
* gsl_siman_metric_t (C type):           Simulated Annealing functions.
                                                              (line  72)
* gsl_siman_params_t (C type):           Simulated Annealing functions.
                                                              (line 107)
* gsl_siman_print_t (C type):            Simulated Annealing functions.
                                                              (line  79)
* gsl_siman_solve (C function):          Simulated Annealing functions.
                                                              (line   6)
* gsl_siman_step_t (C type):             Simulated Annealing functions.
                                                              (line  63)
* gsl_sort (C function):                 Sorting vectors.     (line  22)
* gsl_sort2 (C function):                Sorting vectors.     (line  29)
* gsl_sort_index (C function):           Sorting vectors.     (line  49)
* gsl_sort_largest (C function):         Selecting the k smallest or largest elements.
                                                              (line  26)
* gsl_sort_largest_index (C function):   Selecting the k smallest or largest elements.
                                                              (line  58)
* gsl_sort_smallest (C function):        Selecting the k smallest or largest elements.
                                                              (line  16)
* gsl_sort_smallest_index (C function):  Selecting the k smallest or largest elements.
                                                              (line  48)
* gsl_sort_vector (C function):          Sorting vectors.     (line  38)
* gsl_sort_vector2 (C function):         Sorting vectors.     (line  43)
* gsl_sort_vector_index (C function):    Sorting vectors.     (line  61)
* gsl_sort_vector_largest (C function):  Selecting the k smallest or largest elements.
                                                              (line  35)
* gsl_sort_vector_largest_index (C function): Selecting the k smallest or largest elements.
                                                              (line  69)
* gsl_sort_vector_smallest (C function): Selecting the k smallest or largest elements.
                                                              (line  35)
* gsl_sort_vector_smallest_index (C function): Selecting the k smallest or largest elements.
                                                              (line  69)
* gsl_spblas_dgemm (C function):         Sparse BLAS operations.
                                                              (line  17)
* gsl_spblas_dgemv (C function):         Sparse BLAS operations.
                                                              (line   6)
* gsl_splinalg_itersolve_alloc (C function): Iterating the Sparse Linear System.
                                                              (line   9)
* gsl_splinalg_itersolve_free (C function): Iterating the Sparse Linear System.
                                                              (line  20)
* gsl_splinalg_itersolve_iterate (C function): Iterating the Sparse Linear System.
                                                              (line  31)
* gsl_splinalg_itersolve_name (C function): Iterating the Sparse Linear System.
                                                              (line  26)
* gsl_splinalg_itersolve_normr (C function): Iterating the Sparse Linear System.
                                                              (line  53)
* gsl_splinalg_itersolve_type (C type):  Types of Sparse Iterative Solvers.
                                                              (line   9)
* gsl_splinalg_itersolve_type.gsl_splinalg_itersolve_gmres (C var): Types of Sparse Iterative Solvers.
                                                              (line  11)
* gsl_spline (C type):                   1D Higher-level Interface.
                                                              (line  14)
* gsl_spline2d (C type):                 2D Higher-level Interface.
                                                              (line  14)
* gsl_spline2d_alloc (C function):       2D Higher-level Interface.
                                                              (line  19)
* gsl_spline2d_eval (C function):        2D Higher-level Interface.
                                                              (line  33)
* gsl_spline2d_eval_deriv_x (C function): 2D Higher-level Interface.
                                                              (line  47)
* gsl_spline2d_eval_deriv_xx (C function): 2D Higher-level Interface.
                                                              (line  61)
* gsl_spline2d_eval_deriv_xx_e (C function): 2D Higher-level Interface.
                                                              (line  61)
* gsl_spline2d_eval_deriv_xy (C function): 2D Higher-level Interface.
                                                              (line  75)
* gsl_spline2d_eval_deriv_xy_e (C function): 2D Higher-level Interface.
                                                              (line  75)
* gsl_spline2d_eval_deriv_x_e (C function): 2D Higher-level Interface.
                                                              (line  47)
* gsl_spline2d_eval_deriv_y (C function): 2D Higher-level Interface.
                                                              (line  54)
* gsl_spline2d_eval_deriv_yy (C function): 2D Higher-level Interface.
                                                              (line  68)
* gsl_spline2d_eval_deriv_yy_e (C function): 2D Higher-level Interface.
                                                              (line  68)
* gsl_spline2d_eval_deriv_y_e (C function): 2D Higher-level Interface.
                                                              (line  54)
* gsl_spline2d_eval_e (C function):      2D Higher-level Interface.
                                                              (line  33)
* gsl_spline2d_eval_extrap (C function): 2D Higher-level Interface.
                                                              (line  40)
* gsl_spline2d_eval_extrap_e (C function): 2D Higher-level Interface.
                                                              (line  40)
* gsl_spline2d_free (C function):        2D Higher-level Interface.
                                                              (line  26)
* gsl_spline2d_get (C function):         2D Higher-level Interface.
                                                              (line  85)
* gsl_spline2d_init (C function):        2D Higher-level Interface.
                                                              (line  22)
* gsl_spline2d_min_size (C function):    2D Higher-level Interface.
                                                              (line  30)
* gsl_spline2d_name (C function):        2D Higher-level Interface.
                                                              (line  28)
* gsl_spline2d_set (C function):         2D Higher-level Interface.
                                                              (line  82)
* gsl_spline_alloc (C function):         1D Higher-level Interface.
                                                              (line  19)
* gsl_spline_eval (C function):          1D Higher-level Interface.
                                                              (line  32)
* gsl_spline_eval_deriv (C function):    1D Higher-level Interface.
                                                              (line  37)
* gsl_spline_eval_deriv2 (C function):   1D Higher-level Interface.
                                                              (line  42)
* gsl_spline_eval_deriv2_e (C function): 1D Higher-level Interface.
                                                              (line  42)
* gsl_spline_eval_deriv_e (C function):  1D Higher-level Interface.
                                                              (line  37)
* gsl_spline_eval_e (C function):        1D Higher-level Interface.
                                                              (line  32)
* gsl_spline_eval_integ (C function):    1D Higher-level Interface.
                                                              (line  47)
* gsl_spline_eval_integ_e (C function):  1D Higher-level Interface.
                                                              (line  47)
* gsl_spline_free (C function):          1D Higher-level Interface.
                                                              (line  25)
* gsl_spline_init (C function):          1D Higher-level Interface.
                                                              (line  22)
* gsl_spline_min_size (C function):      1D Higher-level Interface.
                                                              (line  29)
* gsl_spline_name (C function):          1D Higher-level Interface.
                                                              (line  27)
* gsl_spmatrix (C type):                 Overview<8>.         (line  10)
* gsl_spmatrix_add (C function):         Matrix Operations.   (line  45)
* gsl_spmatrix_alloc (C function):       Allocation.          (line  12)
* gsl_spmatrix_alloc_nzmax (C function): Allocation.          (line  28)
* gsl_spmatrix_alloc_nzmax.GSL_SPMATRIX_COO (C macro): Allocation.
                                                              (line  46)
* gsl_spmatrix_alloc_nzmax.GSL_SPMATRIX_CSC (C macro): Allocation.
                                                              (line  50)
* gsl_spmatrix_alloc_nzmax.GSL_SPMATRIX_CSR (C macro): Allocation.
                                                              (line  54)
* gsl_spmatrix_compress (C function):    Compressed Format.   (line  29)
* gsl_spmatrix_csc (C function):         Compressed Format.   (line   9)
* gsl_spmatrix_csr (C function):         Compressed Format.   (line  19)
* gsl_spmatrix_d2sp (C function):        Conversion Between Sparse and Dense Matrices.
                                                              (line  10)
* gsl_spmatrix_dense_add (C function):   Matrix Operations.   (line  53)
* gsl_spmatrix_dense_sub (C function):   Matrix Operations.   (line  65)
* gsl_spmatrix_equal (C function):       Matrix Properties.   (line  28)
* gsl_spmatrix_fprintf (C function):     Reading and Writing Matrices.
                                                              (line  32)
* gsl_spmatrix_fread (C function):       Reading and Writing Matrices.
                                                              (line  18)
* gsl_spmatrix_free (C function):        Allocation.          (line  73)
* gsl_spmatrix_fscanf (C function):      Reading and Writing Matrices.
                                                              (line  46)
* gsl_spmatrix_fwrite (C function):      Reading and Writing Matrices.
                                                              (line   6)
* gsl_spmatrix_get (C function):         Accessing Matrix Elements.
                                                              (line   6)
* gsl_spmatrix_memcpy (C function):      Copying Matrices.    (line   6)
* gsl_spmatrix_minmax (C function):      Finding Maximum and Minimum Elements.
                                                              (line   6)
* gsl_spmatrix_min_index (C function):   Finding Maximum and Minimum Elements.
                                                              (line  16)
* gsl_spmatrix_nnz (C function):         Matrix Properties.   (line  20)
* gsl_spmatrix_norm1 (C function):       Matrix Properties.   (line  39)
* gsl_spmatrix_ptr (C function):         Accessing Matrix Elements.
                                                              (line  23)
* gsl_spmatrix_realloc (C function):     Allocation.          (line  61)
* gsl_spmatrix_scale (C function):       Matrix Operations.   (line   6)
* gsl_spmatrix_scale_columns (C function): Matrix Operations. (line  15)
* gsl_spmatrix_scale_rows (C function):  Matrix Operations.   (line  30)
* gsl_spmatrix_set (C function):         Accessing Matrix Elements.
                                                              (line  15)
* gsl_spmatrix_set_zero (C function):    Initializing Matrix Elements.
                                                              (line  11)
* gsl_spmatrix_sp2d (C function):        Conversion Between Sparse and Dense Matrices.
                                                              (line  18)
* gsl_spmatrix_transpose (C function):   Exchanging Rows and Columns.
                                                              (line  17)
* gsl_spmatrix_transpose_memcpy (C function): Exchanging Rows and Columns.
                                                              (line   6)
* gsl_spmatrix_type (C function):        Matrix Properties.   (line   6)
* gsl_stats_absdev (C function):         Absolute deviation.  (line   6)
* gsl_stats_absdev_m (C function):       Absolute deviation.  (line  22)
* gsl_stats_correlation (C function):    Correlation.         (line   6)
* gsl_stats_covariance (C function):     Covariance.          (line   6)
* gsl_stats_covariance_m (C function):   Covariance.          (line  16)
* gsl_stats_gastwirth_from_sorted_data (C function): Gastwirth Estimator.
                                                              (line  15)
* gsl_stats_kurtosis (C function):       Higher moments skewness and kurtosis.
                                                              (line  35)
* gsl_stats_kurtosis_m_sd (C function):  Higher moments skewness and kurtosis.
                                                              (line  48)
* gsl_stats_lag1_autocorrelation (C function): Autocorrelation.
                                                              (line   6)
* gsl_stats_lag1_autocorrelation_m (C function): Autocorrelation.
                                                              (line  16)
* gsl_stats_mad (C function):            Median Absolute Deviation MAD.
                                                              (line  20)
* gsl_stats_mad0 (C function):           Median Absolute Deviation MAD.
                                                              (line  17)
* gsl_stats_max (C function):            Maximum and Minimum values.
                                                              (line  12)
* gsl_stats_max_index (C function):      Maximum and Minimum values.
                                                              (line  42)
* gsl_stats_mean (C function):           Mean Standard Deviation and Variance.
                                                              (line   6)
* gsl_stats_median (C function):         Median and Percentiles.
                                                              (line  29)
* gsl_stats_median_from_sorted_data (C function): Median and Percentiles.
                                                              (line  13)
* gsl_stats_min (C function):            Maximum and Minimum values.
                                                              (line  24)
* gsl_stats_minmax (C function):         Maximum and Minimum values.
                                                              (line  36)
* gsl_stats_minmax_index (C function):   Maximum and Minimum values.
                                                              (line  60)
* gsl_stats_min_index (C function):      Maximum and Minimum values.
                                                              (line  51)
* gsl_stats_Qn0_from_sorted_data (C function): Q_n Statistic. (line  19)
* gsl_stats_Qn_from_sorted_data (C function): Q_n Statistic.  (line  19)
* gsl_stats_quantile_from_sorted_data (C function): Median and Percentiles.
                                                              (line  38)
* gsl_stats_sd (C function):             Mean Standard Deviation and Variance.
                                                              (line  50)
* gsl_stats_sd_m (C function):           Mean Standard Deviation and Variance.
                                                              (line  50)
* gsl_stats_sd_with_fixed_mean (C function): Mean Standard Deviation and Variance.
                                                              (line  83)
* gsl_stats_select (C function):         Order Statistics.    (line  14)
* gsl_stats_skew (C function):           Higher moments skewness and kurtosis.
                                                              (line   6)
* gsl_stats_skew_m_sd (C function):      Higher moments skewness and kurtosis.
                                                              (line  22)
* gsl_stats_Sn0_from_sorted_data (C function): S_n Statistic. (line  17)
* gsl_stats_Sn_from_sorted_data (C function): S_n Statistic.  (line  21)
* gsl_stats_spearman (C function):       Correlation.         (line  20)
* gsl_stats_trmean_from_sorted_data (C function): Trimmed Mean.
                                                              (line  14)
* gsl_stats_tss (C function):            Mean Standard Deviation and Variance.
                                                              (line  59)
* gsl_stats_tss_m (C function):          Mean Standard Deviation and Variance.
                                                              (line  59)
* gsl_stats_variance (C function):       Mean Standard Deviation and Variance.
                                                              (line  20)
* gsl_stats_variance_m (C function):     Mean Standard Deviation and Variance.
                                                              (line  40)
* gsl_stats_variance_with_fixed_mean (C function): Mean Standard Deviation and Variance.
                                                              (line  72)
* gsl_stats_wabsdev (C function):        Weighted Samples.    (line  96)
* gsl_stats_wabsdev_m (C function):      Weighted Samples.    (line 105)
* gsl_stats_wkurtosis (C function):      Weighted Samples.    (line 129)
* gsl_stats_wkurtosis_m_sd (C function): Weighted Samples.    (line 137)
* gsl_stats_wmean (C function):          Weighted Samples.    (line  14)
* gsl_stats_wsd (C function):            Weighted Samples.    (line  48)
* gsl_stats_wsd_m (C function):          Weighted Samples.    (line  56)
* gsl_stats_wsd_with_fixed_mean (C function): Weighted Samples.
                                                              (line  74)
* gsl_stats_wskew (C function):          Weighted Samples.    (line 113)
* gsl_stats_wskew_m_sd (C function):     Weighted Samples.    (line 121)
* gsl_stats_wtss (C function):           Weighted Samples.    (line  82)
* gsl_stats_wtss_m (C function):         Weighted Samples.    (line  82)
* gsl_stats_wvariance (C function):      Weighted Samples.    (line  24)
* gsl_stats_wvariance_m (C function):    Weighted Samples.    (line  40)
* gsl_stats_wvariance_with_fixed_mean (C function): Weighted Samples.
                                                              (line  62)
* gsl_strerror (C function):             Error Codes.         (line  41)
* gsl_sum_levin_utrunc_accel (C function): Acceleration functions without error estimation.
                                                              (line  39)
* gsl_sum_levin_utrunc_alloc (C function): Acceleration functions without error estimation.
                                                              (line  26)
* gsl_sum_levin_utrunc_free (C function): Acceleration functions without error estimation.
                                                              (line  33)
* gsl_sum_levin_utrunc_workspace (C type): Acceleration functions without error estimation.
                                                              (line  22)
* gsl_sum_levin_u_accel (C function):    Acceleration functions.
                                                              (line  38)
* gsl_sum_levin_u_alloc (C function):    Acceleration functions.
                                                              (line  27)
* gsl_sum_levin_u_free (C function):     Acceleration functions.
                                                              (line  33)
* gsl_sum_levin_u_workspace (C type):    Acceleration functions.
                                                              (line  23)
* gsl_vector (C type):                   Vectors.             (line  17)
* gsl_vector_add (C function):           Vector operations.   (line   6)
* gsl_vector_add_constant (C function):  Vector operations.   (line  40)
* gsl_vector_alloc (C function):         Vector allocation.   (line  14)
* gsl_vector_axpby (C function):         Vector operations.   (line  52)
* gsl_vector_calloc (C function):        Vector allocation.   (line  23)
* gsl_vector_complex_const_imag (C function): Vector views.   (line 115)
* gsl_vector_complex_const_real (C function): Vector views.   (line 103)
* gsl_vector_complex_imag (C function):  Vector views.        (line 115)
* gsl_vector_complex_real (C function):  Vector views.        (line 103)
* gsl_vector_const_ptr (C function):     Accessing vector elements.
                                                              (line  66)
* gsl_vector_const_subvector (C function): Vector views.      (line  31)
* gsl_vector_const_subvector_with_stride (C function): Vector views.
                                                              (line  63)
* gsl_vector_const_view (C type):        Vector views.        (line  12)
* gsl_vector_const_view_array (C function): Vector views.     (line 127)
* gsl_vector_const_view_array_with_stride (C function): Vector views.
                                                              (line 152)
* gsl_vector_div (C function):           Vector operations.   (line  27)
* gsl_vector_equal (C function):         Vector properties.   (line  19)
* gsl_vector_fprintf (C function):       Reading and writing vectors.
                                                              (line  28)
* gsl_vector_fread (C function):         Reading and writing vectors.
                                                              (line  17)
* gsl_vector_free (C function):          Vector allocation.   (line  28)
* gsl_vector_fscanf (C function):        Reading and writing vectors.
                                                              (line  38)
* gsl_vector_fwrite (C function):        Reading and writing vectors.
                                                              (line   9)
* gsl_vector_get (C function):           Accessing vector elements.
                                                              (line  48)
* gsl_vector_isneg (C function):         Vector properties.   (line  10)
* gsl_vector_isnonneg (C function):      Vector properties.   (line  10)
* gsl_vector_isnull (C function):        Vector properties.   (line  10)
* gsl_vector_ispos (C function):         Vector properties.   (line  10)
* gsl_vector_max (C function):           Finding maximum and minimum elements of vectors.
                                                              (line   8)
* gsl_vector_max_index (C function):     Finding maximum and minimum elements of vectors.
                                                              (line  23)
* gsl_vector_memcpy (C function):        Copying vectors.     (line  12)
* gsl_vector_min (C function):           Finding maximum and minimum elements of vectors.
                                                              (line  12)
* gsl_vector_minmax (C function):        Finding maximum and minimum elements of vectors.
                                                              (line  16)
* gsl_vector_minmax_index (C function):  Finding maximum and minimum elements of vectors.
                                                              (line  35)
* gsl_vector_min_index (C function):     Finding maximum and minimum elements of vectors.
                                                              (line  29)
* gsl_vector_mul (C function):           Vector operations.   (line  20)
* gsl_vector_ptr (C function):           Accessing vector elements.
                                                              (line  66)
* gsl_vector_reverse (C function):       Exchanging elements. (line  15)
* gsl_vector_scale (C function):         Vector operations.   (line  34)
* gsl_vector_set (C function):           Accessing vector elements.
                                                              (line  57)
* gsl_vector_set_all (C function):       Initializing vector elements.
                                                              (line   6)
* gsl_vector_set_basis (C function):     Initializing vector elements.
                                                              (line  16)
* gsl_vector_set_zero (C function):      Initializing vector elements.
                                                              (line  11)
* gsl_vector_sub (C function):           Vector operations.   (line  13)
* gsl_vector_subvector (C function):     Vector views.        (line  31)
* gsl_vector_subvector_with_stride (C function): Vector views.
                                                              (line  63)
* gsl_vector_sum (C function):           Vector operations.   (line  47)
* gsl_vector_swap (C function):          Copying vectors.     (line  19)
* gsl_vector_swap_elements (C function): Exchanging elements. (line   9)
* gsl_vector_view (C type):              Vector views.        (line  12)
* gsl_vector_view_array (C function):    Vector views.        (line 127)
* gsl_vector_view_array_with_stride (C function): Vector views.
                                                              (line 152)
* gsl_wavelet (C type):                  Initialization.      (line   6)
* gsl_wavelet2d_nstransform (C function): Wavelet transforms in two dimension.
                                                              (line  67)
* gsl_wavelet2d_nstransform_forward (C function): Wavelet transforms in two dimension.
                                                              (line  67)
* gsl_wavelet2d_nstransform_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  67)
* gsl_wavelet2d_nstransform_matrix (C function): Wavelet transforms in two dimension.
                                                              (line  80)
* gsl_wavelet2d_nstransform_matrix_forward (C function): Wavelet transforms in two dimension.
                                                              (line  80)
* gsl_wavelet2d_nstransform_matrix_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  80)
* gsl_wavelet2d_transform (C function):  Wavelet transforms in two dimension.
                                                              (line  29)
* gsl_wavelet2d_transform_forward (C function): Wavelet transforms in two dimension.
                                                              (line  29)
* gsl_wavelet2d_transform_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  29)
* gsl_wavelet2d_transform_matrix (C function): Wavelet transforms in two dimension.
                                                              (line  56)
* gsl_wavelet2d_transform_matrix_forward (C function): Wavelet transforms in two dimension.
                                                              (line  56)
* gsl_wavelet2d_transform_matrix_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  56)
* gsl_wavelet_alloc (C function):        Initialization.      (line  11)
* gsl_wavelet_free (C function):         Initialization.      (line  57)
* gsl_wavelet_name (C function):         Initialization.      (line  52)
* gsl_wavelet_transform (C function):    Wavelet transforms in one dimension.
                                                              (line   6)
* gsl_wavelet_transform_forward (C function): Wavelet transforms in one dimension.
                                                              (line   6)
* gsl_wavelet_transform_inverse (C function): Wavelet transforms in one dimension.
                                                              (line   6)
* gsl_wavelet_type (C type):             Initialization.      (line  22)
* gsl_wavelet_type.gsl_wavelet_bspline (C var): Initialization.
                                                              (line  39)
* gsl_wavelet_type.gsl_wavelet_bspline_centered (C var): Initialization.
                                                              (line  39)
* gsl_wavelet_type.gsl_wavelet_daubechies (C var): Initialization.
                                                              (line  24)
* gsl_wavelet_type.gsl_wavelet_daubechies_centered (C var): Initialization.
                                                              (line  24)
* gsl_wavelet_type.gsl_wavelet_haar (C var): Initialization.  (line  32)
* gsl_wavelet_type.gsl_wavelet_haar_centered (C var): Initialization.
                                                              (line  32)
* gsl_wavelet_workspace (C type):        Initialization.      (line  61)
* gsl_wavelet_workspace_alloc (C function): Initialization.   (line  66)
* gsl_wavelet_workspace_free (C function): Initialization.    (line  77)
* Gumbel distribution (Type 1):          The Type-1 Gumbel Distribution.
                                                              (line   6)
* Gumbel distribution (Type 2):          The Type-2 Gumbel Distribution.
                                                              (line   6)
* Haar wavelets:                         Initialization.      (line  32)
* Hankel transforms, discrete:           References and Further Reading<27>.
                                                              (line  54)
* HAVE_INLINE:                           ANSI C Compliance.   (line  25)
* hazard function, normal distribution:  Probability functions.
                                                              (line  21)
* HBOOK:                                 References and Further Reading<19>.
                                                              (line   6)
* header files, including:               An Example Program.  (line  28)
* heapsort:                              Examples<6>.         (line 111)
* HEMM, Level-3 BLAS:                    Level 3.             (line  50)
* HEMV, Level-2 BLAS:                    Level 2.             (line  85)
* HER, Level-2 BLAS:                     Level 2.             (line 140)
* HER2, Level-2 BLAS:                    Level 2.             (line 168)
* HER2K, Level-3 BLAS:                   Level 3.             (line 187)
* HERK, Level-3 BLAS:                    Level 3.             (line 144)
* Hermite functions:                     Derivatives of Hermite Polynomials.
                                                              (line  59)
* Hermite functions, derivatives:        Derivatives of Hermite Functions.
                                                              (line   6)
* Hermite functions, zeros:              Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line   6)
* Hermite polynomials:                   Hermite Polynomials and Functions.
                                                              (line  11)
* Hermite polynomials, derivatives:      Derivatives of Hermite Polynomials.
                                                              (line   6)
* Hermite polynomials, zeros:            Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line   6)
* hermitian matrix, complex, eigensystem: Complex Hermitian Matrices.
                                                              (line   9)
* Hessenberg decomposition:              Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  44)
* Hessenberg triangular decomposition:   Hessenberg Decomposition of Real Matrices.
                                                              (line  55)
* histogram statistics:                  Searching histogram ranges.
                                                              (line  23)
* histogram, from ntuple:                Histogramming ntuple values.
                                                              (line  41)
* histograms:                            References and Further Reading<18>.
                                                              (line  19)
* histograms, random sampling from:      Resampling from histograms.
                                                              (line  17)
* Householder linear solver:             Householder Transformations.
                                                              (line  59)
* Householder matrix:                    Givens Rotations.    (line  36)
* Householder transformation:            Givens Rotations.    (line  36)
* how to report:                         No Warranty.         (line  12)
* Hurwitz Zeta Function:                 Hurwitz Zeta Function.
                                                              (line   6)
* HYBRID algorithm, unscaled without derivatives: Algorithms without Derivatives.
                                                              (line  28)
* HYBRID algorithms for nonlinear systems: Algorithms using Derivatives.
                                                              (line  18)
* HYBRIDJ algorithm:                     Algorithms using Derivatives.
                                                              (line  76)
* HYBRIDS algorithm, scaled without derivatives: Algorithms without Derivatives.
                                                              (line  18)
* HYBRIDSJ algorithm:                    Algorithms using Derivatives.
                                                              (line  18)
* hydrogen atom:                         Coulomb Functions.   (line   6)
* hyperbolic cosine, inverse:            Elementary Functions.
                                                              (line  36)
* hyperbolic functions, complex numbers: Inverse Complex Trigonometric Functions.
                                                              (line  65)
* hyperbolic integrals:                  Hyperbolic Integrals.
                                                              (line   6)
* hyperbolic sine, inverse:              Elementary Functions.
                                                              (line  41)
* hyperbolic space:                      Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* hyperbolic tangent, inverse:           Elementary Functions.
                                                              (line  46)
* hypergeometric functions:              Hypergeometric Functions.
                                                              (line   6)
* hypergeometric random variates:        The Hypergeometric Distribution.
                                                              (line   6)
* hypot:                                 Elementary Functions.
                                                              (line  24)
* hypot function, special functions:     Circular Trigonometric Functions.
                                                              (line  16)
* I(x), Bessel Functions:                Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* i(x), Bessel Functions:                Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* identity matrix:                       Accessing matrix elements.
                                                              (line  48)
* identity permutation:                  Permutation allocation.
                                                              (line  24)
* IEEE exceptions:                       Representation of floating point numbers.
                                                              (line 135)
* IEEE floating point:                   References and Further Reading<39>.
                                                              (line  18)
* IEEE format for floating point numbers: IEEE floating-point arithmetic.
                                                              (line  11)
* IEEE infinity, defined as a macro:     Mathematical Constants.
                                                              (line  59)
* IEEE NaN, defined as a macro:          Infinities and Not-a-number.
                                                              (line  16)
* illumination, units of:                Viscosity.           (line  13)
* imperial units:                        Measurement of Time. (line  22)
* Implicit Euler method:                 Stepping Functions.  (line 131)
* Implicit Runge-Kutta method:           Stepping Functions.  (line 140)
* importance sampling, VEGAS:            MISER.               (line 160)
* including GSL header files:            An Example Program.  (line  29)
* incomplete Beta function, normalized:  Incomplete Beta Function.
                                                              (line   6)
* incomplete Fermi-Dirac integral:       Incomplete Fermi-Dirac Integrals.
                                                              (line   6)
* incomplete Gamma function:             Incomplete Gamma Functions.
                                                              (line  14)
* indirect sorting:                      Sorting objects.     (line  58)
* indirect sorting, of vector elements:  Sorting vectors.     (line  49)
* infinity, defined as a macro:          Mathematical Constants.
                                                              (line  60)
* infinity, IEEE format:                 Representation of floating point numbers.
                                                              (line  30)
* info-gsl mailing list:                 GSL is Free Software.
                                                              (line  45)
* initial value problems, differential equations: References and Further Reading<21>.
                                                              (line  11)
* initializing matrices:                 Accessing matrix elements.
                                                              (line  48)
* initializing vectors:                  Accessing vector elements.
                                                              (line  75)
* inline functions:                      ANSI C Compliance.   (line  25)
* integer powers:                        Power Function.      (line   6)
* integrals, exponential:                Exponential Integrals.
                                                              (line   6)
* integration, numerical (quadrature):   References and Further Reading<11>.
                                                              (line  67)
* interpolating quadrature:              Fixed point quadratures.
                                                              (line   6)
* interpolation:                         References and Further Reading<22>.
                                                              (line  47)
* interpolation, using Chebyshev polynomials: References and Further Reading<24>.
                                                              (line  14)
* inverse complex trigonometric functions: Complex Trigonometric Functions.
                                                              (line  35)
* inverse cumulative distribution functions: References.      (line  14)
* inverse hyperbolic cosine:             Elementary Functions.
                                                              (line  36)
* inverse hyperbolic functions, complex numbers: Complex Hyperbolic Functions.
                                                              (line  35)
* inverse hyperbolic sine:               Elementary Functions.
                                                              (line  41)
* inverse hyperbolic tangent:            Elementary Functions.
                                                              (line  46)
* inverse of a matrix, by LU decomposition: LU Decomposition. (line  77)
* inverting a permutation:               Permutation functions.
                                                              (line  11)
* Irregular Cylindrical Bessel Functions: Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* Irregular Modified Bessel Functions, Fractional Order: Irregular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* Irregular Modified Cylindrical Bessel Functions: Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Irregular Modified Spherical Bessel Functions: Irregular Modified Spherical Bessel Functions.
                                                              (line   6)
* Irregular Spherical Bessel Functions:  Irregular Spherical Bessel Functions.
                                                              (line   6)
* iterating through combinations:        Combination functions.
                                                              (line   6)
* iterating through multisets:           Multiset functions.  (line   6)
* iterating through permutations:        Permutation functions.
                                                              (line  17)
* iterative refinement of solutions in linear systems: LU Decomposition.
                                                              (line  64)
* J(x), Bessel Functions:                Regular Cylindrical Bessel Functions.
                                                              (line   6)
* j(x), Bessel Functions:                Regular Spherical Bessel Functions.
                                                              (line   6)
* Jacobi elliptic functions:             Elliptic Functions Jacobi.
                                                              (line   6)
* Jacobi orthogonalization:              Singular Value Decomposition.
                                                              (line  61)
* Jacobian matrix, ODEs:                 Defining the ODE System.
                                                              (line  42)
* Jacobian matrix, root finding:         Overview<3>.         (line  41)
* K(x), Bessel Functions:                Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* k(x), Bessel Functions:                Irregular Modified Spherical Bessel Functions.
                                                              (line   6)
* knots, basis splines:                  Initializing the B-splines solver.
                                                              (line  24)
* kurtosis:                              Absolute deviation.  (line  34)
* Laguerre functions:                    Laguerre Functions.  (line   6)
* Lambert function:                      Lambert W Functions. (line   6)
* Landau distribution:                   The Landau Distribution.
                                                              (line   6)
* LAPACK:                                References and Further Reading<10>.
                                                              (line  18)
* Laplace distribution:                  The Laplace Distribution.
                                                              (line   6)
* large dense linear least squares:      Robust linear regression.
                                                              (line 335)
* large linear least squares, normal equations: Large dense linear systems.
                                                              (line  46)
* large linear least squares, routines:  Large Dense Linear Systems Solution Steps.
                                                              (line  30)
* large linear least squares, steps:     Tall Skinny QR TSQR Approach.
                                                              (line  38)
* large linear least squares, TSQR:      Normal Equations Approach.
                                                              (line  30)
* ldexp:                                 Elementary Functions.
                                                              (line  51)
* LDL decomposition:                     Modified Cholesky Decomposition.
                                                              (line  69)
* LDLT decomposition:                    Modified Cholesky Decomposition.
                                                              (line  69)
* LDLT decomposition, banded:            Banded Cholesky Decomposition.
                                                              (line  98)
* LD_LIBRARY_PATH:                       Linking with an alternative BLAS library.
                                                              (line  25)
* leading dimension, matrices:           Example programs for vectors.
                                                              (line 100)
* least squares fit:                     References and Further Reading<32>.
                                                              (line  24)
* least squares troubleshooting:         Large Dense Linear Least Squares Routines.
                                                              (line 204)
* least squares, covariance of best-fit parameters: High Level Driver.
                                                              (line  40)
* least squares, nonlinear:              References and Further Reading<33>.
                                                              (line  55)
* least squares, regularized:            Multi-parameter regression.
                                                              (line 182)
* least squares, robust:                 Regularized regression.
                                                              (line 411)
* Legendre forms of elliptic integrals:  Definition of Legendre Forms.
                                                              (line   6)
* Legendre functions:                    Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* Legendre polynomials:                  Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* length, computed accurately using hypot: Elementary Functions.
                                                              (line  24)
* length, computed accurately using hypot3: Elementary Functions.
                                                              (line  30)
* Levenberg-Marquardt algorithm:         Solving the Trust Region Subproblem TRS.
                                                              (line  29)
* Levenberg-Marquardt algorithm, geodesic acceleration: Levenberg-Marquardt.
                                                              (line  34)
* Levin u-transform:                     References and Further Reading<25>.
                                                              (line  12)
* Levy distribution:                     The Levy alpha-Stable Distributions.
                                                              (line   6)
* Levy distribution, skew:               The Levy skew alpha-Stable Distribution.
                                                              (line   6)
* libraries, linking with:               Compiling and Linking.
                                                              (line  24)
* libraries, shared:                     Linking with an alternative BLAS library.
                                                              (line  26)
* license of GSL:                        Top.                 (line  12)
* light, units of:                       Viscosity.           (line  14)
* linear algebra:                        References and Further Reading<8>.
                                                              (line  33)
* linear algebra, BLAS:                  References and Further Reading<7>.
                                                              (line  15)
* linear algebra, sparse:                References and Further Reading<37>.
                                                              (line  13)
* linear interpolation:                  1D Interpolation Types.
                                                              (line  10)
* linear least squares, large:           Robust linear regression.
                                                              (line 334)
* linear regression:                     Overview<5>.         (line  60)
* linear systems:                        LU Decomposition.    (line  43)
* linear systems, refinement of solutions: LU Decomposition.  (line  64)
* linking with GSL libraries:            Compiling and Linking.
                                                              (line  24)
* location estimation:                   Order Statistics.    (line  23)
* log1p:                                 Elementary Functions.
                                                              (line  12)
* logarithm and related functions:       Logarithm and Related Functions.
                                                              (line   6)
* logarithm of Beta function:            Beta Functions.      (line  14)
* logarithm of combinatorial factor C(m: Factorials.          (line  50)
* logarithm of complex number:           Elementary Complex Functions.
                                                              (line  33)
* logarithm of cosh function, special functions: Hyperbolic Trigonometric Functions.
                                                              (line  11)
* logarithm of double factorial:         Factorials.          (line  36)
* logarithm of factorial:                Factorials.          (line  27)
* logarithm of Gamma function:           Gamma Functions.     (line  23)
* logarithm of Pochhammer symbol:        Pochhammer Symbol.   (line  16)
* logarithm of sinh function:            Hyperbolic Trigonometric Functions.
                                                              (line   6)
* logarithm of the determinant of a matrix: LU Decomposition. (line 120)
* logarithm, computed accurately near 1: Elementary Functions.
                                                              (line  12)
* Logarithmic random variates:           The Hypergeometric Distribution.
                                                              (line  39)
* Logistic distribution:                 The Logistic Distribution.
                                                              (line   6)
* Lognormal distribution:                The Lognormal Distribution.
                                                              (line   6)
* low discrepancy sequences:             Acknowledgements.    (line  12)
* Low-level CBLAS:                       Autoconf Macros.     (line 107)
* LQ decomposition:                      QR Decomposition with Column Pivoting.
                                                              (line 167)
* LU decomposition:                      Linear Algebra.      (line  15)
* LU decomposition, banded:              Symmetric Banded Format.
                                                              (line  36)
* macros for mathematical constants:     Mathematical Functions.
                                                              (line  14)
* magnitude of complex number:           Properties of complex numbers.
                                                              (line  11)
* mailing list:                          No Warranty.         (line  12)
* mailing list archives:                 Reporting Bugs.      (line  31)
* mailing list for GSL announcements:    GSL is Free Software.
                                                              (line  46)
* mantissa, IEEE format:                 IEEE floating-point arithmetic.
                                                              (line  11)
* mass, units of:                        Volume Area and Length.
                                                              (line  42)
* mathematical constants, defined as macros: Mathematical Functions.
                                                              (line  14)
* mathematical functions, elementary:    Examples.            (line  38)
* Mathieu Function Characteristic Values: Mathieu Function Characteristic Values.
                                                              (line   6)
* Mathieu functions:                     Mathieu Functions.   (line   6)
* matrices:                              Example programs for vectors.
                                                              (line 100)
* matrices, banded:                      Triangular Systems.  (line  43)
* matrices, initializing:                Accessing matrix elements.
                                                              (line  48)
* matrices, range-checking:              Matrix allocation.   (line  40)
* matrices, sparse:                      References and Further Reading<35>.
                                                              (line  22)
* matrix determinant:                    LU Decomposition.    (line 111)
* matrix diagonal:                       Creating row and column views.
                                                              (line  70)
* matrix factorization:                  References and Further Reading<8>.
                                                              (line  33)
* matrix inverse:                        LU Decomposition.    (line  77)
* matrix square root, Cholesky decomposition: Singular Value Decomposition.
                                                              (line  96)
* matrix subdiagonal:                    Creating row and column views.
                                                              (line  84)
* matrix superdiagonal:                  Creating row and column views.
                                                              (line  98)
* matrix, constant:                      Accessing matrix elements.
                                                              (line  47)
* matrix, identity:                      Accessing matrix elements.
                                                              (line  48)
* matrix, operations:                    References and Further Reading<7>.
                                                              (line  15)
* matrix, zero:                          Accessing matrix elements.
                                                              (line  48)
* max:                                   References and Further Reading<14>.
                                                              (line  51)
* maximal phase, Daubechies wavelets:    Initialization.      (line  24)
* maximization, see minimization:        References and Further Reading<29>.
                                                              (line  17)
* maximum of two numbers:                Maximum and Minimum functions.
                                                              (line  10)
* maximum value, from histogram:         Searching histogram ranges.
                                                              (line  23)
* mean:                                  References and Further Reading<14>.
                                                              (line  52)
* mean value, from histogram:            Histogram Statistics.
                                                              (line  28)
* mean, trimmed:                         Robust Location Estimates.
                                                              (line  16)
* mean, truncated:                       Robust Location Estimates.
                                                              (line  16)
* median absolute deviation:             Robust Scale Estimates.
                                                              (line  12)
* Mills’ ratio, inverse:                 Probability functions.
                                                              (line  21)
* min:                                   References and Further Reading<14>.
                                                              (line  52)
* minimization, BFGS algorithm:          Algorithms with Derivatives.
                                                              (line  45)
* minimization, caveats:                 Overview<2>.         (line  45)
* minimization, conjugate gradient algorithm: Algorithms with Derivatives.
                                                              (line  15)
* minimization, multidimensional:        References and Further Reading<31>.
                                                              (line  29)
* minimization, one-dimensional:         References and Further Reading<29>.
                                                              (line  17)
* minimization, overview:                One Dimensional Minimization.
                                                              (line  20)
* minimization, Polak-Ribiere algorithm: Algorithms with Derivatives.
                                                              (line  35)
* minimization, providing a function to minimize: Initializing the Minimizer.
                                                              (line  60)
* minimization, simplex algorithm:       Algorithms without Derivatives<2>.
                                                              (line  14)
* minimization, steepest descent algorithm: Algorithms with Derivatives.
                                                              (line  69)
* minimization, stopping parameters:     Iteration<2>.        (line  56)
* minimum finding, Brent’s method:       Minimization Algorithms.
                                                              (line  34)
* minimum finding, golden section algorithm: Minimization Algorithms.
                                                              (line  15)
* minimum of two numbers:                Maximum and Minimum functions.
                                                              (line  15)
* minimum value, from histogram:         Searching histogram ranges.
                                                              (line  22)
* MINPACK, minimization algorithms:      Algorithms using Derivatives.
                                                              (line  18)
* MISCFUN:                               References and Further Reading<3>.
                                                              (line  15)
* MISER monte carlo integration:         PLAIN Monte Carlo.   (line  66)
* Mixed-radix FFT, complex data:         Radix-2 FFT routines for complex data.
                                                              (line 119)
* Mixed-radix FFT, real data:            Radix-2 FFT routines for real data.
                                                              (line  97)
* Modified Bessel Functions, Fractional Order: Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* Modified Cholesky Decomposition:       Pivoted Cholesky Decomposition.
                                                              (line 110)
* Modified Clenshaw-Curtis quadrature:   Integrands with weight functions.
                                                              (line   6)
* Modified Cylindrical Bessel Functions: Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Modified Givens Rotation, BLAS:        Level 1.             (line 152)
* Modified Newton’s method for nonlinear systems: Algorithms using Derivatives.
                                                              (line 105)
* Modified Spherical Bessel Functions:   Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* Monte Carlo integration:               References and Further Reading<19>.
                                                              (line   9)
* moving maximum:                        Moving Variance and Standard Deviation.
                                                              (line  32)
* moving mean:                           Allocation for Moving Window Statistics.
                                                              (line  30)
* moving median:                         Moving Sum.          (line  20)
* moving median absolute deviation:      Robust Scale Estimation.
                                                              (line  17)
* moving minimum:                        Moving Variance and Standard Deviation.
                                                              (line  32)
* moving quantile range:                 Moving MAD.          (line  31)
* moving standard deviation:             Moving Mean.         (line  25)
* moving sum:                            Moving Minimum and Maximum.
                                                              (line  40)
* moving variance:                       Moving Mean.         (line  25)
* moving window accumulators:            Moving Q_n.          (line  20)
* moving window statistics:              References and Further Reading<16>.
                                                              (line  14)
* moving window, allocation:             Handling Endpoints.  (line  40)
* MRG, multiple recursive random number generator: Random number generator algorithms.
                                                              (line 126)
* MT19937 random number generator:       Random number generator algorithms.
                                                              (line  19)
* multi-parameter regression:            Linear regression without a constant term.
                                                              (line  51)
* multidimensional integration:          References and Further Reading<19>.
                                                              (line   8)
* multidimensional root finding, Broyden algorithm: Algorithms without Derivatives.
                                                              (line  57)
* multidimensional root finding, overview: Multidimensional Root-Finding.
                                                              (line  20)
* multidimensional root finding, providing a function to solve: Initializing the Solver<2>.
                                                              (line  83)
* Multimin, caveats:                     Overview<4>.         (line  47)
* Multinomial distribution:              The Multinomial Distribution.
                                                              (line   6)
* multiplication:                        Elementary Operations.
                                                              (line   6)
* multisets:                             References and Further Reading<6>.
                                                              (line  11)
* multistep methods, ODEs:               Stepping Functions.  (line 164)
* n):                                    Factorials.          (line  43)
* n) <1>:                                Factorials.          (line  50)
* N-dimensional random direction vector: Spherical Vector Distributions.
                                                              (line  44)
* NaN, defined as a macro:               Infinities and Not-a-number.
                                                              (line  16)
* nautical units:                        Imperial Units.      (line  25)
* Negative Binomial distribution, random variates: The Negative Binomial Distribution.
                                                              (line   6)
* Nelder-Mead simplex algorithm for minimization: Algorithms without Derivatives<2>.
                                                              (line  14)
* Newton algorithm, discrete:            Algorithms without Derivatives.
                                                              (line  34)
* Newton algorithm, globally convergent: Algorithms using Derivatives.
                                                              (line 105)
* Newton’s method for finding roots:     Root Finding Algorithms using Derivatives.
                                                              (line  17)
* Newton’s method for systems of nonlinear equations: Algorithms using Derivatives.
                                                              (line  84)
* Niederreiter sequence:                 Acknowledgements.    (line  11)
* NIST Statistical Reference Datasets:   References and Further Reading<33>.
                                                              (line  16)
* non-normalized incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line   6)
* nonlinear equation, solutions of:      References and Further Reading<28>.
                                                              (line  14)
* nonlinear fitting, stopping parameters, convergence: Iteration<5>.
                                                              (line 120)
* nonlinear functions, minimization:     References and Further Reading<29>.
                                                              (line  16)
* nonlinear least squares:               References and Further Reading<33>.
                                                              (line  56)
* nonlinear least squares, dogleg:       Levenberg-Marquardt with Geodesic Acceleration.
                                                              (line  29)
* nonlinear least squares, double dogleg: Dogleg.             (line  26)
* nonlinear least squares, levenberg-marquardt: Solving the Trust Region Subproblem TRS.
                                                              (line  29)
* nonlinear least squares, levenberg-marquardt, geodesic acceleration: Levenberg-Marquardt.
                                                              (line  33)
* nonlinear least squares, overview:     Nonlinear Least-Squares Fitting.
                                                              (line  27)
* nonlinear systems of equations, solution of: References and Further Reading<30>.
                                                              (line  13)
* nonsymmetric matrix, real, eigensystem: Real Nonsymmetric Matrices.
                                                              (line   6)
* Nordsieck form:                        Stepping Functions.  (line 164)
* normalized form, IEEE format:          Representation of floating point numbers.
                                                              (line  14)
* normalized incomplete Beta function:   Incomplete Beta Function.
                                                              (line   6)
* Not-a-number, defined as a macro:      Infinities and Not-a-number.
                                                              (line  16)
* NRM2, Level-1 BLAS:                    Level 1.             (line  42)
* ntuples:                               Example programs for 2D histograms.
                                                              (line  72)
* nuclear physics, constants:            Astronomy and Astrophysics.
                                                              (line  29)
* numerical constants, defined as macros: Mathematical Functions.
                                                              (line  14)
* numerical derivatives:                 References and Further Reading<23>.
                                                              (line  18)
* numerical integration (quadrature):    References and Further Reading<11>.
                                                              (line  67)
* obtaining GSL:                         GSL is Free Software.
                                                              (line  46)
* ODEs, initial value problems:          References and Further Reading<21>.
                                                              (line  10)
* online statistics:                     References and Further Reading<15>.
                                                              (line  39)
* online statistics <1>:                 References and Further Reading<16>.
                                                              (line  13)
* optimization, combinatorial:           References and Further Reading<20>.
                                                              (line  21)
* optimization, see minimization:        References and Further Reading<29>.
                                                              (line  17)
* optimized functions, alternatives:     Portability functions.
                                                              (line  33)
* ordering, matrix elements:             Example programs for vectors.
                                                              (line  99)
* ordinary differential equations, initial value problem: References and Further Reading<21>.
                                                              (line  11)
* oscillatory functions, numerical integration of: QAWO adaptive integration for oscillatory functions.
                                                              (line   6)
* overflow, IEEE exceptions:             Representation of floating point numbers.
                                                              (line 134)
* Pareto distribution:                   The Pareto Distribution.
                                                              (line   6)
* PAW:                                   References and Further Reading<19>.
                                                              (line   6)
* permutations:                          References and Further Reading<4>.
                                                              (line  12)
* physical constants:                    References and Further Reading<38>.
                                                              (line  15)
* physical dimension, matrices:          Example programs for vectors.
                                                              (line 100)
* pi, defined as a macro:                Mathematical Constants.
                                                              (line   9)
* Pivoted Cholesky Decomposition:        Cholesky Decomposition.
                                                              (line 154)
* plain Monte Carlo:                     Interface.           (line  80)
* Pochhammer symbol:                     Pochhammer Symbol.   (line   6)
* Poisson random numbers:                The Poisson Distribution.
                                                              (line   6)
* Polak-Ribiere algorithm, minimization: Algorithms with Derivatives.
                                                              (line  35)
* polar form of complex numbers:         Complex Numbers.     (line  21)
* polar to rectangular conversion:       Conversion Functions.
                                                              (line   6)
* polygamma functions:                   Psi Digamma Function.
                                                              (line   6)
* polynomial evaluation:                 Polynomials.         (line  13)
* polynomial interpolation:              1D Interpolation Types.
                                                              (line  15)
* polynomials, roots of:                 References and Further Reading.
                                                              (line  32)
* power function:                        Power Function.      (line   6)
* power of complex number:               Elementary Complex Functions.
                                                              (line  17)
* power, units of:                       Mass and Weight.     (line  50)
* precision, IEEE arithmetic:            Representation of floating point numbers.
                                                              (line 135)
* predictor-corrector method, ODEs:      Stepping Functions.  (line 164)
* prefixes:                              Force and Energy.    (line  22)
* pressure:                              Thermal Energy and Power.
                                                              (line  22)
* Prince-Dormand, Runge-Kutta method:    Stepping Functions.  (line 126)
* printers units:                        Speed and Nautical Units.
                                                              (line  25)
* probability distribution, from histogram: Resampling from histograms.
                                                              (line  18)
* probability distributions, from histograms: Reading and writing histograms.
                                                              (line  67)
* projection of ntuples:                 Histogramming ntuple values.
                                                              (line  41)
* psi function:                          Psi Digamma Function.
                                                              (line   6)
* QAG quadrature algorithm:              QAG adaptive integration.
                                                              (line   6)
* QAGI quadrature algorithm:             QAGI adaptive integration on infinite intervals.
                                                              (line   6)
* QAGP quadrature algorithm:             QAGP adaptive integration with known singular points.
                                                              (line   6)
* QAGS quadrature algorithm:             QAGS adaptive integration with singularities.
                                                              (line   6)
* QAWC quadrature algorithm:             QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* QAWF quadrature algorithm:             QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* QAWO quadrature algorithm:             QAWO adaptive integration for oscillatory functions.
                                                              (line   6)
* QAWS quadrature algorithm:             QAWS adaptive integration for singular functions.
                                                              (line   6)
* QL decomposition:                      LQ Decomposition.    (line  65)
* Qn statistic:                          S_n Statistic.       (line  35)
* QNG quadrature algorithm:              QNG non-adaptive Gauss-Kronrod integration.
                                                              (line   6)
* QR decomposition:                      LU Decomposition.    (line 136)
* QR decomposition with column pivoting: Triangle on Top of Diagonal.
                                                              (line  54)
* QUADPACK:                              References and Further Reading<11>.
                                                              (line  66)
* quadratic equation, solving:           Divided Difference Representation of Polynomials.
                                                              (line  81)
* quadrature:                            References and Further Reading<11>.
                                                              (line  67)
* quadrature, fixed point:               Fixed point quadratures.
                                                              (line   6)
* quadrature, interpolating:             Fixed point quadratures.
                                                              (line   6)
* quantile functions:                    References.          (line  13)
* quasi-random sequences:                Acknowledgements.    (line  12)
* R250 shift-register random number generator: Other random number generators.
                                                              (line  62)
* Racah coefficients:                    Coupling Coefficients.
                                                              (line   6)
* Radial Mathieu Functions:              Radial Mathieu Functions.
                                                              (line   6)
* radioactivity:                         Light and Illumination.
                                                              (line  34)
* Radix-2 FFT for real data:             Overview of real data FFTs.
                                                              (line  49)
* Radix-2 FFT, complex data:             Overview of complex data FFTs.
                                                              (line  72)
* rand, BSD random number generator:     Unix random number generators.
                                                              (line  17)
* rand48 random number generator:        Unix random number generators.
                                                              (line  59)
* random number distributions:           References.          (line  14)
* random number generators:              References and Further Reading<12>.
                                                              (line  33)
* random sampling from histograms:       Resampling from histograms.
                                                              (line  18)
* RANDU random number generator:         Other random number generators.
                                                              (line 113)
* RANF random number generator:          Other random number generators.
                                                              (line  22)
* range:                                 References and Further Reading<14>.
                                                              (line  52)
* range-checking for matrices:           Matrix allocation.   (line  39)
* range-checking for vectors:            Vector allocation.   (line  36)
* RANLUX random number generator:        Random number generator algorithms.
                                                              (line  74)
* RANLXD random number generator:        Random number generator algorithms.
                                                              (line  67)
* RANLXS random number generator:        Random number generator algorithms.
                                                              (line  47)
* RANMAR random number generator:        Other random number generators.
                                                              (line  55)
* RANMAR random number generator <1>:    Other random number generators.
                                                              (line 123)
* Rayleigh distribution:                 The Rayleigh Distribution.
                                                              (line   6)
* Rayleigh Tail distribution:            The Rayleigh Tail Distribution.
                                                              (line   6)
* real nonsymmetric matrix, eigensystem: Real Nonsymmetric Matrices.
                                                              (line   6)
* real symmetric matrix, eigensystem:    Real Symmetric Matrices.
                                                              (line   6)
* Reciprocal Gamma function:             Gamma Functions.     (line  54)
* rectangular to polar conversion:       Conversion Functions.
                                                              (line   6)
* recursive stratified sampling, MISER:  PLAIN Monte Carlo.   (line  65)
* reduction of angular variables:        Restriction Functions.
                                                              (line   6)
* refinement of solutions in linear systems: LU Decomposition.
                                                              (line  64)
* regression, least squares:             References and Further Reading<32>.
                                                              (line  24)
* regression, ridge:                     Multi-parameter regression.
                                                              (line 183)
* regression, robust:                    Regularized regression.
                                                              (line 412)
* regression, Tikhonov:                  Multi-parameter regression.
                                                              (line 183)
* Regular Bessel Functions, Fractional Order: Regular Bessel Function—Fractional Order.
                                                              (line   6)
* Regular Bessel Functions, Zeros of:    Zeros of Regular Bessel Functions.
                                                              (line   6)
* Regular Cylindrical Bessel Functions:  Regular Cylindrical Bessel Functions.
                                                              (line   6)
* Regular Modified Bessel Functions, Fractional Order: Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* Regular Modified Cylindrical Bessel Functions: Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Regular Modified Spherical Bessel Functions: Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* Regular Spherical Bessel Functions:    Regular Spherical Bessel Functions.
                                                              (line   6)
* Regulated Gamma function:              Gamma Functions.     (line  43)
* relative Pochhammer symbol:            Pochhammer Symbol.   (line  31)
* reporting bugs in GSL:                 No Warranty.         (line  12)
* representations of complex numbers:    Complex Numbers.     (line  21)
* resampling from histograms:            Reading and writing histograms.
                                                              (line  68)
* residual, in nonlinear systems of equations: Search Stopping Parameters<2>.
                                                              (line  34)
* reversing a permutation:               Permutation functions.
                                                              (line   6)
* ridge regression:                      Multi-parameter regression.
                                                              (line 183)
* Riemann Zeta Function:                 Riemann Zeta Function.
                                                              (line   6)
* RK2, Runge-Kutta method:               Stepping Functions.  (line 104)
* RK4, Runge-Kutta method:               Stepping Functions.  (line 108)
* RKF45, Runge-Kutta-Fehlberg method:    Stepping Functions.  (line 115)
* robust location estimators:            Order Statistics.    (line  23)
* robust regression:                     Regularized regression.
                                                              (line 412)
* robust scale estimators:               Gastwirth Estimator. (line  25)
* rolling maximum:                       Moving Variance and Standard Deviation.
                                                              (line  31)
* rolling mean:                          Allocation for Moving Window Statistics.
                                                              (line  29)
* rolling median:                        Moving Sum.          (line  19)
* rolling median absolute deviation:     Robust Scale Estimation.
                                                              (line  17)
* rolling minimum:                       Moving Variance and Standard Deviation.
                                                              (line  32)
* rolling quantile range:                Moving MAD.          (line  30)
* rolling standard deviation:            Moving Mean.         (line  24)
* rolling sum:                           Moving Minimum and Maximum.
                                                              (line  39)
* rolling variance:                      Moving Mean.         (line  25)
* rolling window accumulators:           Moving Q_n.          (line  19)
* root finding:                          References and Further Reading<28>.
                                                              (line  15)
* root finding, bisection algorithm:     Root Bracketing Algorithms.
                                                              (line  18)
* root finding, Brent’s method:          Root Bracketing Algorithms.
                                                              (line  54)
* root finding, caveats:                 Overview.            (line  39)
* root finding, false position algorithm: Root Bracketing Algorithms.
                                                              (line  35)
* root finding, initial guess:           Providing the function to solve.
                                                              (line 121)
* root finding, Newton’s method:         Root Finding Algorithms using Derivatives.
                                                              (line  17)
* root finding, overview:                One Dimensional Root-Finding.
                                                              (line  19)
* root finding, providing a function to solve: Initializing the Solver.
                                                              (line  75)
* root finding, search bounds:           Providing the function to solve.
                                                              (line 122)
* root finding, secant method:           Root Finding Algorithms using Derivatives.
                                                              (line  32)
* root finding, Steffenson’s method:     Root Finding Algorithms using Derivatives.
                                                              (line  64)
* root finding, stopping parameters:     Iteration.           (line  49)
* root finding, stopping parameters <1>: Iteration<3>.        (line  58)
* roots:                                 References and Further Reading<28>.
                                                              (line  15)
* ROTG, Level-1 BLAS:                    Level 1.             (line 130)
* rounding mode:                         Representation of floating point numbers.
                                                              (line 135)
* Runge-Kutta Cash-Karp method:          Stepping Functions.  (line 121)
* Runge-Kutta methods, ordinary differential equations: Stepping Functions.
                                                              (line 104)
* Runge-Kutta Prince-Dormand method:     Stepping Functions.  (line 126)
* running statistics:                    References and Further Reading<15>.
                                                              (line  40)
* safe comparison of floating point numbers: Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* safeguarded step-length algorithm:     Minimization Algorithms.
                                                              (line  52)
* sampling from histograms:              Reading and writing histograms.
                                                              (line  68)
* sampling from histograms <1>:          Resampling from histograms.
                                                              (line  18)
* SAXPY, Level-1 BLAS:                   Level 1.             (line 105)
* SCAL, Level-1 BLAS:                    Level 1.             (line 117)
* scale estimation:                      Gastwirth Estimator. (line  25)
* schedule, cooling:                     Simulated Annealing algorithm.
                                                              (line  20)
* se(q,x), Mathieu function:             Angular Mathieu Functions.
                                                              (line   6)
* secant method for finding roots:       Root Finding Algorithms using Derivatives.
                                                              (line  32)
* selection function, ntuples:           Histogramming ntuple values.
                                                              (line  13)
* series, acceleration:                  References and Further Reading<25>.
                                                              (line  12)
* shared libraries:                      Linking with an alternative BLAS library.
                                                              (line  26)
* shell prompt:                          Further Information. (line  27)
* Shi(x):                                Hyperbolic Integrals.
                                                              (line   6)
* shift-register random number generator: Other random number generators.
                                                              (line  62)
* Si(x):                                 Trigonometric Integrals.
                                                              (line   6)
* sign bit, IEEE format:                 IEEE floating-point arithmetic.
                                                              (line  11)
* sign of the determinant of a matrix:   LU Decomposition.    (line 129)
* simplex algorithm, minimization:       Algorithms without Derivatives<2>.
                                                              (line  14)
* simulated annealing:                   References and Further Reading<20>.
                                                              (line  21)
* sin, of complex number:                Complex Trigonometric Functions.
                                                              (line   6)
* sine function, special functions:      Circular Trigonometric Functions.
                                                              (line   6)
* single precision, IEEE format:         Representation of floating point numbers.
                                                              (line  34)
* singular functions, numerical integration of: QAWS adaptive integration for singular functions.
                                                              (line   6)
* singular points, specifying positions in quadrature: QAGP adaptive integration with known singular points.
                                                              (line   6)
* singular value decomposition:          Complete Orthogonal Decomposition.
                                                              (line 122)
* Skew Levy distribution:                The Levy skew alpha-Stable Distribution.
                                                              (line   6)
* skewness:                              Absolute deviation.  (line  35)
* slope, see numerical derivative:       References and Further Reading<23>.
                                                              (line  17)
* Sn statistic:                          Median Absolute Deviation MAD.
                                                              (line  30)
* Sobol sequence:                        Acknowledgements.    (line  12)
* solution of:                           LU Decomposition.    (line  43)
* solution of linear system by Householder transformations: Householder Transformations.
                                                              (line  60)
* solution of linear systems, Ax=b:      References and Further Reading<8>.
                                                              (line  33)
* solving a nonlinear equation:          References and Further Reading<28>.
                                                              (line  15)
* solving nonlinear systems of equations: References and Further Reading<30>.
                                                              (line  13)
* sorting:                               Examples<6>.         (line 112)
* sorting eigenvalues and eigenvectors:  Sorting Eigenvalues and Eigenvectors.
                                                              (line   6)
* sorting vector elements:               Sorting vectors.     (line  22)
* source code, reuse in applications:    Deprecated Functions.
                                                              (line  13)
* sparse BLAS:                           References and Further Reading<36>.
                                                              (line  14)
* sparse BLAS, references:               Sparse BLAS operations.
                                                              (line  22)
* sparse linear algebra:                 References and Further Reading<37>.
                                                              (line  14)
* sparse linear algebra, examples:       Iterating the Sparse Linear System.
                                                              (line  59)
* sparse linear algebra, iterative solvers: Overview<9>.      (line  20)
* sparse linear algebra, overview:       Sparse Linear Algebra.
                                                              (line  14)
* sparse linear algebra, references:     Examples<35>.        (line 136)
* sparse matrices:                       References and Further Reading<35>.
                                                              (line  23)
* sparse matrices, accessing elements:   Allocation.          (line  80)
* sparse matrices, allocation:           Overview<8>.         (line  61)
* sparse matrices, BLAS operations:      Sparse BLAS Support. (line  14)
* sparse matrices, compressed column storage: Coordinate Storage COO.
                                                              (line  39)
* sparse matrices, compressed row storage: Compressed Sparse Column CSC.
                                                              (line  27)
* sparse matrices, compressed sparse column: Coordinate Storage COO.
                                                              (line  40)
* sparse matrices, compressed sparse row: Compressed Sparse Column CSC.
                                                              (line  28)
* sparse matrices, compression:          Finding Maximum and Minimum Elements.
                                                              (line  27)
* sparse matrices, conversion:           Compressed Format.   (line  38)
* sparse matrices, coordinate format:    Sparse Matrix Storage Formats.
                                                              (line  18)
* sparse matrices, copying:              Reading and Writing Matrices.
                                                              (line  55)
* sparse matrices, data types:           Sparse Matrices.     (line  18)
* sparse matrices, examples:             Conversion Between Sparse and Dense Matrices.
                                                              (line  26)
* sparse matrices, exchanging rows and columns: Copying Matrices.
                                                              (line  15)
* sparse matrices, initializing elements: Accessing Matrix Elements.
                                                              (line  33)
* sparse matrices, iterative solvers:    Overview<9>.         (line  20)
* sparse matrices, min/max elements:     Matrix Properties.   (line  48)
* sparse matrices, operations:           Exchanging Rows and Columns.
                                                              (line  30)
* sparse matrices, overview:             Compressed Sparse Row CSR.
                                                              (line  26)
* sparse matrices, properties:           Matrix Operations.   (line  76)
* sparse matrices, reading:              Initializing Matrix Elements.
                                                              (line  21)
* sparse matrices, references:           Examples<34>.        (line 133)
* sparse matrices, storage formats:      Data types<2>.       (line  56)
* sparse matrices, triplet format:       Sparse Matrix Storage Formats.
                                                              (line  18)
* sparse matrices, writing:              Initializing Matrix Elements.
                                                              (line  20)
* sparse, iterative solvers:             Overview<9>.         (line  19)
* special functions:                     References and Further Reading<2>.
                                                              (line  28)
* special functions <1>:                 Hyperbolic Trigonometric Functions.
                                                              (line   6)
* Spherical Bessel Functions:            Regular Spherical Bessel Functions.
                                                              (line   6)
* spherical harmonics:                   Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* spherical random variates, 2D:         Spherical Vector Distributions.
                                                              (line  10)
* spherical random variates, 3D:         Spherical Vector Distributions.
                                                              (line  33)
* spherical random variates, N-dimensional: Spherical Vector Distributions.
                                                              (line  44)
* spline:                                References and Further Reading<22>.
                                                              (line  46)
* splines, basis:                        References and Further Reading<34>.
                                                              (line  37)
* square root of a matrix, Cholesky decomposition: Singular Value Decomposition.
                                                              (line  97)
* square root of complex number:         Elementary Complex Functions.
                                                              (line   6)
* standard deviation:                    References and Further Reading<14>.
                                                              (line  52)
* standard deviation, from histogram:    Histogram Statistics.
                                                              (line  35)
* standards conformance, ANSI C:         Conventions used in this manual.
                                                              (line  20)
* Statistical Reference Datasets (StRD): References and Further Reading<33>.
                                                              (line  16)
* statistics:                            References and Further Reading<14>.
                                                              (line  52)
* statistics, from histogram:            Searching histogram ranges.
                                                              (line  23)
* statistics, moving window:             References and Further Reading<16>.
                                                              (line  14)
* steepest descent algorithm, minimization: Algorithms with Derivatives.
                                                              (line  69)
* Steffenson’s method for finding roots: Root Finding Algorithms using Derivatives.
                                                              (line  64)
* stratified sampling in Monte Carlo integration: References and Further Reading<19>.
                                                              (line   9)
* stride, of vector index:               Example programs for blocks.
                                                              (line  27)
* Student t-distribution:                The t-distribution.  (line  14)
* subdiagonal, of a matrix:              Creating row and column views.
                                                              (line  84)
* summation, acceleration:               References and Further Reading<25>.
                                                              (line  12)
* superdiagonal, matrix:                 Creating row and column views.
                                                              (line  98)
* SVD:                                   Complete Orthogonal Decomposition.
                                                              (line 123)
* SWAP, Level-1 BLAS:                    Level 1.             (line  83)
* swapping permutation elements:         Accessing permutation elements.
                                                              (line  18)
* SYMM, Level-3 BLAS:                    Level 3.             (line  27)
* symmetric matrices, banded:            General Banded Format.
                                                              (line  26)
* symmetric matrix, real, eigensystem:   Real Symmetric Matrices.
                                                              (line   6)
* SYMV, Level-2 BLAS:                    Level 2.             (line  70)
* synchrotron functions:                 Synchrotron Functions.
                                                              (line   6)
* SYR, Level-2 BLAS:                     Level 2.             (line 127)
* SYR2, Level-2 BLAS:                    Level 2.             (line 154)
* SYR2K, Level-3 BLAS:                   Level 3.             (line 162)
* SYRK, Level-3 BLAS:                    Level 3.             (line 121)
* systems of equations, nonlinear:       References and Further Reading<30>.
                                                              (line  12)
* t-distribution:                        The t-distribution.  (line  14)
* t-test:                                References and Further Reading<14>.
                                                              (line  52)
* tangent of complex number:             Complex Trigonometric Functions.
                                                              (line  16)
* Tausworthe random number generator:    Random number generator algorithms.
                                                              (line 144)
* Taylor coefficients, computation of:   Factorials.          (line  57)
* testing combination for validity:      Combination properties.
                                                              (line  20)
* testing multiset for validity:         Multiset properties. (line  20)
* testing permutation for validity:      Permutation properties.
                                                              (line  15)
* thermal energy, units of:              Mass and Weight.     (line  49)
* Tikhonov regression:                   Multi-parameter regression.
                                                              (line 183)
* time units:                            Atomic and Nuclear Physics.
                                                              (line  79)
* trailing dimension, matrices:          Example programs for vectors.
                                                              (line 100)
* transformation, Householder:           Givens Rotations.    (line  35)
* transforms, Hankel:                    References and Further Reading<27>.
                                                              (line  53)
* transforms, wavelet:                   References and Further Reading<26>.
                                                              (line  24)
* transport functions:                   Transport Functions. (line   6)
* traveling salesman problem:            Trivial example.     (line 114)
* triangular systems:                    Tridiagonal Systems. (line  70)
* tridiagonal decomposition:             LDLT Decomposition.  (line  63)
* tridiagonal decomposition <1>:         Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  43)
* tridiagonal systems:                   Householder solver for linear systems.
                                                              (line  21)
* trigonometric functions:               Trigonometric Functions.
                                                              (line   6)
* trigonometric functions of complex numbers: Elementary Complex Functions.
                                                              (line  50)
* trigonometric integrals:               Trigonometric Integrals.
                                                              (line   6)
* trimmed mean:                          Robust Location Estimates.
                                                              (line  16)
* TRMM, Level-3 BLAS:                    Level 3.             (line  68)
* TRMV, Level-2 BLAS:                    Level 2.             (line  25)
* TRSM, Level-3 BLAS:                    Level 3.             (line  94)
* TRSV, Level-2 BLAS:                    Level 2.             (line  48)
* truncated mean:                        Robust Location Estimates.
                                                              (line  16)
* TSP:                                   Trivial example.     (line 115)
* TT800 random number generator:         Other random number generators.
                                                              (line  79)
* two dimensional Gaussian distribution: The Bivariate Gaussian Distribution.
                                                              (line   6)
* two dimensional Gaussian distribution <1>: The Multivariate Gaussian Distribution.
                                                              (line   6)
* two dimensional histograms:            Example programs for histograms.
                                                              (line  65)
* two-sided exponential distribution:    The Laplace Distribution.
                                                              (line   6)
* Type 1 Gumbel distribution, random variates: The Type-1 Gumbel Distribution.
                                                              (line   6)
* Type 2 Gumbel distribution:            The Type-2 Gumbel Distribution.
                                                              (line   6)
* u-transform for series:                References and Further Reading<25>.
                                                              (line  12)
* underflow, IEEE exceptions:            Representation of floating point numbers.
                                                              (line 135)
* uniform distribution:                  The Flat Uniform Distribution.
                                                              (line   6)
* units of:                              Thermal Energy and Power.
                                                              (line  21)
* units of <1>:                          Pressure.            (line  33)
* units of <2>:                          Light and Illumination.
                                                              (line  33)
* units of <3>:                          Radioactivity.       (line  17)
* units, conversion of:                  References and Further Reading<38>.
                                                              (line  14)
* units, imperial:                       Measurement of Time. (line  21)
* Unix random number generators, rand:   Unix random number generators.
                                                              (line  17)
* Unix random number generators, rand48: Unix random number generators.
                                                              (line  17)
* unnormalized incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line   6)
* unweighted linear fits:                References and Further Reading<32>.
                                                              (line  23)
* value function, ntuples:               Histogramming ntuple values.
                                                              (line  27)
* Van der Pol oscillator, example:       Examples<22>.        (line   6)
* variance:                              References and Further Reading<14>.
                                                              (line  52)
* variance, from histogram:              Histogram Statistics.
                                                              (line  35)
* variance-covariance matrix, linear fits: Overview<5>.       (line  48)
* VAX random number generator:           Other random number generators.
                                                              (line  93)
* vector, operations:                    References and Further Reading<7>.
                                                              (line  15)
* vector, sorting elements of:           Sorting vectors.     (line  22)
* vectors:                               Example programs for blocks.
                                                              (line  28)
* vectors, initializing:                 Accessing vector elements.
                                                              (line  76)
* vectors, range-checking:               Vector allocation.   (line  36)
* VEGAS Monte Carlo integration:         MISER.               (line 161)
* viscosity:                             Pressure.            (line  34)
* volume units:                          Printers Units.      (line  13)
* W function:                            Lambert W Functions. (line   6)
* warning options:                       Handling floating point exceptions.
                                                              (line  31)
* warranty (none):                       Obtaining GSL.       (line  25)
* wavelet transforms:                    References and Further Reading<26>.
                                                              (line  25)
* website, developer information:        Reporting Bugs.      (line  31)
* Weibull distribution:                  The Weibull Distribution.
                                                              (line   6)
* weight, units of:                      Volume Area and Length.
                                                              (line  41)
* weighted linear fits:                  References and Further Reading<32>.
                                                              (line  24)
* Wigner coefficients:                   Coupling Coefficients.
                                                              (line   6)
* Wishart random variates:               The Logarithmic Distribution.
                                                              (line  23)
* Y(x), Bessel Functions:                Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* y(x), Bessel Functions:                Irregular Spherical Bessel Functions.
                                                              (line   6)
* zero finding:                          References and Further Reading<28>.
                                                              (line  15)
* zero matrix:                           Accessing matrix elements.
                                                              (line  48)
* zero, IEEE format:                     Representation of floating point numbers.
                                                              (line  30)
* Zeros of Regular Bessel Functions:     Zeros of Regular Bessel Functions.
                                                              (line   6)
* Zeta functions:                        Zeta Functions.      (line   6)
* Ziggurat method:                       The Gaussian Distribution.
                                                              (line  27)



Tag Table:
Node: Top324
Ref: index doc527
Ref: 0527
Node: Introduction26393
Ref: intro doc26484
Ref: 126484
Ref: intro gnu-scientific-library26484
Ref: 226484
Ref: intro introduction26484
Ref: 326484
Node: Routines available in GSL27045
Ref: intro routines-available-in-gsl27149
Ref: 427149
Node: GSL is Free Software30186
Ref: intro gsl-is-free-software30312
Ref: 530312
Ref: GSL is Free Software-Footnote-132525
Ref: GSL is Free Software-Footnote-232643
Node: Obtaining GSL32692
Ref: intro obtaining-gsl32804
Ref: 632804
Node: No Warranty33716
Ref: intro no-warranty33822
Ref: 733822
Ref: No Warranty-Footnote-134228
Node: Reporting Bugs34346
Ref: intro reporting-bugs34458
Ref: 834458
Ref: Reporting Bugs-Footnote-135302
Node: Further Information35360
Ref: intro further-information35492
Ref: 935492
Node: Conventions used in this manual36377
Ref: intro conventions-used-in-this-manual36486
Ref: a36486
Node: Using the Library37140
Ref: usage doc37242
Ref: b37242
Ref: usage using-the-library37242
Ref: c37242
Node: An Example Program37726
Ref: usage an-example-program37829
Ref: d37829
Ref: An Example Program-Footnote-138470
Node: Compiling and Linking38581
Ref: usage compiling-and-linking38709
Ref: e38709
Node: Linking programs with the library39690
Ref: usage linking-programs-with-the-library39831
Ref: f39831
Ref: Linking programs with the library-Footnote-141124
Ref: Linking programs with the library-Footnote-241164
Node: Linking with an alternative BLAS library41225
Ref: usage linking-with-an-alternative-blas-library41366
Ref: 1041366
Node: Shared Libraries42319
Ref: usage shared-libraries42446
Ref: 1242446
Ref: Shared Libraries-Footnote-143811
Node: ANSI C Compliance43866
Ref: usage ansi-c-compliance43988
Ref: 1343988
Node: Inline functions44958
Ref: usage inline-functions45075
Ref: 1445075
Ref: usage sec-inline-functions45075
Ref: 1545075
Node: Long double46537
Ref: usage long-double46658
Ref: 1746658
Node: Portability functions48165
Ref: usage id548301
Ref: 1848301
Ref: usage portability-functions48301
Ref: 1948301
Node: Alternative optimized functions49600
Ref: usage alternative-optimized-functions49760
Ref: 1b49760
Node: Support for different numeric types51062
Ref: usage support-for-different-numeric-types51223
Ref: 1c51223
Node: Compatibility with C++54167
Ref: usage compatibility-with-c54315
Ref: 1e54315
Node: Aliasing of arrays54748
Ref: usage aliasing-of-arrays54874
Ref: 1f54874
Ref: usage id654874
Ref: 2054874
Node: Thread-safety55503
Ref: usage thread-safety55627
Ref: 2155627
Node: Deprecated Functions56592
Ref: usage deprecated-functions56708
Ref: 2256708
Node: Code Reuse57221
Ref: usage code-reuse57315
Ref: 2357315
Node: Error Handling57874
Ref: err doc57986
Ref: 2457986
Ref: err error-handling57986
Ref: 2557986
Node: Error Reporting58609
Ref: err error-reporting58696
Ref: 2658696
Node: Error Codes60509
Ref: err error-codes60619
Ref: 2760619
Ref: err c GSL_EDOM61067
Ref: 2861067
Ref: err c GSL_ERANGE61279
Ref: 2961279
Ref: err c GSL_ENOMEM61482
Ref: 2a61482
Ref: err c GSL_EINVAL61784
Ref: 2b61784
Ref: err c gsl_strerror62093
Ref: 2c62093
Node: Error Handlers62450
Ref: err error-handlers62592
Ref: 2d62592
Ref: err c gsl_error_handler_t63531
Ref: 2e63531
Ref: err c gsl_set_error_handler64450
Ref: 2f64450
Ref: err c gsl_set_error_handler_off65542
Ref: 3065542
Node: Using GSL error reporting in your own functions66178
Ref: err using-gsl-error-reporting-in-your-own-functions66317
Ref: 3166317
Ref: err c GSL_ERROR66892
Ref: 3266892
Ref: err c GSL_ERROR_VAL67643
Ref: 3367643
Node: Examples68142
Ref: err examples68258
Ref: 3468258
Node: Mathematical Functions69351
Ref: math doc69461
Ref: 3669461
Ref: math mathematical-functions69461
Ref: 3769461
Node: Mathematical Constants70113
Ref: math mathematical-constants70231
Ref: 3870231
Node: Infinities and Not-a-number72385
Ref: math infinities-and-not-a-number72532
Ref: 3972532
Ref: math c GSL_POSINF72597
Ref: 3a72597
Ref: math c GSL_NEGINF72758
Ref: 3b72758
Ref: math c GSL_NAN72919
Ref: 3c72919
Ref: math c gsl_isnan73079
Ref: 3d73079
Ref: math c gsl_isinf73188
Ref: 3e73188
Ref: math c gsl_finite73370
Ref: 3f73370
Ref: Infinities and Not-a-number-Footnote-173563
Node: Elementary Functions74003
Ref: math elementary-functions74148
Ref: 4074148
Ref: math c gsl_log1p74528
Ref: 4174528
Ref: math c gsl_expm174757
Ref: 4274757
Ref: math c gsl_hypot74986
Ref: 1a74986
Ref: math c gsl_hypot375222
Ref: 4375222
Ref: math c gsl_acosh75415
Ref: 4475415
Ref: math c gsl_asinh75600
Ref: 4575600
Ref: math c gsl_atanh75785
Ref: 4675785
Ref: math c gsl_ldexp75970
Ref: 4775970
Ref: math c gsl_frexp76152
Ref: 4876152
Node: Small integer powers76547
Ref: math small-integer-powers76692
Ref: 4976692
Ref: math c gsl_pow_int77007
Ref: 4a77007
Ref: math c gsl_pow_uint77058
Ref: 4b77058
Ref: math c gsl_pow_277453
Ref: 4d77453
Ref: math c gsl_pow_377501
Ref: 4e77501
Ref: math c gsl_pow_477549
Ref: 4f77549
Ref: math c gsl_pow_577597
Ref: 5077597
Ref: math c gsl_pow_677645
Ref: 5177645
Ref: math c gsl_pow_777693
Ref: 5277693
Ref: math c gsl_pow_877741
Ref: 5377741
Ref: math c gsl_pow_977789
Ref: 5477789
Node: Testing the Sign of Numbers78233
Ref: math testing-the-sign-of-numbers78390
Ref: 5578390
Ref: math c GSL_SIGN78455
Ref: 5678455
Node: Testing for Odd and Even Numbers78677
Ref: math testing-for-odd-and-even-numbers78843
Ref: 5778843
Ref: math c GSL_IS_ODD78918
Ref: 5878918
Ref: math c GSL_IS_EVEN79073
Ref: 5979073
Node: Maximum and Minimum functions79279
Ref: math maximum-and-minimum-functions79466
Ref: 5a79466
Ref: math c GSL_MAX79728
Ref: 5b79728
Ref: math c GSL_MIN79868
Ref: 5c79868
Ref: math c GSL_MAX_DBL80008
Ref: 5d80008
Ref: math c GSL_MIN_DBL80442
Ref: 5e80442
Ref: math c GSL_MAX_INT80876
Ref: 5f80876
Ref: math c GSL_MIN_INT80935
Ref: 6080935
Ref: math c GSL_MAX_LDBL81276
Ref: 6181276
Ref: math c GSL_MIN_LDBL81370
Ref: 6281370
Node: Approximate Comparison of Floating Point Numbers81755
Ref: math approximate-comparison-of-floating-point-numbers81901
Ref: 6381901
Ref: math c gsl_fcmp82317
Ref: 6482317
Node: Complex Numbers83162
Ref: complex doc83269
Ref: 6583269
Ref: complex complex-numbers83269
Ref: 6683269
Ref: Complex Numbers-Footnote-184493
Node: Representation of complex numbers84561
Ref: complex representation-of-complex-numbers84677
Ref: 6784677
Node: Complex number macros86590
Ref: complex complex-number-macros86740
Ref: 6a86740
Ref: complex c GSL_REAL86870
Ref: 6886870
Ref: complex c GSL_IMAG86894
Ref: 6986894
Ref: complex c GSL_SET_COMPLEX87396
Ref: 6b87396
Node: Assigning complex numbers87671
Ref: complex assigning-complex-numbers87817
Ref: 6c87817
Ref: complex c gsl_complex_rect87878
Ref: 6d87878
Ref: complex c gsl_complex_polar88142
Ref: 6e88142
Node: Properties of complex numbers88389
Ref: complex properties-of-complex-numbers88542
Ref: 6f88542
Ref: complex c gsl_complex_arg88611
Ref: 7088611
Ref: complex c gsl_complex_abs88784
Ref: 7188784
Ref: complex c gsl_complex_abs288925
Ref: 7288925
Ref: complex c gsl_complex_logabs89077
Ref: 7389077
Node: Complex arithmetic operators89440
Ref: complex complex-arithmetic-operators89596
Ref: 7489596
Ref: complex c gsl_complex_add89663
Ref: 7589663
Ref: complex c gsl_complex_sub89838
Ref: 7689838
Ref: complex c gsl_complex_mul90020
Ref: 7790020
Ref: complex c gsl_complex_div90198
Ref: 7890198
Ref: complex c gsl_complex_add_real90378
Ref: 7990378
Ref: complex c gsl_complex_sub_real90568
Ref: 7a90568
Ref: complex c gsl_complex_mul_real90765
Ref: 7b90765
Ref: complex c gsl_complex_div_real90958
Ref: 7c90958
Ref: complex c gsl_complex_add_imag91153
Ref: 7d91153
Ref: complex c gsl_complex_sub_imag91340
Ref: 7e91340
Ref: complex c gsl_complex_mul_imag91534
Ref: 7f91534
Ref: complex c gsl_complex_div_imag91727
Ref: 8091727
Ref: complex c gsl_complex_conjugate91921
Ref: 8191921
Ref: complex c gsl_complex_inverse92091
Ref: 8292091
Ref: complex c gsl_complex_negative92279
Ref: 8392279
Node: Elementary Complex Functions92443
Ref: complex elementary-complex-functions92601
Ref: 8492601
Ref: complex c gsl_complex_sqrt92668
Ref: 8592668
Ref: complex c gsl_complex_sqrt_real92933
Ref: 8692933
Ref: complex c gsl_complex_pow93118
Ref: 8793118
Ref: complex c gsl_complex_pow_real93395
Ref: 8893395
Ref: complex c gsl_complex_exp93577
Ref: 8993577
Ref: complex c gsl_complex_log93737
Ref: 8a93737
Ref: complex c gsl_complex_log1093960
Ref: 8b93960
Ref: complex c gsl_complex_log_b94134
Ref: 8c94134
Node: Complex Trigonometric Functions94401
Ref: complex complex-trigonometric-functions94570
Ref: 8d94570
Ref: complex c gsl_complex_sin94643
Ref: 8e94643
Ref: complex c gsl_complex_cos94826
Ref: 8f94826
Ref: complex c gsl_complex_tan95008
Ref: 9095008
Ref: complex c gsl_complex_sec95182
Ref: 9195182
Ref: complex c gsl_complex_csc95349
Ref: 9295349
Ref: complex c gsl_complex_cot95518
Ref: 9395518
Node: Inverse Complex Trigonometric Functions95688
Ref: complex inverse-complex-trigonometric-functions95857
Ref: 9495857
Ref: complex c gsl_complex_arcsin95946
Ref: 9595946
Ref: complex c gsl_complex_arcsin_real96185
Ref: 9696185
Ref: complex c gsl_complex_arccos96628
Ref: 9796628
Ref: complex c gsl_complex_arccos_real96869
Ref: 9896869
Ref: complex c gsl_complex_arctan97278
Ref: 9997278
Ref: complex c gsl_complex_arcsec97514
Ref: 9a97514
Ref: complex c gsl_complex_arcsec_real97693
Ref: 9b97693
Ref: complex c gsl_complex_arccsc97869
Ref: 9c97869
Ref: complex c gsl_complex_arccsc_real98050
Ref: 9d98050
Ref: complex c gsl_complex_arccot98228
Ref: 9e98228
Node: Complex Hyperbolic Functions98410
Ref: complex complex-hyperbolic-functions98584
Ref: 9f98584
Ref: complex c gsl_complex_sinh98651
Ref: a098651
Ref: complex c gsl_complex_cosh98842
Ref: a198842
Ref: complex c gsl_complex_tanh99035
Ref: a299035
Ref: complex c gsl_complex_sech99224
Ref: a399224
Ref: complex c gsl_complex_csch99405
Ref: a499405
Ref: complex c gsl_complex_coth99588
Ref: a599588
Node: Inverse Complex Hyperbolic Functions99772
Ref: complex inverse-complex-hyperbolic-functions99937
Ref: a699937
Ref: complex c gsl_complex_arcsinh100022
Ref: a7100022
Ref: complex c gsl_complex_arccosh100268
Ref: a8100268
Ref: complex c gsl_complex_arccosh_real100648
Ref: a9100648
Ref: complex c gsl_complex_arctanh100822
Ref: aa100822
Ref: complex c gsl_complex_arctanh_real101077
Ref: ab101077
Ref: complex c gsl_complex_arcsech101252
Ref: ac101252
Ref: complex c gsl_complex_arccsch101445
Ref: ad101445
Ref: complex c gsl_complex_arccoth101640
Ref: ae101640
Node: References and Further Reading101836
Ref: complex references-and-further-reading101964
Ref: af101964
Node: Polynomials103248
Ref: poly doc103350
Ref: b0103350
Ref: poly polynomials103350
Ref: b1103350
Node: Polynomial Evaluation104023
Ref: poly polynomial-evaluation104150
Ref: b2104150
Ref: poly c gsl_poly_eval104437
Ref: b3104437
Ref: poly c gsl_poly_complex_eval104639
Ref: b4104639
Ref: poly c gsl_complex_poly_complex_eval104862
Ref: b5104862
Ref: poly c gsl_poly_eval_derivs105101
Ref: b6105101
Node: Divided Difference Representation of Polynomials105508
Ref: poly divided-difference-representation-of-polynomials105663
Ref: b7105663
Ref: poly c gsl_poly_dd_init106990
Ref: b8106990
Ref: poly c gsl_poly_dd_eval107514
Ref: b9107514
Ref: poly c gsl_poly_dd_taylor107908
Ref: ba107908
Ref: poly c gsl_poly_dd_hermite_init108549
Ref: bb108549
Node: Quadratic Equations109720
Ref: poly quadratic-equations109869
Ref: bc109869
Ref: poly c gsl_poly_solve_quadratic109918
Ref: bd109918
Ref: poly c gsl_poly_complex_solve_quadratic111174
Ref: be111174
Node: Cubic Equations111775
Ref: poly cubic-equations111904
Ref: bf111904
Ref: poly c gsl_poly_solve_cubic111945
Ref: c0111945
Ref: poly c gsl_poly_complex_solve_cubic112919
Ref: c1112919
Node: General Polynomial Equations113457
Ref: poly general-polynomial-equations113578
Ref: c2113578
Ref: poly c gsl_poly_complex_workspace113918
Ref: c3113918
Ref: poly c gsl_poly_complex_workspace_alloc114048
Ref: c4114048
Ref: poly c gsl_poly_complex_workspace_free114570
Ref: c6114570
Ref: poly c gsl_poly_complex_solve114753
Ref: c5114753
Node: Examples<2>116039
Ref: poly examples116178
Ref: c7116178
Node: References and Further Reading<2>117367
Ref: poly references-and-further-reading117469
Ref: c8117469
Node: Special Functions118387
Ref: specfunc doc118494
Ref: c9118494
Ref: specfunc special-functions118494
Ref: ca118494
Node: Usage120486
Ref: specfunc usage120579
Ref: cb120579
Node: The gsl_sf_result struct121658
Ref: specfunc the-gsl-sf-result-struct121765
Ref: cc121765
Ref: specfunc c gsl_sf_result122153
Ref: cd122153
Ref: specfunc c gsl_sf_result_e10122835
Ref: ce122835
Node: Modes123005
Ref: specfunc modes123137
Ref: cf123137
Ref: specfunc c gsl_mode_t123613
Ref: d0123613
Ref: specfunc c gsl_mode_t GSL_PREC_DOUBLE123635
Ref: d1123635
Ref: specfunc c gsl_mode_t GSL_PREC_SINGLE123758
Ref: d2123758
Ref: specfunc c gsl_mode_t GSL_PREC_APPROX123876
Ref: d3123876
Node: Airy Functions and Derivatives124078
Ref: specfunc airy-functions-and-derivatives124202
Ref: d4124202
Node: Airy Functions124761
Ref: specfunc airy-functions124881
Ref: d5124881
Ref: specfunc c gsl_sf_airy_Ai124924
Ref: d6124924
Ref: specfunc c gsl_sf_airy_Ai_e124988
Ref: d7124988
Ref: specfunc c gsl_sf_airy_Bi125189
Ref: d8125189
Ref: specfunc c gsl_sf_airy_Bi_e125253
Ref: d9125253
Ref: specfunc c gsl_sf_airy_Ai_scaled125454
Ref: da125454
Ref: specfunc c gsl_sf_airy_Ai_scaled_e125525
Ref: db125525
Ref: specfunc c gsl_sf_airy_Bi_scaled125802
Ref: dc125802
Ref: specfunc c gsl_sf_airy_Bi_scaled_e125873
Ref: dd125873
Node: Derivatives of Airy Functions126149
Ref: specfunc derivatives-of-airy-functions126301
Ref: de126301
Ref: specfunc c gsl_sf_airy_Ai_deriv126374
Ref: df126374
Ref: specfunc c gsl_sf_airy_Ai_deriv_e126444
Ref: e0126444
Ref: specfunc c gsl_sf_airy_Bi_deriv126663
Ref: e1126663
Ref: specfunc c gsl_sf_airy_Bi_deriv_e126733
Ref: e2126733
Ref: specfunc c gsl_sf_airy_Ai_deriv_scaled126952
Ref: e3126952
Ref: specfunc c gsl_sf_airy_Ai_deriv_scaled_e127039
Ref: e4127039
Ref: specfunc c gsl_sf_airy_Bi_deriv_scaled127321
Ref: e5127321
Ref: specfunc c gsl_sf_airy_Bi_deriv_scaled_e127408
Ref: e6127408
Node: Zeros of Airy Functions127689
Ref: specfunc zeros-of-airy-functions127865
Ref: e7127865
Ref: specfunc c gsl_sf_airy_zero_Ai127926
Ref: e8127926
Ref: specfunc c gsl_sf_airy_zero_Ai_e127984
Ref: e9127984
Ref: specfunc c gsl_sf_airy_zero_Bi128178
Ref: ea128178
Ref: specfunc c gsl_sf_airy_zero_Bi_e128236
Ref: eb128236
Node: Zeros of Derivatives of Airy Functions128430
Ref: specfunc zeros-of-derivatives-of-airy-functions128568
Ref: ec128568
Ref: specfunc c gsl_sf_airy_zero_Ai_deriv128659
Ref: ed128659
Ref: specfunc c gsl_sf_airy_zero_Ai_deriv_e128723
Ref: ee128723
Ref: specfunc c gsl_sf_airy_zero_Bi_deriv128935
Ref: ef128935
Ref: specfunc c gsl_sf_airy_zero_Bi_deriv_e128999
Ref: f0128999
Node: Bessel Functions129211
Ref: specfunc bessel-functions129347
Ref: f1129347
Node: Regular Cylindrical Bessel Functions130407
Ref: specfunc regular-cylindrical-bessel-functions130544
Ref: f2130544
Ref: specfunc c gsl_sf_bessel_J0130631
Ref: f3130631
Ref: specfunc c gsl_sf_bessel_J0_e130680
Ref: f4130680
Ref: specfunc c gsl_sf_bessel_J1130851
Ref: f5130851
Ref: specfunc c gsl_sf_bessel_J1_e130900
Ref: f6130900
Ref: specfunc c gsl_sf_bessel_Jn131070
Ref: f7131070
Ref: specfunc c gsl_sf_bessel_Jn_e131126
Ref: f8131126
Ref: specfunc c gsl_sf_bessel_Jn_array131319
Ref: f9131319
Node: Irregular Cylindrical Bessel Functions131773
Ref: specfunc irregular-cylindrical-bessel-functions131964
Ref: fa131964
Ref: specfunc c gsl_sf_bessel_Y0132055
Ref: fb132055
Ref: specfunc c gsl_sf_bessel_Y0_e132104
Ref: fc132104
Ref: specfunc c gsl_sf_bessel_Y1132286
Ref: fd132286
Ref: specfunc c gsl_sf_bessel_Y1_e132335
Ref: fe132335
Ref: specfunc c gsl_sf_bessel_Yn132516
Ref: ff132516
Ref: specfunc c gsl_sf_bessel_Yn_e132572
Ref: 100132572
Ref: specfunc c gsl_sf_bessel_Yn_array132777
Ref: 101132777
Node: Regular Modified Cylindrical Bessel Functions133278
Ref: specfunc regular-modified-cylindrical-bessel-functions133480
Ref: 102133480
Ref: specfunc c gsl_sf_bessel_I0133585
Ref: 103133585
Ref: specfunc c gsl_sf_bessel_I0_e133634
Ref: 104133634
Ref: specfunc c gsl_sf_bessel_I1133814
Ref: 105133814
Ref: specfunc c gsl_sf_bessel_I1_e133863
Ref: 106133863
Ref: specfunc c gsl_sf_bessel_In134042
Ref: 107134042
Ref: specfunc c gsl_sf_bessel_In_e134098
Ref: 108134098
Ref: specfunc c gsl_sf_bessel_In_array134301
Ref: 109134301
Ref: specfunc c gsl_sf_bessel_I0_scaled134845
Ref: 10a134845
Ref: specfunc c gsl_sf_bessel_I0_scaled_e134901
Ref: 10b134901
Ref: specfunc c gsl_sf_bessel_I1_scaled135115
Ref: 10c135115
Ref: specfunc c gsl_sf_bessel_I1_scaled_e135171
Ref: 10d135171
Ref: specfunc c gsl_sf_bessel_In_scaled135384
Ref: 10e135384
Ref: specfunc c gsl_sf_bessel_In_scaled_e135447
Ref: 10f135447
Ref: specfunc c gsl_sf_bessel_In_scaled_array135674
Ref: 110135674
Node: Irregular Modified Cylindrical Bessel Functions136234
Ref: specfunc irregular-modified-cylindrical-bessel-functions136432
Ref: 111136432
Ref: specfunc c gsl_sf_bessel_K0136541
Ref: 112136541
Ref: specfunc c gsl_sf_bessel_K0_e136590
Ref: 113136590
Ref: specfunc c gsl_sf_bessel_K1136783
Ref: 114136783
Ref: specfunc c gsl_sf_bessel_K1_e136832
Ref: 115136832
Ref: specfunc c gsl_sf_bessel_Kn137024
Ref: 116137024
Ref: specfunc c gsl_sf_bessel_Kn_e137080
Ref: 117137080
Ref: specfunc c gsl_sf_bessel_Kn_array137296
Ref: 118137296
Ref: specfunc c gsl_sf_bessel_K0_scaled137878
Ref: 119137878
Ref: specfunc c gsl_sf_bessel_K0_scaled_e137934
Ref: 11a137934
Ref: specfunc c gsl_sf_bessel_K1_scaled138155
Ref: 11b138155
Ref: specfunc c gsl_sf_bessel_K1_scaled_e138211
Ref: 11c138211
Ref: specfunc c gsl_sf_bessel_Kn_scaled138431
Ref: 11d138431
Ref: specfunc c gsl_sf_bessel_Kn_scaled_e138494
Ref: 11e138494
Ref: specfunc c gsl_sf_bessel_Kn_scaled_array138730
Ref: 11f138730
Node: Regular Spherical Bessel Functions139324
Ref: specfunc regular-spherical-bessel-functions139513
Ref: 120139513
Ref: specfunc c gsl_sf_bessel_j0139596
Ref: 121139596
Ref: specfunc c gsl_sf_bessel_j0_e139645
Ref: 122139645
Ref: specfunc c gsl_sf_bessel_j1139826
Ref: 123139826
Ref: specfunc c gsl_sf_bessel_j1_e139875
Ref: 124139875
Ref: specfunc c gsl_sf_bessel_j2140069
Ref: 125140069
Ref: specfunc c gsl_sf_bessel_j2_e140118
Ref: 126140118
Ref: specfunc c gsl_sf_bessel_jl140325
Ref: 127140325
Ref: specfunc c gsl_sf_bessel_jl_e140381
Ref: 128140381
Ref: specfunc c gsl_sf_bessel_jl_array140600
Ref: 129140600
Ref: specfunc c gsl_sf_bessel_jl_steed_array141065
Ref: 12a141065
Node: Irregular Spherical Bessel Functions141621
Ref: specfunc irregular-spherical-bessel-functions141806
Ref: 12b141806
Ref: specfunc c gsl_sf_bessel_y0141893
Ref: 12c141893
Ref: specfunc c gsl_sf_bessel_y0_e141942
Ref: 12d141942
Ref: specfunc c gsl_sf_bessel_y1142126
Ref: 12e142126
Ref: specfunc c gsl_sf_bessel_y1_e142175
Ref: 12f142175
Ref: specfunc c gsl_sf_bessel_y2142372
Ref: 130142372
Ref: specfunc c gsl_sf_bessel_y2_e142421
Ref: 131142421
Ref: specfunc c gsl_sf_bessel_yl142633
Ref: 132142633
Ref: specfunc c gsl_sf_bessel_yl_e142689
Ref: 133142689
Ref: specfunc c gsl_sf_bessel_yl_array142897
Ref: 134142897
Node: Regular Modified Spherical Bessel Functions143351
Ref: specfunc regular-modified-spherical-bessel-functions143547
Ref: 135143547
Ref: specfunc c gsl_sf_bessel_i0_scaled143808
Ref: 136143808
Ref: specfunc c gsl_sf_bessel_i0_scaled_e143864
Ref: 137143864
Ref: specfunc c gsl_sf_bessel_i1_scaled144077
Ref: 138144077
Ref: specfunc c gsl_sf_bessel_i1_scaled_e144133
Ref: 139144133
Ref: specfunc c gsl_sf_bessel_i2_scaled144345
Ref: 13a144345
Ref: specfunc c gsl_sf_bessel_i2_scaled_e144401
Ref: 13b144401
Ref: specfunc c gsl_sf_bessel_il_scaled144613
Ref: 13c144613
Ref: specfunc c gsl_sf_bessel_il_scaled_e144676
Ref: 13d144676
Ref: specfunc c gsl_sf_bessel_il_scaled_array144901
Ref: 13e144901
Node: Irregular Modified Spherical Bessel Functions145387
Ref: specfunc irregular-modified-spherical-bessel-functions145589
Ref: 13f145589
Ref: specfunc c gsl_sf_bessel_k0_scaled145867
Ref: 140145867
Ref: specfunc c gsl_sf_bessel_k0_scaled_e145923
Ref: 141145923
Ref: specfunc c gsl_sf_bessel_k1_scaled146144
Ref: 142146144
Ref: specfunc c gsl_sf_bessel_k1_scaled_e146200
Ref: 143146200
Ref: specfunc c gsl_sf_bessel_k2_scaled146420
Ref: 144146420
Ref: specfunc c gsl_sf_bessel_k2_scaled_e146476
Ref: 145146476
Ref: specfunc c gsl_sf_bessel_kl_scaled146697
Ref: 146146697
Ref: specfunc c gsl_sf_bessel_kl_scaled_e146760
Ref: 147146760
Ref: specfunc c gsl_sf_bessel_kl_scaled_array146994
Ref: 148146994
Node: Regular Bessel Function—Fractional Order147487
Ref: specfunc regular-bessel-function-fractional-order147691
Ref: 149147691
Ref: specfunc c gsl_sf_bessel_Jnu147788
Ref: 14a147788
Ref: specfunc c gsl_sf_bessel_Jnu_e147849
Ref: 14b147849
Ref: specfunc c gsl_sf_bessel_sequence_Jnu_e148052
Ref: 14c148052
Node: Irregular Bessel Functions—Fractional Order148510
Ref: specfunc irregular-bessel-functions-fractional-order148721
Ref: 14d148721
Ref: specfunc c gsl_sf_bessel_Ynu148826
Ref: 14e148826
Ref: specfunc c gsl_sf_bessel_Ynu_e148887
Ref: 14f148887
Node: Regular Modified Bessel Functions—Fractional Order149092
Ref: specfunc regular-modified-bessel-functions-fractional-order149315
Ref: 150149315
Ref: specfunc c gsl_sf_bessel_Inu149434
Ref: 151149434
Ref: specfunc c gsl_sf_bessel_Inu_e149495
Ref: 152149495
Ref: specfunc c gsl_sf_bessel_Inu_scaled149710
Ref: 153149710
Ref: specfunc c gsl_sf_bessel_Inu_scaled_e149778
Ref: 154149778
Node: Irregular Modified Bessel Functions—Fractional Order150017
Ref: specfunc irregular-modified-bessel-functions-fractional-order150228
Ref: 155150228
Ref: specfunc c gsl_sf_bessel_Knu150351
Ref: 156150351
Ref: specfunc c gsl_sf_bessel_Knu_e150412
Ref: 157150412
Ref: specfunc c gsl_sf_bessel_lnKnu150629
Ref: 158150629
Ref: specfunc c gsl_sf_bessel_lnKnu_e150692
Ref: 159150692
Ref: specfunc c gsl_sf_bessel_Knu_scaled150938
Ref: 15a150938
Ref: specfunc c gsl_sf_bessel_Knu_scaled_e151006
Ref: 15b151006
Node: Zeros of Regular Bessel Functions151253
Ref: specfunc zeros-of-regular-bessel-functions151403
Ref: 15c151403
Ref: specfunc c gsl_sf_bessel_zero_J0151486
Ref: 15d151486
Ref: specfunc c gsl_sf_bessel_zero_J0_e151546
Ref: 15e151546
Ref: specfunc c gsl_sf_bessel_zero_J1151755
Ref: 15f151755
Ref: specfunc c gsl_sf_bessel_zero_J1_e151815
Ref: 160151815
Ref: specfunc c gsl_sf_bessel_zero_Jnu152024
Ref: 161152024
Ref: specfunc c gsl_sf_bessel_zero_Jnu_e152096
Ref: 162152096
Node: Clausen Functions152403
Ref: specfunc clausen-functions152526
Ref: 163152526
Ref: specfunc c gsl_sf_clausen152841
Ref: 165152841
Ref: specfunc c gsl_sf_clausen_e152888
Ref: 166152888
Node: Coulomb Functions153017
Ref: specfunc coulomb-functions153145
Ref: 167153145
Node: Normalized Hydrogenic Bound States153470
Ref: specfunc normalized-hydrogenic-bound-states153590
Ref: 168153590
Ref: specfunc c gsl_sf_hydrogenicR_1153673
Ref: 169153673
Ref: specfunc c gsl_sf_hydrogenicR_1_e153736
Ref: 16a153736
Ref: specfunc c gsl_sf_hydrogenicR153968
Ref: 16b153968
Ref: specfunc c gsl_sf_hydrogenicR_e154053
Ref: 16c154053
Node: Coulomb Wave Functions154571
Ref: specfunc coulomb-wave-functions154744
Ref: 16e154744
Ref: specfunc c gsl_sf_coulomb_wave_FG_e155404
Ref: 16f155404
Ref: specfunc c gsl_sf_coulomb_wave_F_array156224
Ref: 170156224
Ref: specfunc c gsl_sf_coulomb_wave_FG_array156599
Ref: 171156599
Ref: specfunc c gsl_sf_coulomb_wave_FGp_array157083
Ref: 172157083
Ref: specfunc c gsl_sf_coulomb_wave_sphF_array157716
Ref: 173157716
Node: Coulomb Wave Function Normalization Constant158215
Ref: specfunc coulomb-wave-function-normalization-constant158345
Ref: 174158345
Ref: specfunc c gsl_sf_coulomb_CL_e158531
Ref: 175158531
Ref: specfunc c gsl_sf_coulomb_CL_array158731
Ref: 176158731
Node: Coupling Coefficients158969
Ref: specfunc coupling-coefficients159095
Ref: 177159095
Node: 3-j Symbols159696
Ref: specfunc j-symbols159786
Ref: 178159786
Ref: specfunc c gsl_sf_coupling_3j159823
Ref: 179159823
Ref: specfunc c gsl_sf_coupling_3j_e159946
Ref: 17a159946
Node: 6-j Symbols160331
Ref: specfunc id1160441
Ref: 17b160441
Ref: specfunc c gsl_sf_coupling_6j160478
Ref: 17c160478
Ref: specfunc c gsl_sf_coupling_6j_e160601
Ref: 17d160601
Node: 9-j Symbols160980
Ref: specfunc id2161070
Ref: 17e161070
Ref: specfunc c gsl_sf_coupling_9j161107
Ref: 17f161107
Ref: specfunc c gsl_sf_coupling_9j_e161276
Ref: 180161276
Node: Dawson Function161715
Ref: specfunc dawson-function161839
Ref: 181161839
Ref: specfunc c gsl_sf_dawson162111
Ref: 182162111
Ref: specfunc c gsl_sf_dawson_e162157
Ref: 183162157
Node: Debye Functions162311
Ref: specfunc debye-functions162425
Ref: 184162425
Ref: specfunc c gsl_sf_debye_1162723
Ref: 185162723
Ref: specfunc c gsl_sf_debye_1_e162770
Ref: 186162770
Ref: specfunc c gsl_sf_debye_2162908
Ref: 187162908
Ref: specfunc c gsl_sf_debye_2_e162955
Ref: 188162955
Ref: specfunc c gsl_sf_debye_3163094
Ref: 189163094
Ref: specfunc c gsl_sf_debye_3_e163141
Ref: 18a163141
Ref: specfunc c gsl_sf_debye_4163279
Ref: 18b163279
Ref: specfunc c gsl_sf_debye_4_e163326
Ref: 18c163326
Ref: specfunc c gsl_sf_debye_5163465
Ref: 18d163465
Ref: specfunc c gsl_sf_debye_5_e163512
Ref: 18e163512
Ref: specfunc c gsl_sf_debye_6163650
Ref: 18f163650
Ref: specfunc c gsl_sf_debye_6_e163697
Ref: 190163697
Node: Dilogarithm163835
Ref: specfunc dilog-function163955
Ref: 164163955
Ref: specfunc dilogarithm163955
Ref: 191163955
Node: Real Argument164208
Ref: specfunc real-argument164295
Ref: 192164295
Ref: specfunc c gsl_sf_dilog164338
Ref: 193164338
Ref: specfunc c gsl_sf_dilog_e164383
Ref: 194164383
Node: Complex Argument164914
Ref: specfunc complex-argument165001
Ref: 195165001
Ref: specfunc c gsl_sf_complex_dilog_e165050
Ref: 196165050
Node: Elementary Operations165410
Ref: specfunc elementary-operations165533
Ref: 197165533
Ref: specfunc c gsl_sf_multiply165768
Ref: 198165768
Ref: specfunc c gsl_sf_multiply_e165826
Ref: 199165826
Ref: specfunc c gsl_sf_multiply_err_e166052
Ref: 19a166052
Node: Elliptic Integrals166394
Ref: specfunc elliptic-integrals166531
Ref: 19b166531
Node: Definition of Legendre Forms166968
Ref: specfunc definition-of-legendre-forms167088
Ref: 19c167088
Node: Definition of Carlson Forms167783
Ref: specfunc definition-of-carlson-forms167956
Ref: 19d167956
Node: Legendre Form of Complete Elliptic Integrals168469
Ref: specfunc legendre-form-of-complete-elliptic-integrals168660
Ref: 19e168660
Ref: specfunc c gsl_sf_ellint_Kcomp168765
Ref: 19f168765
Ref: specfunc c gsl_sf_ellint_Kcomp_e168834
Ref: 1a0168834
Ref: specfunc c gsl_sf_ellint_Ecomp169167
Ref: 1a1169167
Ref: specfunc c gsl_sf_ellint_Ecomp_e169236
Ref: 1a2169236
Ref: specfunc c gsl_sf_ellint_Pcomp169569
Ref: 1a3169569
Ref: specfunc c gsl_sf_ellint_Pcomp_e169658
Ref: 1a4169658
Node: Legendre Form of Incomplete Elliptic Integrals170070
Ref: specfunc legendre-form-of-incomplete-elliptic-integrals170247
Ref: 1a5170247
Ref: specfunc c gsl_sf_ellint_F170356
Ref: 1a6170356
Ref: specfunc c gsl_sf_ellint_F_e170443
Ref: 1a7170443
Ref: specfunc c gsl_sf_ellint_E170790
Ref: 1a8170790
Ref: specfunc c gsl_sf_ellint_E_e170877
Ref: 1a9170877
Ref: specfunc c gsl_sf_ellint_P171224
Ref: 1aa171224
Ref: specfunc c gsl_sf_ellint_P_e171321
Ref: 1ab171321
Ref: specfunc c gsl_sf_ellint_D171748
Ref: 1ac171748
Ref: specfunc c gsl_sf_ellint_D_e171835
Ref: 1ad171835
Node: Carlson Forms172194
Ref: specfunc carlson-forms172318
Ref: 1ae172318
Ref: specfunc c gsl_sf_ellint_RC172361
Ref: 1af172361
Ref: specfunc c gsl_sf_ellint_RC_e172447
Ref: 1b0172447
Ref: specfunc c gsl_sf_ellint_RD172695
Ref: 1b1172695
Ref: specfunc c gsl_sf_ellint_RD_e172791
Ref: 1b2172791
Ref: specfunc c gsl_sf_ellint_RF173051
Ref: 1b3173051
Ref: specfunc c gsl_sf_ellint_RF_e173147
Ref: 1b4173147
Ref: specfunc c gsl_sf_ellint_RJ173407
Ref: 1b5173407
Ref: specfunc c gsl_sf_ellint_RJ_e173513
Ref: 1b6173513
Node: Elliptic Functions Jacobi173785
Ref: specfunc elliptic-functions-jacobi173916
Ref: 1b7173916
Ref: specfunc c gsl_sf_elljac_e174134
Ref: 1b8174134
Node: Error Functions174366
Ref: specfunc error-functions174500
Ref: 1b9174500
Node: Error Function174819
Ref: specfunc error-function174923
Ref: 1ba174923
Ref: specfunc c gsl_sf_erf174968
Ref: 1bb174968
Ref: specfunc c gsl_sf_erf_e175011
Ref: 1bc175011
Node: Complementary Error Function175194
Ref: specfunc complementary-error-function175339
Ref: 1bd175339
Ref: specfunc c gsl_sf_erfc175412
Ref: 1be175412
Ref: specfunc c gsl_sf_erfc_e175456
Ref: 1bf175456
Node: Log Complementary Error Function175655
Ref: specfunc log-complementary-error-function175807
Ref: 1c0175807
Ref: specfunc c gsl_sf_log_erfc175888
Ref: 1c1175888
Ref: specfunc c gsl_sf_log_erfc_e175936
Ref: 1c2175936
Node: Probability functions176107
Ref: specfunc probability-functions176222
Ref: 1c3176222
Ref: specfunc c gsl_sf_erf_Z176400
Ref: 1c4176400
Ref: specfunc c gsl_sf_erf_Z_e176445
Ref: 1c5176445
Ref: specfunc c gsl_sf_erf_Q176625
Ref: 1c6176625
Ref: specfunc c gsl_sf_erf_Q_e176670
Ref: 1c7176670
Ref: specfunc c gsl_sf_hazard177159
Ref: 1c8177159
Ref: specfunc c gsl_sf_hazard_e177205
Ref: 1c9177205
Node: Exponential Functions177357
Ref: specfunc exponential-functions177487
Ref: 1ca177487
Node: Exponential Function177744
Ref: specfunc exponential-function177862
Ref: 1cb177862
Ref: specfunc c gsl_sf_exp177919
Ref: 1cc177919
Ref: specfunc c gsl_sf_exp_e177962
Ref: 1cd177962
Ref: specfunc c gsl_sf_exp_e10_e178134
Ref: 1ce178134
Ref: specfunc c gsl_sf_exp_mult178462
Ref: 1cf178462
Ref: specfunc c gsl_sf_exp_mult_e178520
Ref: 1d0178520
Ref: specfunc c gsl_sf_exp_mult_e10_e178738
Ref: 1d1178738
Node: Relative Exponential Functions179002
Ref: specfunc relative-exponential-functions179163
Ref: 1d2179163
Ref: specfunc c gsl_sf_expm1179240
Ref: 1d3179240
Ref: specfunc c gsl_sf_expm1_e179285
Ref: 1d4179285
Ref: specfunc c gsl_sf_exprel179459
Ref: 1d5179459
Ref: specfunc c gsl_sf_exprel_e179505
Ref: 1d6179505
Ref: specfunc c gsl_sf_exprel_2179829
Ref: 1d7179829
Ref: specfunc c gsl_sf_exprel_2_e179877
Ref: 1d8179877
Ref: specfunc c gsl_sf_exprel_n180213
Ref: 1d9180213
Ref: specfunc c gsl_sf_exprel_n_e180268
Ref: 1da180268
Node: Exponentiation With Error Estimate180760
Ref: specfunc exponentiation-with-error-estimate180892
Ref: 1db180892
Ref: specfunc c gsl_sf_exp_err_e180977
Ref: 1dc180977
Ref: specfunc c gsl_sf_exp_err_e10_e181170
Ref: 1dd181170
Ref: specfunc c gsl_sf_exp_mult_err_e181471
Ref: 1de181471
Ref: specfunc c gsl_sf_exp_mult_err_e10_e181757
Ref: 1df181757
Node: Exponential Integrals182140
Ref: specfunc exponential-integrals182275
Ref: 1e0182275
Node: Exponential Integral182638
Ref: specfunc exponential-integral182730
Ref: 1e1182730
Ref: specfunc c gsl_sf_expint_E1182787
Ref: 1e2182787
Ref: specfunc c gsl_sf_expint_E1_e182836
Ref: 1e3182836
Ref: specfunc c gsl_sf_expint_E2183025
Ref: 1e4183025
Ref: specfunc c gsl_sf_expint_E2_e183074
Ref: 1e5183074
Ref: specfunc c gsl_sf_expint_En183282
Ref: 1e6183282
Ref: specfunc c gsl_sf_expint_En_e183338
Ref: 1e7183338
Node: Ei x183573
Ref: specfunc ei-x183694
Ref: 1e8183694
Ref: specfunc c gsl_sf_expint_Ei183721
Ref: 1e9183721
Ref: specfunc c gsl_sf_expint_Ei_e183770
Ref: 1ea183770
Node: Hyperbolic Integrals184019
Ref: specfunc hyperbolic-integrals184126
Ref: 1eb184126
Ref: specfunc c gsl_sf_Shi184183
Ref: 1ec184183
Ref: specfunc c gsl_sf_Shi_e184226
Ref: 1ed184226
Ref: specfunc c gsl_sf_Chi184377
Ref: 1ee184377
Ref: specfunc c gsl_sf_Chi_e184420
Ref: 1ef184420
Node: Ei_3 x184692
Ref: specfunc ei-3-x184818
Ref: 1f0184818
Ref: specfunc c gsl_sf_expint_3184849
Ref: 1f1184849
Ref: specfunc c gsl_sf_expint_3_e184897
Ref: 1f2184897
Node: Trigonometric Integrals185098
Ref: specfunc trigonometric-integrals185223
Ref: 1f3185223
Ref: specfunc c gsl_sf_Si185286
Ref: 1f4185286
Ref: specfunc c gsl_sf_Si_e185334
Ref: 1f5185334
Ref: specfunc c gsl_sf_Ci185487
Ref: 1f6185487
Ref: specfunc c gsl_sf_Ci_e185535
Ref: 1f7185535
Node: Arctangent Integral185713
Ref: specfunc arctangent-integral185823
Ref: 1f8185823
Ref: specfunc c gsl_sf_atanint185878
Ref: 1f9185878
Ref: specfunc c gsl_sf_atanint_e185925
Ref: 1fa185925
Node: Fermi-Dirac Function186118
Ref: specfunc fermi-dirac-function186256
Ref: 1fb186256
Node: Complete Fermi-Dirac Integrals186492
Ref: specfunc complete-fermi-dirac-integrals186621
Ref: 1fc186621
Ref: specfunc c gsl_sf_fermi_dirac_m1186932
Ref: 1fd186932
Ref: specfunc c gsl_sf_fermi_dirac_m1_e186986
Ref: 1fe186986
Ref: specfunc c gsl_sf_fermi_dirac_0187223
Ref: 1ff187223
Ref: specfunc c gsl_sf_fermi_dirac_0_e187276
Ref: 200187276
Ref: specfunc c gsl_sf_fermi_dirac_1187500
Ref: 201187500
Ref: specfunc c gsl_sf_fermi_dirac_1_e187553
Ref: 202187553
Ref: specfunc c gsl_sf_fermi_dirac_2187773
Ref: 203187773
Ref: specfunc c gsl_sf_fermi_dirac_2_e187826
Ref: 204187826
Ref: specfunc c gsl_sf_fermi_dirac_int188054
Ref: 205188054
Ref: specfunc c gsl_sf_fermi_dirac_int_e188116
Ref: 206188116
Ref: specfunc c gsl_sf_fermi_dirac_mhalf188376
Ref: 207188376
Ref: specfunc c gsl_sf_fermi_dirac_mhalf_e188433
Ref: 208188433
Ref: specfunc c gsl_sf_fermi_dirac_half188604
Ref: 209188604
Ref: specfunc c gsl_sf_fermi_dirac_half_e188660
Ref: 20a188660
Ref: specfunc c gsl_sf_fermi_dirac_3half188829
Ref: 20b188829
Ref: specfunc c gsl_sf_fermi_dirac_3half_e188886
Ref: 20c188886
Node: Incomplete Fermi-Dirac Integrals189056
Ref: specfunc incomplete-fermi-dirac-integrals189185
Ref: 20d189185
Ref: specfunc c gsl_sf_fermi_dirac_inc_0189400
Ref: 20e189400
Ref: specfunc c gsl_sf_fermi_dirac_inc_0_e189467
Ref: 20f189467
Node: Gamma and Beta Functions189696
Ref: specfunc gamma-and-beta-functions189833
Ref: 210189833
Node: Gamma Functions190272
Ref: specfunc gamma-functions190368
Ref: 211190368
Ref: specfunc c gsl_sf_gamma190705
Ref: 212190705
Ref: specfunc c gsl_sf_gamma_e190750
Ref: 213190750
Ref: specfunc c gsl_sf_lngamma191139
Ref: 214191139
Ref: specfunc c gsl_sf_lngamma_e191186
Ref: 215191186
Ref: specfunc c gsl_sf_lngamma_sgn_e191565
Ref: 216191565
Ref: specfunc c gsl_sf_gammastar192085
Ref: 217192085
Ref: specfunc c gsl_sf_gammastar_e192134
Ref: 218192134
Ref: specfunc c gsl_sf_gammainv192510
Ref: 219192510
Ref: specfunc c gsl_sf_gammainv_e192558
Ref: 21a192558
Ref: specfunc c gsl_sf_lngamma_complex_e192744
Ref: 21b192744
Node: Factorials193432
Ref: specfunc factorials193554
Ref: 21c193554
Ref: specfunc c gsl_sf_fact193880
Ref: 21d193880
Ref: specfunc c gsl_sf_fact_e193930
Ref: 21e193930
Ref: specfunc c gsl_sf_doublefact194258
Ref: 21f194258
Ref: specfunc c gsl_sf_doublefact_e194314
Ref: 220194314
Ref: specfunc c gsl_sf_lnfact194619
Ref: 221194619
Ref: specfunc c gsl_sf_lnfact_e194671
Ref: 222194671
Ref: specfunc c gsl_sf_lndoublefact194997
Ref: 223194997
Ref: specfunc c gsl_sf_lndoublefact_e195055
Ref: 224195055
Ref: specfunc c gsl_sf_choose195246
Ref: 225195246
Ref: specfunc c gsl_sf_choose_e195314
Ref: 226195314
Ref: specfunc c gsl_sf_lnchoose195507
Ref: 227195507
Ref: specfunc c gsl_sf_lnchoose_e195577
Ref: 228195577
Ref: specfunc c gsl_sf_taylorcoeff195816
Ref: 229195816
Ref: specfunc c gsl_sf_taylorcoeff_e195874
Ref: 22a195874
Node: Pochhammer Symbol196052
Ref: specfunc id3196185
Ref: 22b196185
Ref: specfunc pochhammer-symbol196185
Ref: 22c196185
Ref: specfunc c gsl_sf_poch196236
Ref: 22d196236
Ref: specfunc c gsl_sf_poch_e196290
Ref: 22e196290
Ref: specfunc c gsl_sf_lnpoch196664
Ref: 22f196664
Ref: specfunc c gsl_sf_lnpoch_e196720
Ref: 230196720
Ref: specfunc c gsl_sf_lnpoch_sgn_e196928
Ref: 231196928
Ref: specfunc c gsl_sf_pochrel197294
Ref: 232197294
Ref: specfunc c gsl_sf_pochrel_e197351
Ref: 233197351
Node: Incomplete Gamma Functions197559
Ref: specfunc incomplete-gamma-functions197696
Ref: 234197696
Ref: specfunc c gsl_sf_gamma_inc197765
Ref: 235197765
Ref: specfunc c gsl_sf_gamma_inc_e197824
Ref: 236197824
Ref: specfunc c gsl_sf_gamma_inc_Q198073
Ref: 237198073
Ref: specfunc c gsl_sf_gamma_inc_Q_e198134
Ref: 238198134
Ref: specfunc c gsl_sf_gamma_inc_P198385
Ref: 239198385
Ref: specfunc c gsl_sf_gamma_inc_P_e198446
Ref: 23a198446
Node: Beta Functions198817
Ref: specfunc beta-functions198961
Ref: 23b198961
Ref: specfunc c gsl_sf_beta199006
Ref: 23c199006
Ref: specfunc c gsl_sf_beta_e199060
Ref: 23d199060
Ref: specfunc c gsl_sf_lnbeta199293
Ref: 23e199293
Ref: specfunc c gsl_sf_lnbeta_e199349
Ref: 23f199349
Node: Incomplete Beta Function199569
Ref: specfunc incomplete-beta-function199678
Ref: 240199678
Ref: specfunc c gsl_sf_beta_inc199743
Ref: 241199743
Ref: specfunc c gsl_sf_beta_inc_e199811
Ref: 242199811
Node: Gegenbauer Functions200304
Ref: specfunc gegenbauer-functions200454
Ref: 243200454
Ref: specfunc c gsl_sf_gegenpoly_1200735
Ref: 244200735
Ref: specfunc c gsl_sf_gegenpoly_2200801
Ref: 245200801
Ref: specfunc c gsl_sf_gegenpoly_3200867
Ref: 246200867
Ref: specfunc c gsl_sf_gegenpoly_1_e200933
Ref: 247200933
Ref: specfunc c gsl_sf_gegenpoly_2_e201031
Ref: 248201031
Ref: specfunc c gsl_sf_gegenpoly_3_e201129
Ref: 249201129
Ref: specfunc c gsl_sf_gegenpoly_n201358
Ref: 24a201358
Ref: specfunc c gsl_sf_gegenpoly_n_e201431
Ref: 24b201431
Ref: specfunc c gsl_sf_gegenpoly_array201730
Ref: 24c201730
Node: Hermite Polynomials and Functions202003
Ref: specfunc hermite-polynomials-and-functions202153
Ref: 24d202153
Node: Hermite Polynomials202689
Ref: specfunc hermite-polynomials202822
Ref: 24e202822
Ref: specfunc c gsl_sf_hermite203387
Ref: 24f203387
Ref: specfunc c gsl_sf_hermite_e203453
Ref: 250203453
Ref: specfunc c gsl_sf_hermite_array203780
Ref: 251203780
Ref: specfunc c gsl_sf_hermite_series204069
Ref: 252204069
Ref: specfunc c gsl_sf_hermite_series_e204169
Ref: 253204169
Ref: specfunc c gsl_sf_hermite_prob204450
Ref: 254204450
Ref: specfunc c gsl_sf_hermite_prob_e204521
Ref: 255204521
Ref: specfunc c gsl_sf_hermite_prob_array204856
Ref: 256204856
Ref: specfunc c gsl_sf_hermite_prob_series205156
Ref: 257205156
Ref: specfunc c gsl_sf_hermite_prob_series_e205261
Ref: 258205261
Node: Derivatives of Hermite Polynomials205551
Ref: specfunc derivatives-of-hermite-polynomials205710
Ref: 259205710
Ref: specfunc c gsl_sf_hermite_deriv205795
Ref: 25a205795
Ref: specfunc c gsl_sf_hermite_deriv_e205890
Ref: 25b205890
Ref: specfunc c gsl_sf_hermite_array_deriv206172
Ref: 25c206172
Ref: specfunc c gsl_sf_hermite_deriv_array206624
Ref: 25d206624
Ref: specfunc c gsl_sf_hermite_prob_deriv207079
Ref: 25e207079
Ref: specfunc c gsl_sf_hermite_prob_deriv_e207179
Ref: 25f207179
Ref: specfunc c gsl_sf_hermite_prob_array_deriv207469
Ref: 260207469
Ref: specfunc c gsl_sf_hermite_prob_deriv_array207930
Ref: 261207930
Node: Hermite Functions208394
Ref: specfunc hermite-functions208566
Ref: 262208566
Ref: specfunc c gsl_sf_hermite_func209448
Ref: 263209448
Ref: specfunc c gsl_sf_hermite_func_e209519
Ref: 264209519
Ref: specfunc c gsl_sf_hermite_func_fast209818
Ref: 265209818
Ref: specfunc c gsl_sf_hermite_func_fast_e209904
Ref: 266209904
Ref: specfunc c gsl_sf_hermite_func_array210238
Ref: 267210238
Ref: specfunc c gsl_sf_hermite_func_series210624
Ref: 268210624
Ref: specfunc c gsl_sf_hermite_func_series_e210729
Ref: 269210729
Node: Derivatives of Hermite Functions211009
Ref: specfunc derivatives-of-hermite-functions211197
Ref: 26a211197
Ref: specfunc c gsl_sf_hermite_func_der211278
Ref: 26b211278
Ref: specfunc c gsl_sf_hermite_func_der_e211376
Ref: 26c211376
Node: Zeros of Hermite Polynomials and Hermite Functions211652
Ref: specfunc zeros-of-hermite-polynomials-and-hermite-functions211814
Ref: 26d211814
Ref: specfunc c gsl_sf_hermite_zero212240
Ref: 26e212240
Ref: specfunc c gsl_sf_hermite_zero_e212308
Ref: 26f212308
Ref: specfunc c gsl_sf_hermite_prob_zero212536
Ref: 270212536
Ref: specfunc c gsl_sf_hermite_prob_zero_e212609
Ref: 271212609
Ref: specfunc c gsl_sf_hermite_func_zero212845
Ref: 272212845
Ref: specfunc c gsl_sf_hermite_func_zero_e212918
Ref: 273212918
Node: Hypergeometric Functions213142
Ref: specfunc hypergeometric-functions213290
Ref: 274213290
Ref: specfunc c gsl_sf_hyperg_0F1213507
Ref: 275213507
Ref: specfunc c gsl_sf_hyperg_0F1_e213567
Ref: 276213567
Ref: specfunc c gsl_sf_hyperg_1F1_int213737
Ref: 277213737
Ref: specfunc c gsl_sf_hyperg_1F1_int_e213805
Ref: 278213805
Ref: specfunc c gsl_sf_hyperg_1F1214063
Ref: 279214063
Ref: specfunc c gsl_sf_hyperg_1F1_e214133
Ref: 27a214133
Ref: specfunc c gsl_sf_hyperg_U_int214393
Ref: 27b214393
Ref: specfunc c gsl_sf_hyperg_U_int_e214459
Ref: 27c214459
Ref: specfunc c gsl_sf_hyperg_U_int_e10_e214690
Ref: 27d214690
Ref: specfunc c gsl_sf_hyperg_U215022
Ref: 27e215022
Ref: specfunc c gsl_sf_hyperg_U_e215090
Ref: 27f215090
Ref: specfunc c gsl_sf_hyperg_U_e10_e215273
Ref: 280215273
Ref: specfunc c gsl_sf_hyperg_2F1215551
Ref: 281215551
Ref: specfunc c gsl_sf_hyperg_2F1_e215641
Ref: 282215641
Ref: specfunc c gsl_sf_hyperg_2F1_conj216127
Ref: 283216127
Ref: specfunc c gsl_sf_hyperg_2F1_conj_e216224
Ref: 284216224
Ref: specfunc c gsl_sf_hyperg_2F1_renorm216495
Ref: 285216495
Ref: specfunc c gsl_sf_hyperg_2F1_renorm_e216592
Ref: 286216592
Ref: specfunc c gsl_sf_hyperg_2F1_conj_renorm216848
Ref: 287216848
Ref: specfunc c gsl_sf_hyperg_2F1_conj_renorm_e216952
Ref: 288216952
Ref: specfunc c gsl_sf_hyperg_2F0217238
Ref: 289217238
Ref: specfunc c gsl_sf_hyperg_2F0_e217308
Ref: 28a217308
Node: Laguerre Functions217641
Ref: specfunc id4217775
Ref: 28b217775
Ref: specfunc laguerre-functions217775
Ref: 16d217775
Ref: specfunc c gsl_sf_laguerre_1218376
Ref: 28c218376
Ref: specfunc c gsl_sf_laguerre_2218436
Ref: 28d218436
Ref: specfunc c gsl_sf_laguerre_3218496
Ref: 28e218496
Ref: specfunc c gsl_sf_laguerre_1_e218556
Ref: 28f218556
Ref: specfunc c gsl_sf_laguerre_2_e218648
Ref: 290218648
Ref: specfunc c gsl_sf_laguerre_3_e218740
Ref: 291218740
Ref: specfunc c gsl_sf_laguerre_n218966
Ref: 292218966
Ref: specfunc c gsl_sf_laguerre_n_e219061
Ref: 293219061
Node: Lambert W Functions219263
Ref: specfunc lambert-w-functions219415
Ref: 294219415
Ref: specfunc c gsl_sf_lambert_W0219874
Ref: 295219874
Ref: specfunc c gsl_sf_lambert_W0_e219924
Ref: 296219924
Ref: specfunc c gsl_sf_lambert_Wm1220078
Ref: 297220078
Ref: specfunc c gsl_sf_lambert_Wm1_e220129
Ref: 298220129
Node: Legendre Functions and Spherical Harmonics220299
Ref: specfunc legendre-functions-and-spherical-harmonics220464
Ref: 299220464
Node: Legendre Polynomials220892
Ref: specfunc legendre-polynomials221056
Ref: 29a221056
Ref: specfunc c gsl_sf_legendre_P1221113
Ref: 29b221113
Ref: specfunc c gsl_sf_legendre_P2221164
Ref: 29c221164
Ref: specfunc c gsl_sf_legendre_P3221215
Ref: 29d221215
Ref: specfunc c gsl_sf_legendre_P1_e221266
Ref: 29e221266
Ref: specfunc c gsl_sf_legendre_P2_e221339
Ref: 29f221339
Ref: specfunc c gsl_sf_legendre_P3_e221412
Ref: 2a0221412
Ref: specfunc c gsl_sf_legendre_Pl221602
Ref: 2a1221602
Ref: specfunc c gsl_sf_legendre_Pl_e221660
Ref: 2a2221660
Ref: specfunc c gsl_sf_legendre_Pl_array221909
Ref: 2a3221909
Ref: specfunc c gsl_sf_legendre_Pl_deriv_array222006
Ref: 2a4222006
Ref: specfunc c gsl_sf_legendre_Q0222277
Ref: 2a5222277
Ref: specfunc c gsl_sf_legendre_Q0_e222328
Ref: 2a6222328
Ref: specfunc c gsl_sf_legendre_Q1222489
Ref: 2a7222489
Ref: specfunc c gsl_sf_legendre_Q1_e222540
Ref: 2a8222540
Ref: specfunc c gsl_sf_legendre_Ql222701
Ref: 2a9222701
Ref: specfunc c gsl_sf_legendre_Ql_e222759
Ref: 2aa222759
Node: Associated Legendre Polynomials and Spherical Harmonics222946
Ref: specfunc associated-legendre-polynomials-and-spherical-harmonics223136
Ref: 2ab223136
Ref: specfunc c gsl_sf_legendre_t225684
Ref: 2ac225684
Ref: specfunc c gsl_sf_legendre_array226749
Ref: 2ad226749
Ref: specfunc c gsl_sf_legendre_array_e226888
Ref: 2ae226888
Ref: specfunc c gsl_sf_legendre_deriv_array227761
Ref: 2b0227761
Ref: specfunc c gsl_sf_legendre_deriv_array_e227945
Ref: 2b1227945
Ref: specfunc c gsl_sf_legendre_deriv_alt_array228761
Ref: 2b2228761
Ref: specfunc c gsl_sf_legendre_deriv_alt_array_e228949
Ref: 2b3228949
Ref: specfunc c gsl_sf_legendre_deriv2_array229741
Ref: 2b4229741
Ref: specfunc c gsl_sf_legendre_deriv2_array_e229966
Ref: 2b5229966
Ref: specfunc c gsl_sf_legendre_deriv2_alt_array230921
Ref: 2b6230921
Ref: specfunc c gsl_sf_legendre_deriv2_alt_array_e231150
Ref: 2b7231150
Ref: specfunc c gsl_sf_legendre_nlm232152
Ref: 2b8232152
Ref: specfunc c gsl_sf_legendre_array_n232386
Ref: 2af232386
Ref: specfunc c gsl_sf_legendre_array_index232826
Ref: 2b9232826
Ref: specfunc c gsl_sf_legendre_Plm233233
Ref: 2ba233233
Ref: specfunc c gsl_sf_legendre_Plm_e233299
Ref: 2bb233299
Ref: specfunc c gsl_sf_legendre_sphPlm233513
Ref: 2bc233513
Ref: specfunc c gsl_sf_legendre_sphPlm_e233582
Ref: 2bd233582
Ref: specfunc c gsl_sf_legendre_Plm_array234024
Ref: 2be234024
Ref: specfunc c gsl_sf_legendre_Plm_deriv_array234129
Ref: 2bf234129
Ref: specfunc c gsl_sf_legendre_sphPlm_array234448
Ref: 2c0234448
Ref: specfunc c gsl_sf_legendre_sphPlm_deriv_array234556
Ref: 2c1234556
Ref: specfunc c gsl_sf_legendre_array_size234878
Ref: 2c2234878
Node: Conical Functions235048
Ref: specfunc conical-functions235255
Ref: 2c3235255
Ref: specfunc c gsl_sf_conicalP_half235436
Ref: 2c4235436
Ref: specfunc c gsl_sf_conicalP_half_e235504
Ref: 2c5235504
Ref: specfunc c gsl_sf_conicalP_mhalf235722
Ref: 2c6235722
Ref: specfunc c gsl_sf_conicalP_mhalf_e235791
Ref: 2c7235791
Ref: specfunc c gsl_sf_conicalP_0236009
Ref: 2c8236009
Ref: specfunc c gsl_sf_conicalP_0_e236074
Ref: 2c9236074
Ref: specfunc c gsl_sf_conicalP_1236265
Ref: 2ca236265
Ref: specfunc c gsl_sf_conicalP_1_e236330
Ref: 2cb236330
Ref: specfunc c gsl_sf_conicalP_sph_reg236521
Ref: 2cc236521
Ref: specfunc c gsl_sf_conicalP_sph_reg_e236609
Ref: 2cd236609
Ref: specfunc c gsl_sf_conicalP_cyl_reg236851
Ref: 2ce236851
Ref: specfunc c gsl_sf_conicalP_cyl_reg_e236939
Ref: 2cf236939
Node: Radial Functions for Hyperbolic Space237179
Ref: specfunc radial-functions-for-hyperbolic-space237322
Ref: 2d0237322
Ref: specfunc c gsl_sf_legendre_H3d_0237680
Ref: 2d1237680
Ref: specfunc c gsl_sf_legendre_H3d_0_e237751
Ref: 2d2237751
Ref: specfunc c gsl_sf_legendre_H3d_1238170
Ref: 2d3238170
Ref: specfunc c gsl_sf_legendre_H3d_1_e238241
Ref: 2d4238241
Ref: specfunc c gsl_sf_legendre_H3d238753
Ref: 2d5238753
Ref: specfunc c gsl_sf_legendre_H3d_e238839
Ref: 2d6238839
Ref: specfunc c gsl_sf_legendre_H3d_array239196
Ref: 2d7239196
Node: Logarithm and Related Functions239428
Ref: specfunc logarithm-and-related-functions239591
Ref: 2d8239591
Ref: specfunc c gsl_sf_log239864
Ref: 2d9239864
Ref: specfunc c gsl_sf_log_e239907
Ref: 2da239907
Ref: specfunc c gsl_sf_log_abs240058
Ref: 2db240058
Ref: specfunc c gsl_sf_log_abs_e240105
Ref: 2dc240105
Ref: specfunc c gsl_sf_complex_log_e240281
Ref: 2dd240281
Ref: specfunc c gsl_sf_log_1plusx240632
Ref: 2de240632
Ref: specfunc c gsl_sf_log_1plusx_e240682
Ref: 2df240682
Ref: specfunc c gsl_sf_log_1plusx_mx240872
Ref: 2e0240872
Ref: specfunc c gsl_sf_log_1plusx_mx_e240925
Ref: 2e1240925
Node: Mathieu Functions241132
Ref: specfunc mathieu-functions241267
Ref: 2e2241267
Node: Mathieu Function Workspace242490
Ref: specfunc mathieu-function-workspace242618
Ref: 2e3242618
Ref: specfunc c gsl_sf_mathieu_workspace242857
Ref: 2e4242857
Ref: specfunc c gsl_sf_mathieu_alloc242943
Ref: 2e5242943
Ref: specfunc c gsl_sf_mathieu_free243294
Ref: 2e6243294
Node: Mathieu Function Characteristic Values243424
Ref: specfunc mathieu-function-characteristic-values243586
Ref: 2e7243586
Ref: specfunc c gsl_sf_mathieu_a243679
Ref: 2e8243679
Ref: specfunc c gsl_sf_mathieu_a_e243732
Ref: 2e9243732
Ref: specfunc c gsl_sf_mathieu_b243820
Ref: 2ea243820
Ref: specfunc c gsl_sf_mathieu_b_e243873
Ref: 2eb243873
Ref: specfunc c gsl_sf_mathieu_a_array244101
Ref: 2ec244101
Ref: specfunc c gsl_sf_mathieu_b_array244258
Ref: 2ed244258
Node: Angular Mathieu Functions244644
Ref: specfunc angular-mathieu-functions244804
Ref: 2ee244804
Ref: specfunc c gsl_sf_mathieu_ce244871
Ref: 2ef244871
Ref: specfunc c gsl_sf_mathieu_ce_e244935
Ref: 2f0244935
Ref: specfunc c gsl_sf_mathieu_se245034
Ref: 2f1245034
Ref: specfunc c gsl_sf_mathieu_se_e245098
Ref: 2f2245098
Ref: specfunc c gsl_sf_mathieu_ce_array245301
Ref: 2f3245301
Ref: specfunc c gsl_sf_mathieu_se_array245459
Ref: 2f4245459
Node: Radial Mathieu Functions245850
Ref: specfunc radial-mathieu-functions245963
Ref: 2f5245963
Ref: specfunc c gsl_sf_mathieu_Mc246028
Ref: 2f6246028
Ref: specfunc c gsl_sf_mathieu_Mc_e246099
Ref: 2f7246099
Ref: specfunc c gsl_sf_mathieu_Ms246205
Ref: 2f8246205
Ref: specfunc c gsl_sf_mathieu_Ms_e246276
Ref: 2f9246276
Ref: specfunc c gsl_sf_mathieu_Mc_array246767
Ref: 2fa246767
Ref: specfunc c gsl_sf_mathieu_Ms_array246932
Ref: 2fb246932
Node: Power Function247327
Ref: specfunc power-function247451
Ref: 2fc247451
Ref: specfunc c gsl_sf_pow_int247670
Ref: 2fd247670
Ref: specfunc c gsl_sf_pow_int_e247724
Ref: 4c247724
Node: Psi Digamma Function248304
Ref: specfunc psi-digamma-function248432
Ref: 2fe248432
Node: Digamma Function248826
Ref: specfunc digamma-function248926
Ref: 2ff248926
Ref: specfunc c gsl_sf_psi_int248975
Ref: 300248975
Ref: specfunc c gsl_sf_psi_int_e249019
Ref: 301249019
Ref: specfunc c gsl_sf_psi249244
Ref: 302249244
Ref: specfunc c gsl_sf_psi_e249287
Ref: 303249287
Ref: specfunc c gsl_sf_psi_1piy249451
Ref: 304249451
Ref: specfunc c gsl_sf_psi_1piy_e249499
Ref: 305249499
Node: Trigamma Function249683
Ref: specfunc trigamma-function249810
Ref: 306249810
Ref: specfunc c gsl_sf_psi_1_int249861
Ref: 307249861
Ref: specfunc c gsl_sf_psi_1_int_e249907
Ref: 308249907
Ref: specfunc c gsl_sf_psi_1250065
Ref: 309250065
Ref: specfunc c gsl_sf_psi_1_e250110
Ref: 30a250110
Node: Polygamma Function250269
Ref: specfunc polygamma-function250371
Ref: 30b250371
Ref: specfunc c gsl_sf_psi_n250424
Ref: 30c250424
Ref: specfunc c gsl_sf_psi_n_e250476
Ref: 30d250476
Node: Synchrotron Functions250652
Ref: specfunc synchrotron-functions250785
Ref: 30e250785
Ref: specfunc c gsl_sf_synchrotron_1250941
Ref: 30f250941
Ref: specfunc c gsl_sf_synchrotron_1_e250994
Ref: 310250994
Ref: specfunc c gsl_sf_synchrotron_2251188
Ref: 311251188
Ref: specfunc c gsl_sf_synchrotron_2_e251241
Ref: 312251241
Node: Transport Functions251419
Ref: specfunc transport-functions251555
Ref: 313251555
Ref: specfunc c gsl_sf_transport_2251794
Ref: 314251794
Ref: specfunc c gsl_sf_transport_2_e251845
Ref: 315251845
Ref: specfunc c gsl_sf_transport_3251979
Ref: 316251979
Ref: specfunc c gsl_sf_transport_3_e252030
Ref: 317252030
Ref: specfunc c gsl_sf_transport_4252164
Ref: 318252164
Ref: specfunc c gsl_sf_transport_4_e252215
Ref: 319252215
Ref: specfunc c gsl_sf_transport_5252349
Ref: 31a252349
Ref: specfunc c gsl_sf_transport_5_e252400
Ref: 31b252400
Node: Trigonometric Functions252534
Ref: specfunc trigonometric-functions252663
Ref: 31c252663
Node: Circular Trigonometric Functions253161
Ref: specfunc circular-trigonometric-functions253308
Ref: 31d253308
Ref: specfunc c gsl_sf_sin253389
Ref: 31e253389
Ref: specfunc c gsl_sf_sin_e253432
Ref: 31f253432
Ref: specfunc c gsl_sf_cos253554
Ref: 320253554
Ref: specfunc c gsl_sf_cos_e253597
Ref: 321253597
Ref: specfunc c gsl_sf_hypot253721
Ref: 322253721
Ref: specfunc c gsl_sf_hypot_e253776
Ref: 323253776
Ref: specfunc c gsl_sf_sinc253972
Ref: 324253972
Ref: specfunc c gsl_sf_sinc_e254016
Ref: 325254016
Node: Trigonometric Functions for Complex Arguments254181
Ref: specfunc trigonometric-functions-for-complex-arguments254371
Ref: 326254371
Ref: specfunc c gsl_sf_complex_sin_e254478
Ref: 327254478
Ref: specfunc c gsl_sf_complex_cos_e254733
Ref: 328254733
Ref: specfunc c gsl_sf_complex_logsin_e254995
Ref: 329254995
Node: Hyperbolic Trigonometric Functions255285
Ref: specfunc hyperbolic-trigonometric-functions255463
Ref: 32a255463
Ref: specfunc c gsl_sf_lnsinh255548
Ref: 32b255548
Ref: specfunc c gsl_sf_lnsinh_e255594
Ref: 32c255594
Ref: specfunc c gsl_sf_lncosh255718
Ref: 32d255718
Ref: specfunc c gsl_sf_lncosh_e255764
Ref: 32e255764
Node: Conversion Functions255899
Ref: specfunc conversion-functions256053
Ref: 32f256053
Ref: specfunc c gsl_sf_polar_to_rect256110
Ref: 330256110
Ref: specfunc c gsl_sf_rect_to_polar256412
Ref: 331256412
Node: Restriction Functions256792
Ref: specfunc restriction-functions256956
Ref: 332256956
Ref: specfunc c gsl_sf_angle_restrict_symm257015
Ref: 333257015
Ref: specfunc c gsl_sf_angle_restrict_symm_e257078
Ref: 334257078
Ref: specfunc c gsl_sf_angle_restrict_pos257404
Ref: 335257404
Ref: specfunc c gsl_sf_angle_restrict_pos_e257466
Ref: 336257466
Node: Trigonometric Functions With Error Estimates257772
Ref: specfunc trigonometric-functions-with-error-estimates257907
Ref: 337257907
Ref: specfunc c gsl_sf_sin_err_e258012
Ref: 338258012
Ref: specfunc c gsl_sf_cos_err_e258369
Ref: 339258369
Node: Zeta Functions258728
Ref: specfunc zeta-functions258849
Ref: 33a258849
Node: Riemann Zeta Function259175
Ref: specfunc riemann-zeta-function259288
Ref: 33b259288
Ref: specfunc c gsl_sf_zeta_int259447
Ref: 33c259447
Ref: specfunc c gsl_sf_zeta_int_e259492
Ref: 33d259492
Ref: specfunc c gsl_sf_zeta259664
Ref: 33e259664
Ref: specfunc c gsl_sf_zeta_e259708
Ref: 33f259708
Node: Riemann Zeta Function Minus One259881
Ref: specfunc riemann-zeta-function-minus-one260024
Ref: 340260024
Ref: specfunc c gsl_sf_zetam1_int260286
Ref: 341260286
Ref: specfunc c gsl_sf_zetam1_int_e260333
Ref: 342260333
Ref: specfunc c gsl_sf_zetam1260485
Ref: 343260485
Ref: specfunc c gsl_sf_zetam1_e260531
Ref: 344260531
Node: Hurwitz Zeta Function260684
Ref: specfunc hurwitz-zeta-function260818
Ref: 345260818
Ref: specfunc c gsl_sf_hzeta260962
Ref: 346260962
Ref: specfunc c gsl_sf_hzeta_e261017
Ref: 347261017
Node: Eta Function261194
Ref: specfunc eta-function261288
Ref: 348261288
Ref: specfunc c gsl_sf_eta_int261398
Ref: 349261398
Ref: specfunc c gsl_sf_eta_int_e261442
Ref: 34a261442
Ref: specfunc c gsl_sf_eta261594
Ref: 34b261594
Ref: specfunc c gsl_sf_eta_e261637
Ref: 34c261637
Node: Examples<3>261790
Ref: specfunc examples261921
Ref: 34d261921
Node: References and Further Reading<3>263431
Ref: specfunc references-and-further-reading263539
Ref: 34e263539
Node: Vectors and Matrices265078
Ref: vectors doc265186
Ref: 34f265186
Ref: vectors vectors-and-matrices265186
Ref: 350265186
Node: Data types265785
Ref: vectors data-types265868
Ref: 351265868
Node: Blocks268190
Ref: vectors blocks268289
Ref: 352268289
Ref: vectors c gsl_block268536
Ref: 353268536
Node: Block allocation269163
Ref: vectors block-allocation269258
Ref: 354269258
Ref: vectors c gsl_block_alloc269776
Ref: 355269776
Ref: vectors c gsl_block_calloc270352
Ref: 356270352
Ref: vectors c gsl_block_free270527
Ref: 357270527
Node: Reading and writing blocks270744
Ref: vectors reading-and-writing-blocks270875
Ref: 358270875
Ref: vectors c gsl_block_fwrite271049
Ref: 359271049
Ref: vectors c gsl_block_fread271470
Ref: 35a271470
Ref: vectors c gsl_block_fprintf272038
Ref: 35b272038
Ref: vectors c gsl_block_fscanf272545
Ref: 35c272545
Node: Example programs for blocks273003
Ref: vectors example-programs-for-blocks273109
Ref: 35d273109
Node: Vectors273630
Ref: vectors vectors273727
Ref: 35e273727
Ref: vectors c gsl_vector274295
Ref: 35f274295
Node: Vector allocation275656
Ref: vectors vector-allocation275752
Ref: 360275752
Ref: vectors c gsl_vector_alloc276274
Ref: 361276274
Ref: vectors c gsl_vector_calloc276752
Ref: 362276752
Ref: vectors c gsl_vector_free276955
Ref: 363276955
Node: Accessing vector elements277351
Ref: vectors accessing-vector-elements277484
Ref: 364277484
Ref: vectors c GSL_RANGE_CHECK_OFF278146
Ref: 367278146
Ref: vectors c GSL_C99_INLINE278771
Ref: 368278771
Ref: vectors c gsl_check_range279123
Ref: 369279123
Ref: vectors c gsl_vector_get279693
Ref: 365279693
Ref: vectors c gsl_vector_set280084
Ref: 366280084
Ref: vectors c gsl_vector_ptr280487
Ref: 36a280487
Ref: vectors c gsl_vector_const_ptr280550
Ref: 36b280550
Ref: Accessing vector elements-Footnote-1281013
Node: Initializing vector elements281260
Ref: vectors initializing-vector-elements281403
Ref: 36c281403
Ref: vectors c gsl_vector_set_all281474
Ref: 36d281474
Ref: vectors c gsl_vector_set_zero281641
Ref: 36e281641
Ref: vectors c gsl_vector_set_basis281781
Ref: 36f281781
Node: Reading and writing vectors282023
Ref: vectors reading-and-writing-vectors282153
Ref: 370282153
Ref: vectors c gsl_vector_fwrite282330
Ref: 371282330
Ref: vectors c gsl_vector_fread282754
Ref: 372282754
Ref: vectors c gsl_vector_fprintf283326
Ref: 373283326
Ref: vectors c gsl_vector_fscanf283836
Ref: 374283836
Node: Vector views284298
Ref: vectors vector-views284415
Ref: 375284415
Ref: vectors c gsl_vector_view284733
Ref: 376284733
Ref: vectors c gsl_vector_const_view284759
Ref: 377284759
Ref: vectors c gsl_vector_subvector285752
Ref: 378285752
Ref: vectors c gsl_vector_const_subvector285866
Ref: 379285866
Ref: vectors c gsl_vector_subvector_with_stride287309
Ref: 37a287309
Ref: vectors c gsl_vector_const_subvector_with_stride287460
Ref: 37b287460
Ref: vectors c gsl_vector_complex_real289244
Ref: 37c289244
Ref: vectors c gsl_vector_complex_const_real289344
Ref: 37d289344
Ref: vectors c gsl_vector_complex_imag289753
Ref: 37e289753
Ref: vectors c gsl_vector_complex_const_imag289853
Ref: 37f289853
Ref: vectors c gsl_vector_view_array290267
Ref: 380290267
Ref: vectors c gsl_vector_const_view_array290366
Ref: 381290366
Ref: vectors c gsl_vector_view_array_with_stride291365
Ref: 382291365
Ref: vectors c gsl_vector_const_view_array_with_stride291501
Ref: 383291501
Node: Copying vectors292652
Ref: vectors copying-vectors292761
Ref: 384292761
Ref: vectors c gsl_vector_memcpy293106
Ref: 385293106
Ref: vectors c gsl_vector_swap293352
Ref: 386293352
Node: Exchanging elements293574
Ref: vectors exchanging-elements293688
Ref: 387293688
Ref: vectors c gsl_vector_swap_elements293829
Ref: 388293829
Ref: vectors c gsl_vector_reverse294043
Ref: 389294043
Node: Vector operations294185
Ref: vectors vector-operations294331
Ref: 38a294331
Ref: vectors c gsl_vector_add294380
Ref: 38b294380
Ref: vectors c gsl_vector_sub294712
Ref: 38c294712
Ref: vectors c gsl_vector_mul295052
Ref: 38d295052
Ref: vectors c gsl_vector_div295391
Ref: 38e295391
Ref: vectors c gsl_vector_scale295726
Ref: 38f295726
Ref: vectors c gsl_vector_add_constant295969
Ref: 390295969
Ref: vectors c gsl_vector_sum296228
Ref: 391296228
Ref: vectors c gsl_vector_axpby296389
Ref: 392296389
Node: Finding maximum and minimum elements of vectors296669
Ref: vectors finding-maximum-and-minimum-elements-of-vectors296813
Ref: 393296813
Ref: vectors c gsl_vector_max296983
Ref: 394296983
Ref: vectors c gsl_vector_min297116
Ref: 395297116
Ref: vectors c gsl_vector_minmax297249
Ref: 396297249
Ref: vectors c gsl_vector_max_index297510
Ref: 397297510
Ref: vectors c gsl_vector_min_index297754
Ref: 398297754
Ref: vectors c gsl_vector_minmax_index297998
Ref: 399297998
Node: Vector properties298369
Ref: vectors vector-properties298524
Ref: 39a298524
Ref: vectors c gsl_vector_isnull298726
Ref: 39b298726
Ref: vectors c gsl_vector_ispos298784
Ref: 39c298784
Ref: vectors c gsl_vector_isneg298841
Ref: 39d298841
Ref: vectors c gsl_vector_isnonneg298898
Ref: 39e298898
Ref: vectors c gsl_vector_equal299143
Ref: 39f299143
Node: Example programs for vectors299375
Ref: vectors example-programs-for-vectors299474
Ref: 3a0299474
Node: Matrices301729
Ref: vectors matrices301811
Ref: 3a1301811
Ref: vectors c gsl_matrix302052
Ref: 3a2302052
Node: Matrix allocation304835
Ref: vectors matrix-allocation304932
Ref: 3a3304932
Ref: vectors c gsl_matrix_alloc305442
Ref: 3a4305442
Ref: vectors c gsl_matrix_calloc306022
Ref: 3a5306022
Ref: vectors c gsl_matrix_free306282
Ref: 3a6306282
Node: Accessing matrix elements306678
Ref: vectors accessing-matrix-elements306812
Ref: 3a7306812
Ref: vectors c gsl_matrix_get307428
Ref: 3a8307428
Ref: vectors c gsl_matrix_set307864
Ref: 3a9307864
Ref: vectors c gsl_matrix_ptr308312
Ref: 3aa308312
Ref: vectors c gsl_matrix_const_ptr308385
Ref: 3ab308385
Node: Initializing matrix elements308850
Ref: vectors initializing-matrix-elements308995
Ref: 3ac308995
Ref: vectors c gsl_matrix_set_all309066
Ref: 3ad309066
Ref: vectors c gsl_matrix_set_zero309233
Ref: 3ae309233
Ref: vectors c gsl_matrix_set_identity309373
Ref: 3af309373
Node: Reading and writing matrices309704
Ref: vectors reading-and-writing-matrices309836
Ref: 3b0309836
Ref: vectors c gsl_matrix_fwrite310016
Ref: 3b1310016
Ref: vectors c gsl_matrix_fread310440
Ref: 3b2310440
Ref: vectors c gsl_matrix_fprintf311016
Ref: 3b3311016
Ref: vectors c gsl_matrix_fscanf311526
Ref: 3b4311526
Node: Matrix views311991
Ref: vectors matrix-views312124
Ref: 3b5312124
Ref: vectors c gsl_matrix_view312163
Ref: 3b6312163
Ref: vectors c gsl_matrix_const_view312189
Ref: 3b7312189
Ref: vectors c gsl_matrix_submatrix313022
Ref: 3b8313022
Ref: vectors c gsl_matrix_const_submatrix313155
Ref: 3b9313155
Ref: vectors c gsl_matrix_view_array314821
Ref: 3ba314821
Ref: vectors c gsl_matrix_const_view_array314932
Ref: 3bb314932
Ref: vectors c gsl_matrix_view_array_with_tda316050
Ref: 3bc316050
Ref: vectors c gsl_matrix_const_view_array_with_tda316182
Ref: 3bd316182
Ref: vectors c gsl_matrix_view_vector317459
Ref: 3be317459
Ref: vectors c gsl_matrix_const_view_vector317572
Ref: 3bf317572
Ref: vectors c gsl_matrix_view_vector_with_tda318728
Ref: 3c0318728
Ref: vectors c gsl_matrix_const_view_vector_with_tda318872
Ref: 3c1318872
Node: Creating row and column views320186
Ref: vectors creating-row-and-column-views320307
Ref: 3c2320307
Ref: vectors c gsl_matrix_row320729
Ref: 3c3320729
Ref: vectors c gsl_matrix_const_row320822
Ref: 3c4320822
Ref: vectors c gsl_matrix_column321305
Ref: 3c5321305
Ref: vectors c gsl_matrix_const_column321401
Ref: 3c6321401
Ref: vectors c gsl_matrix_subrow321896
Ref: 3c7321896
Ref: vectors c gsl_matrix_const_subrow322017
Ref: 3c8322017
Ref: vectors c gsl_matrix_subcolumn322683
Ref: 3c9322683
Ref: vectors c gsl_matrix_const_subcolumn322807
Ref: 3ca322807
Ref: vectors c gsl_matrix_diagonal323491
Ref: 3cb323491
Ref: vectors c gsl_matrix_const_diagonal323579
Ref: 3cc323579
Ref: vectors c gsl_matrix_subdiagonal324130
Ref: 3cd324130
Ref: vectors c gsl_matrix_const_subdiagonal324231
Ref: 3ce324231
Ref: vectors c gsl_matrix_superdiagonal324766
Ref: 3cf324766
Ref: vectors c gsl_matrix_const_superdiagonal324869
Ref: 3d0324869
Node: Copying matrices325412
Ref: vectors copying-matrices325545
Ref: 3d1325545
Ref: vectors c gsl_matrix_memcpy325592
Ref: 3d2325592
Ref: vectors c gsl_matrix_swap325837
Ref: 3d3325837
Node: Copying rows and columns326063
Ref: vectors copying-rows-and-columns326194
Ref: 3d4326194
Ref: vectors c gsl_matrix_get_row326676
Ref: 3d5326676
Ref: vectors c gsl_matrix_get_col326976
Ref: 3d6326976
Ref: vectors c gsl_matrix_set_row327282
Ref: 3d7327282
Ref: vectors c gsl_matrix_set_col327582
Ref: 3d8327582
Node: Exchanging rows and columns327888
Ref: vectors exchanging-rows-and-columns328020
Ref: 3d9328020
Ref: vectors c gsl_matrix_swap_rows328172
Ref: 3da328172
Ref: vectors c gsl_matrix_swap_columns328378
Ref: 3db328378
Ref: vectors c gsl_matrix_swap_rowcol328590
Ref: 3dc328590
Ref: vectors c gsl_matrix_transpose_memcpy328871
Ref: 3dd328871
Ref: vectors c gsl_matrix_transpose329313
Ref: 3de329313
Ref: vectors c gsl_matrix_complex_conjtrans_memcpy329560
Ref: 3df329560
Node: Matrix operations330051
Ref: vectors matrix-operations330207
Ref: 3e0330207
Ref: vectors c gsl_matrix_add330327
Ref: 3e1330327
Ref: vectors c gsl_matrix_sub330674
Ref: 3e2330674
Ref: vectors c gsl_matrix_mul_elements331028
Ref: 3e3331028
Ref: vectors c gsl_matrix_div_elements331400
Ref: 3e4331400
Ref: vectors c gsl_matrix_scale331769
Ref: 3e5331769
Ref: vectors c gsl_matrix_scale_columns332018
Ref: 3e6332018
Ref: vectors c gsl_matrix_scale_rows332404
Ref: 3e7332404
Ref: vectors c gsl_matrix_add_constant332781
Ref: 3e8332781
Node: Finding maximum and minimum elements of matrices333046
Ref: vectors finding-maximum-and-minimum-elements-of-matrices333192
Ref: 3e9333192
Ref: vectors c gsl_matrix_max333367
Ref: 3ea333367
Ref: vectors c gsl_matrix_min333500
Ref: 3eb333500
Ref: vectors c gsl_matrix_minmax333633
Ref: 3ec333633
Ref: vectors c gsl_matrix_max_index333894
Ref: 3ed333894
Ref: vectors c gsl_matrix_min_index334272
Ref: 3ee334272
Ref: vectors c gsl_matrix_minmax_index334650
Ref: 3ef334650
Node: Matrix properties335129
Ref: vectors matrix-properties335287
Ref: 3f0335287
Ref: vectors c gsl_matrix_isnull335491
Ref: 3f1335491
Ref: vectors c gsl_matrix_ispos335549
Ref: 3f2335549
Ref: vectors c gsl_matrix_isneg335606
Ref: 3f3335606
Ref: vectors c gsl_matrix_isnonneg335663
Ref: 3f4335663
Ref: vectors c gsl_matrix_equal336009
Ref: 3f6336009
Ref: vectors c gsl_matrix_norm1336242
Ref: 3f7336242
Node: Example programs for matrices336489
Ref: vectors example-programs-for-matrices336632
Ref: 3f8336632
Node: References and Further Reading<4>340654
Ref: vectors references-and-further-reading340771
Ref: 3f9340771
Node: Permutations341142
Ref: permutation doc341245
Ref: 3fa341245
Ref: permutation permutations341245
Ref: 3fb341245
Node: The Permutation struct342451
Ref: permutation the-permutation-struct342554
Ref: 3fc342554
Ref: permutation c gsl_permutation342609
Ref: 3fd342609
Node: Permutation allocation343038
Ref: permutation permutation-allocation343180
Ref: 3fe343180
Ref: permutation c gsl_permutation_alloc343235
Ref: 3ff343235
Ref: permutation c gsl_permutation_calloc343716
Ref: 400343716
Ref: permutation c gsl_permutation_init344027
Ref: 401344027
Ref: permutation c gsl_permutation_free344204
Ref: 402344204
Ref: permutation c gsl_permutation_memcpy344350
Ref: 403344350
Node: Accessing permutation elements344624
Ref: permutation accessing-permutation-elements344766
Ref: 404344766
Ref: permutation c gsl_permutation_get344913
Ref: 405344913
Ref: permutation c gsl_permutation_swap345326
Ref: 406345326
Node: Permutation properties345548
Ref: permutation permutation-properties345689
Ref: 407345689
Ref: permutation c gsl_permutation_size345744
Ref: 408345744
Ref: permutation c gsl_permutation_data345884
Ref: 409345884
Ref: permutation c gsl_permutation_valid346056
Ref: 40a346056
Node: Permutation functions346299
Ref: permutation permutation-functions346431
Ref: 40b346431
Ref: permutation c gsl_permutation_reverse346484
Ref: 40c346484
Ref: permutation c gsl_permutation_inverse346629
Ref: 40d346629
Ref: permutation c gsl_permutation_next346848
Ref: 40e346848
Ref: permutation c gsl_permutation_prev347322
Ref: 40f347322
Node: Applying Permutations347657
Ref: permutation applying-permutations347799
Ref: 410347799
Ref: permutation c gsl_permute347961
Ref: 411347961
Ref: permutation c gsl_permute_inverse348210
Ref: 412348210
Ref: permutation c gsl_permute_vector348482
Ref: 413348482
Ref: permutation c gsl_permute_vector_inverse348970
Ref: 414348970
Ref: permutation c gsl_permute_matrix349580
Ref: 415349580
Ref: permutation c gsl_permutation_mul350108
Ref: 416350108
Node: Reading and writing permutations350490
Ref: permutation reading-and-writing-permutations350638
Ref: 417350638
Ref: permutation c gsl_permutation_fwrite350826
Ref: 418350826
Ref: permutation c gsl_permutation_fread351257
Ref: 419351257
Ref: permutation c gsl_permutation_fprintf351837
Ref: 41a351837
Ref: permutation c gsl_permutation_fscanf352384
Ref: 41b352384
Ref: Reading and writing permutations-Footnote-1352894
Node: Permutations in cyclic form353014
Ref: permutation permutations-in-cyclic-form353152
Ref: 41c353152
Ref: permutation c gsl_permutation_linear_to_canonical355187
Ref: 41d355187
Ref: permutation c gsl_permutation_canonical_to_linear355436
Ref: 41e355436
Ref: permutation c gsl_permutation_inversions355703
Ref: 41f355703
Ref: permutation c gsl_permutation_linear_cycles356107
Ref: 420356107
Ref: permutation c gsl_permutation_canonical_cycles356304
Ref: 421356304
Node: Examples<4>356507
Ref: permutation examples356646
Ref: 422356646
Node: References and Further Reading<5>359064
Ref: permutation references-and-further-reading359167
Ref: 423359167
Node: Combinations359698
Ref: combination doc359790
Ref: 424359790
Ref: combination combinations359790
Ref: 425359790
Node: The Combination struct360571
Ref: combination the-combination-struct360674
Ref: 426360674
Ref: combination c gsl_combination360731
Ref: 427360731
Node: Combination allocation361218
Ref: combination combination-allocation361360
Ref: 428361360
Ref: combination c gsl_combination_alloc361417
Ref: 429361417
Ref: combination c gsl_combination_calloc361962
Ref: 42a361962
Ref: combination c gsl_combination_init_first362332
Ref: 42b362332
Ref: combination c gsl_combination_init_last362542
Ref: 42c362542
Ref: combination c gsl_combination_free362764
Ref: 42d362764
Ref: combination c gsl_combination_memcpy362910
Ref: 42e362910
Node: Accessing combination elements363184
Ref: combination accessing-combination-elements363326
Ref: 42f363326
Ref: combination c gsl_combination_get363476
Ref: 430363476
Node: Combination properties363889
Ref: combination combination-properties364030
Ref: 431364030
Ref: combination c gsl_combination_n364087
Ref: 432364087
Ref: combination c gsl_combination_k364218
Ref: 433364218
Ref: combination c gsl_combination_data364378
Ref: 434364378
Ref: combination c gsl_combination_valid364550
Ref: 435364550
Node: Combination functions364815
Ref: combination combination-functions364958
Ref: 436364958
Ref: combination c gsl_combination_next365013
Ref: 437365013
Ref: combination c gsl_combination_prev365484
Ref: 438365484
Node: Reading and writing combinations365819
Ref: combination reading-and-writing-combinations365951
Ref: 439365951
Ref: combination c gsl_combination_fwrite366141
Ref: 43a366141
Ref: combination c gsl_combination_fread366572
Ref: 43b366572
Ref: combination c gsl_combination_fprintf367173
Ref: 43c367173
Ref: combination c gsl_combination_fscanf367720
Ref: 43d367720
Ref: Reading and writing combinations-Footnote-1368237
Node: Examples<5>368357
Ref: combination examples368501
Ref: 43e368501
Node: References and Further Reading<6>369632
Ref: combination references-and-further-reading369735
Ref: 43f369735
Node: Multisets370015
Ref: multiset doc370102
Ref: 440370102
Ref: multiset multisets370102
Ref: 441370102
Node: The Multiset struct370947
Ref: multiset the-multiset-struct371041
Ref: 442371041
Ref: multiset c gsl_multiset371092
Ref: 443371092
Node: Multiset allocation371561
Ref: multiset multiset-allocation371691
Ref: 444371691
Ref: multiset c gsl_multiset_alloc371742
Ref: 445371742
Ref: multiset c gsl_multiset_calloc372271
Ref: 446372271
Ref: multiset c gsl_multiset_init_first372639
Ref: 447372639
Ref: multiset c gsl_multiset_init_last372840
Ref: 448372840
Ref: multiset c gsl_multiset_free373046
Ref: 449373046
Ref: multiset c gsl_multiset_memcpy373183
Ref: 44a373183
Node: Accessing multiset elements373439
Ref: multiset accessing-multiset-elements373569
Ref: 44b373569
Ref: multiset c gsl_multiset_get373710
Ref: 44c373710
Node: Multiset properties374114
Ref: multiset multiset-properties374243
Ref: 44d374243
Ref: multiset c gsl_multiset_n374294
Ref: 44e374294
Ref: multiset c gsl_multiset_k374427
Ref: 44f374427
Ref: multiset c gsl_multiset_data374578
Ref: 450374578
Ref: multiset c gsl_multiset_valid374741
Ref: 451374741
Node: Multiset functions374984
Ref: multiset multiset-functions375115
Ref: 452375115
Ref: multiset c gsl_multiset_next375164
Ref: 453375164
Ref: multiset c gsl_multiset_prev375630
Ref: 454375630
Node: Reading and writing multisets375958
Ref: multiset reading-and-writing-multisets376081
Ref: 455376081
Ref: multiset c gsl_multiset_fwrite376262
Ref: 456376262
Ref: multiset c gsl_multiset_fread376684
Ref: 457376684
Ref: multiset c gsl_multiset_fprintf377263
Ref: 458377263
Ref: multiset c gsl_multiset_fscanf377801
Ref: 459377801
Ref: Reading and writing multisets-Footnote-1378296
Node: Examples<6>378416
Ref: multiset examples378512
Ref: 45a378512
Node: Sorting380503
Ref: sort doc380590
Ref: 45b380590
Ref: sort sorting380590
Ref: 45c380590
Node: Sorting objects381481
Ref: sort sorting-objects381565
Ref: 45d381565
Ref: sort c gsl_heapsort382103
Ref: 45e382103
Ref: sort c gsl_heapsort gsl_comparison_fn_t382471
Ref: 45f382471
Ref: sort c gsl_heapsort_index383694
Ref: 460383694
Node: Sorting vectors384534
Ref: sort sorting-vectors384671
Ref: 461384671
Ref: sort c gsl_sort385706
Ref: 462385706
Ref: sort c gsl_sort2385947
Ref: 463385947
Ref: sort c gsl_sort_vector386366
Ref: 464386366
Ref: sort c gsl_sort_vector2386522
Ref: 465386522
Ref: sort c gsl_sort_index386767
Ref: 466386767
Ref: sort c gsl_sort_vector_index387421
Ref: 467387421
Node: Selecting the k smallest or largest elements388081
Ref: sort selecting-the-k-smallest-or-largest-elements388221
Ref: 468388221
Ref: sort c gsl_sort_smallest388930
Ref: 469388930
Ref: sort c gsl_sort_largest389419
Ref: 46a389419
Ref: sort c gsl_sort_vector_smallest389879
Ref: 46b389879
Ref: sort c gsl_sort_vector_largest389978
Ref: 46c389978
Ref: sort c gsl_sort_smallest_index390404
Ref: 46d390404
Ref: sort c gsl_sort_largest_index390941
Ref: 46e390941
Ref: sort c gsl_sort_vector_smallest_index391483
Ref: 46f391483
Ref: sort c gsl_sort_vector_largest_index391585
Ref: 470391585
Node: Computing the rank391931
Ref: sort computing-the-rank392067
Ref: 471392067
Node: Examples<7>393140
Ref: sort examples393265
Ref: 472393265
Node: References and Further Reading<7>394776
Ref: sort references-and-further-reading394874
Ref: 473394874
Node: BLAS Support395285
Ref: blas doc395377
Ref: 474395377
Ref: blas blas-support395377
Ref: 475395377
Ref: blas chap-blas-support395377
Ref: 11395377
Ref: BLAS Support-Footnote-1399327
Node: GSL BLAS Interface399500
Ref: blas gsl-blas-interface399588
Ref: 477399588
Node: Level 1399932
Ref: blas level-1400011
Ref: 478400011
Ref: blas c gsl_blas_sdsdot400042
Ref: 479400042
Ref: blas c gsl_blas_sdot400323
Ref: 47a400323
Ref: blas c gsl_blas_dsdot400435
Ref: 47b400435
Ref: blas c gsl_blas_ddot400549
Ref: 47c400549
Ref: blas c gsl_blas_cdotu400807
Ref: 47d400807
Ref: blas c gsl_blas_zdotu400946
Ref: 47e400946
Ref: blas c gsl_blas_cdotc401230
Ref: 47f401230
Ref: blas c gsl_blas_zdotc401369
Ref: 480401369
Ref: blas c gsl_blas_snrm2401663
Ref: 481401663
Ref: blas c gsl_blas_dnrm2401726
Ref: 482401726
Ref: blas c gsl_blas_scnrm2401895
Ref: 483401895
Ref: blas c gsl_blas_dznrm2401967
Ref: 484401967
Ref: blas c gsl_blas_sasum402185
Ref: 485402185
Ref: blas c gsl_blas_dasum402248
Ref: 486402248
Ref: blas c gsl_blas_scasum402414
Ref: 487402414
Ref: blas c gsl_blas_dzasum402486
Ref: 488402486
Ref: blas c gsl_blas_isamax402733
Ref: 489402733
Ref: blas c gsl_blas_idamax402805
Ref: 48a402805
Ref: blas c gsl_blas_icamax402871
Ref: 48b402871
Ref: blas c gsl_blas_izamax402961
Ref: 48c402961
Ref: blas c gsl_blas_sswap403444
Ref: 48d403444
Ref: blas c gsl_blas_dswap403530
Ref: 48e403530
Ref: blas c gsl_blas_cswap403594
Ref: 48f403594
Ref: blas c gsl_blas_zswap403696
Ref: 490403696
Ref: blas c gsl_blas_scopy403883
Ref: 491403883
Ref: blas c gsl_blas_dcopy403975
Ref: 492403975
Ref: blas c gsl_blas_ccopy404045
Ref: 493404045
Ref: blas c gsl_blas_zcopy404153
Ref: 494404153
Ref: blas c gsl_blas_saxpy404353
Ref: 495404353
Ref: blas c gsl_blas_daxpy404458
Ref: 496404458
Ref: blas c gsl_blas_caxpy404552
Ref: 497404552
Ref: blas c gsl_blas_zaxpy404691
Ref: 498404691
Ref: blas c gsl_blas_sscal404921
Ref: 499404921
Ref: blas c gsl_blas_dscal404990
Ref: 49a404990
Ref: blas c gsl_blas_cscal405054
Ref: 49b405054
Ref: blas c gsl_blas_zscal405159
Ref: 49c405159
Ref: blas c gsl_blas_csscal405252
Ref: 49d405252
Ref: blas c gsl_blas_zdscal405340
Ref: 49e405340
Ref: blas c gsl_blas_srotg405521
Ref: 49f405521
Ref: blas c gsl_blas_drotg405609
Ref: 4a0405609
Ref: blas c gsl_blas_srot405950
Ref: 4a1405950
Ref: blas c gsl_blas_drot406053
Ref: 4a2406053
Ref: blas c gsl_blas_srotmg406288
Ref: 4a3406288
Ref: blas c gsl_blas_drotmg406390
Ref: 4a4406390
Ref: blas c gsl_blas_srotm406688
Ref: 4a5406688
Ref: blas c gsl_blas_drotm406791
Ref: 4a6406791
Node: Level 2406946
Ref: blas level-2407041
Ref: 4a7407041
Ref: blas c gsl_blas_sgemv407072
Ref: 4a8407072
Ref: blas c gsl_blas_dgemv407252
Ref: 4a9407252
Ref: blas c gsl_blas_cgemv407416
Ref: 4aa407416
Ref: blas c gsl_blas_zgemv407666
Ref: 4ab407666
Ref: blas c gsl_blas_strmv408102
Ref: 4ac408102
Ref: blas c gsl_blas_dtrmv408268
Ref: 4ad408268
Ref: blas c gsl_blas_ctrmv408412
Ref: 4ae408412
Ref: blas c gsl_blas_ztrmv408594
Ref: 4af408594
Ref: blas c gsl_blas_strsv409442
Ref: 4b0409442
Ref: blas c gsl_blas_dtrsv409608
Ref: 4b1409608
Ref: blas c gsl_blas_ctrsv409752
Ref: 4b2409752
Ref: blas c gsl_blas_ztrsv409934
Ref: 4b3409934
Ref: blas c gsl_blas_ssymv410731
Ref: 4b4410731
Ref: blas c gsl_blas_dsymv410904
Ref: 4b5410904
Ref: blas c gsl_blas_chemv411535
Ref: 4b6411535
Ref: blas c gsl_blas_zhemv411778
Ref: 4b7411778
Ref: blas c gsl_blas_sger412566
Ref: 4b8412566
Ref: blas c gsl_blas_dger412697
Ref: 4b9412697
Ref: blas c gsl_blas_cgeru412811
Ref: 4ba412811
Ref: blas c gsl_blas_zgeru412995
Ref: 4bb412995
Ref: blas c gsl_blas_cgerc413258
Ref: 4bc413258
Ref: blas c gsl_blas_zgerc413442
Ref: 4bd413442
Ref: blas c gsl_blas_ssyr413715
Ref: 4be413715
Ref: blas c gsl_blas_dsyr413838
Ref: 4bf413838
Ref: blas c gsl_blas_cher414413
Ref: 4c0414413
Ref: blas c gsl_blas_zher414552
Ref: 4c1414552
Ref: blas c gsl_blas_ssyr2415219
Ref: 4c2415219
Ref: blas c gsl_blas_dsyr2415380
Ref: 4c3415380
Ref: blas c gsl_blas_cher2415992
Ref: 4c4415992
Ref: blas c gsl_blas_zher2416205
Ref: 4c5416205
Node: Level 3416939
Ref: blas level-3417018
Ref: 4c6417018
Ref: blas c gsl_blas_sgemm417049
Ref: 4c7417049
Ref: blas c gsl_blas_dgemm417265
Ref: 4c8417265
Ref: blas c gsl_blas_cgemm417455
Ref: 4c9417455
Ref: blas c gsl_blas_zgemm417731
Ref: 4ca417731
Ref: blas c gsl_blas_ssymm418251
Ref: 4cb418251
Ref: blas c gsl_blas_dsymm418443
Ref: 4cc418443
Ref: blas c gsl_blas_csymm418619
Ref: 4cd418619
Ref: blas c gsl_blas_zsymm418881
Ref: 4ce418881
Ref: blas c gsl_blas_chemm419606
Ref: 4cf419606
Ref: blas c gsl_blas_zhemm419868
Ref: 4d0419868
Ref: blas c gsl_blas_strmm420669
Ref: 4d1420669
Ref: blas c gsl_blas_dtrmm420867
Ref: 4d2420867
Ref: blas c gsl_blas_ctrmm421054
Ref: 4d3421054
Ref: blas c gsl_blas_ztrmm421296
Ref: 4d4421296
Ref: blas c gsl_blas_strsm422307
Ref: 4d5422307
Ref: blas c gsl_blas_dtrsm422505
Ref: 4d6422505
Ref: blas c gsl_blas_ctrsm422692
Ref: 4d7422692
Ref: blas c gsl_blas_ztrsm422934
Ref: 4d8422934
Ref: blas c gsl_blas_ssyrk423966
Ref: 4d9423966
Ref: blas c gsl_blas_dsyrk424137
Ref: 4da424137
Ref: blas c gsl_blas_csyrk424298
Ref: 4db424298
Ref: blas c gsl_blas_zsyrk424531
Ref: 4dc424531
Ref: blas c gsl_blas_cherk425312
Ref: 4dd425312
Ref: blas c gsl_blas_zherk425499
Ref: 4de425499
Ref: blas c gsl_blas_ssyr2k426342
Ref: 4df426342
Ref: blas c gsl_blas_dsyr2k426541
Ref: 4e0426541
Ref: blas c gsl_blas_csyr2k426724
Ref: 4e1426724
Ref: blas c gsl_blas_zsyr2k426993
Ref: 4e2426993
Ref: blas c gsl_blas_cher2k427850
Ref: 4e3427850
Ref: blas c gsl_blas_zher2k428101
Ref: 4e4428101
Node: Examples<8>429031
Ref: blas examples429161
Ref: 4e5429161
Node: References and Further Reading<8>430433
Ref: blas references-and-further-reading430536
Ref: 4e6430536
Ref: blas sec-blas-references430536
Ref: 4e7430536
Node: Linear Algebra431788
Ref: linalg doc431885
Ref: 4e8431885
Ref: linalg linear-algebra431885
Ref: 4e9431885
Node: LU Decomposition433189
Ref: linalg lu-decomposition433282
Ref: 4ea433282
Ref: linalg sec-lu-decomposition433282
Ref: 4eb433282
Ref: linalg c gsl_linalg_LU_decomp433942
Ref: 4ec433942
Ref: linalg c gsl_linalg_complex_LU_decomp434040
Ref: 4ed434040
Ref: linalg c gsl_linalg_LU_solve435205
Ref: 4ee435205
Ref: linalg c gsl_linalg_complex_LU_solve435338
Ref: 4ef435338
Ref: linalg c gsl_linalg_LU_svx435751
Ref: 4f0435751
Ref: linalg c gsl_linalg_complex_LU_svx435861
Ref: 4f1435861
Ref: linalg c gsl_linalg_LU_refine436262
Ref: 4f2436262
Ref: linalg c gsl_linalg_complex_LU_refine436445
Ref: 4f3436445
Ref: linalg c gsl_linalg_LU_invert436949
Ref: 4f4436949
Ref: linalg c gsl_linalg_complex_LU_invert437068
Ref: 4f5437068
Ref: linalg c gsl_linalg_LU_invx437827
Ref: 4f6437827
Ref: linalg c gsl_linalg_complex_LU_invx437917
Ref: 4f7437917
Ref: linalg c gsl_linalg_LU_det438643
Ref: 4f8438643
Ref: linalg c gsl_linalg_complex_LU_det438711
Ref: 4f9438711
Ref: linalg c gsl_linalg_LU_lndet439057
Ref: 4fa439057
Ref: linalg c gsl_linalg_complex_LU_lndet439115
Ref: 4fb439115
Ref: linalg c gsl_linalg_LU_sgndet439474
Ref: 4fc439474
Ref: linalg c gsl_linalg_complex_LU_sgndet439542
Ref: 4fd439542
Node: QR Decomposition439805
Ref: linalg linalg-qr439944
Ref: 4fe439944
Ref: linalg qr-decomposition439944
Ref: 4ff439944
Ref: linalg c gsl_linalg_QR_decomp_r442458
Ref: 502442458
Ref: linalg c gsl_linalg_complex_QR_decomp_r442530
Ref: 503442530
Ref: linalg c gsl_linalg_QR_solve_r443257
Ref: 504443257
Ref: linalg c gsl_linalg_complex_QR_solve_r443387
Ref: 505443387
Ref: linalg c gsl_linalg_QR_lssolve_r443872
Ref: 506443872
Ref: linalg c gsl_linalg_complex_QR_lssolve_r444032
Ref: 507444032
Ref: linalg c gsl_linalg_QR_QTvec_r445101
Ref: 508445101
Ref: linalg c gsl_linalg_complex_QR_QHvec_r445228
Ref: 509445228
Ref: linalg c gsl_linalg_QR_QTmat_r445868
Ref: 50a445868
Ref: linalg c gsl_linalg_QR_unpack_r446411
Ref: 50b446411
Ref: linalg c gsl_linalg_complex_QR_unpack_r446536
Ref: 50c446536
Ref: linalg c gsl_linalg_QR_rcond447173
Ref: 50d447173
Node: Level 2 Interface447791
Ref: linalg level-2-interface447899
Ref: 50e447899
Ref: linalg c gsl_linalg_QR_decomp448159
Ref: 50f448159
Ref: linalg c gsl_linalg_complex_QR_decomp448231
Ref: 510448231
Ref: linalg c gsl_linalg_QR_solve449261
Ref: 511449261
Ref: linalg c gsl_linalg_complex_QR_solve449391
Ref: 512449391
Ref: linalg c gsl_linalg_QR_svx449822
Ref: 514449822
Ref: linalg c gsl_linalg_complex_QR_svx449929
Ref: 515449929
Ref: linalg c gsl_linalg_QR_lssolve450328
Ref: 513450328
Ref: linalg c gsl_linalg_complex_QR_lssolve450492
Ref: 516450492
Ref: linalg c gsl_linalg_QR_QTvec451288
Ref: 517451288
Ref: linalg c gsl_linalg_complex_QR_QHvec451397
Ref: 518451397
Ref: linalg c gsl_linalg_QR_Qvec451934
Ref: 519451934
Ref: linalg c gsl_linalg_complex_QR_Qvec452042
Ref: 51a452042
Ref: linalg c gsl_linalg_QR_QTmat452514
Ref: 51b452514
Ref: linalg c gsl_linalg_QR_Rsolve452968
Ref: 51c452968
Ref: linalg c gsl_linalg_QR_Rsvx453272
Ref: 51d453272
Ref: linalg c gsl_linalg_QR_unpack453688
Ref: 51e453688
Ref: linalg c gsl_linalg_QR_QRsolve454023
Ref: 51f454023
Ref: linalg c gsl_linalg_QR_update454337
Ref: 520454337
Ref: linalg c gsl_linalg_R_solve454754
Ref: 521454754
Ref: linalg c gsl_linalg_R_svx454958
Ref: 522454958
Node: Triangle on Top of Rectangle455215
Ref: linalg triangle-on-top-of-rectangle455359
Ref: 523455359
Ref: linalg c gsl_linalg_QR_UR_decomp455963
Ref: 524455963
Node: Triangle on Top of Triangle456362
Ref: linalg triangle-on-top-of-triangle456519
Ref: 525456519
Ref: linalg c gsl_linalg_QR_UU_decomp456972
Ref: 526456972
Ref: linalg c gsl_linalg_QR_UU_lssolve457388
Ref: 527457388
Ref: linalg c gsl_linalg_QR_UU_QTec458398
Ref: 528458398
Node: Triangle on Top of Trapezoidal458846
Ref: linalg triangle-on-top-of-trapezoidal459002
Ref: 529459002
Ref: linalg c gsl_linalg_QR_UZ_decomp459659
Ref: 52a459659
Node: Triangle on Top of Diagonal460070
Ref: linalg triangle-on-top-of-diagonal460190
Ref: 52b460190
Ref: linalg c gsl_linalg_QR_UD_decomp460735
Ref: 52c460735
Ref: linalg c gsl_linalg_QR_UD_lssolve461179
Ref: 52d461179
Node: QR Decomposition with Column Pivoting462186
Ref: linalg linalg-qrpt462325
Ref: 500462325
Ref: linalg qr-decomposition-with-column-pivoting462325
Ref: 52e462325
Ref: linalg c gsl_linalg_QRPT_decomp463477
Ref: 52f463477
Ref: linalg c gsl_linalg_QRPT_decomp2464840
Ref: 530464840
Ref: linalg c gsl_linalg_QRPT_solve465248
Ref: 531465248
Ref: linalg c gsl_linalg_QRPT_svx465656
Ref: 532465656
Ref: linalg c gsl_linalg_QRPT_lssolve466067
Ref: 533466067
Ref: linalg c gsl_linalg_QRPT_lssolve2466935
Ref: 534466935
Ref: linalg c gsl_linalg_QRPT_QRsolve467912
Ref: 536467912
Ref: linalg c gsl_linalg_QRPT_update468287
Ref: 537468287
Ref: linalg c gsl_linalg_QRPT_Rsolve468805
Ref: 538468805
Ref: linalg c gsl_linalg_QRPT_Rsvx469060
Ref: 539469060
Ref: linalg c gsl_linalg_QRPT_rank469418
Ref: 535469418
Ref: linalg c gsl_linalg_QRPT_rcond469876
Ref: 53a469876
Node: LQ Decomposition470332
Ref: linalg lq-decomposition470471
Ref: 53b470471
Ref: linalg c gsl_linalg_LQ_decomp471297
Ref: 53c471297
Ref: linalg c gsl_linalg_LQ_lssolve472097
Ref: 53d472097
Ref: linalg c gsl_linalg_LQ_unpack472709
Ref: 53e472709
Ref: linalg c gsl_linalg_LQ_QTvec473044
Ref: 53f473044
Node: QL Decomposition473272
Ref: linalg ql-decomposition473407
Ref: 540473407
Ref: linalg c gsl_linalg_QL_decomp473885
Ref: 541473885
Ref: linalg c gsl_linalg_QL_unpack474284
Ref: 542474284
Node: Complete Orthogonal Decomposition474619
Ref: linalg complete-orthogonal-decomposition474766
Ref: 543474766
Ref: linalg linalg-cod474766
Ref: 501474766
Ref: linalg c gsl_linalg_COD_decomp476391
Ref: 544476391
Ref: linalg c gsl_linalg_COD_decomp_e476557
Ref: 545476557
Ref: linalg c gsl_linalg_COD_lssolve477636
Ref: 546477636
Ref: linalg c gsl_linalg_COD_lssolve2478547
Ref: 547478547
Ref: linalg c gsl_linalg_COD_unpack479569
Ref: 548479569
Ref: linalg c gsl_linalg_COD_matZ480069
Ref: 549480069
Node: Singular Value Decomposition480572
Ref: linalg singular-value-decomposition480725
Ref: 54a480725
Ref: linalg c gsl_linalg_SV_decomp482352
Ref: 54b482352
Ref: linalg c gsl_linalg_SV_decomp_mod483145
Ref: 54c483145
Ref: linalg c gsl_linalg_SV_decomp_jacobi483526
Ref: 54d483526
Ref: linalg c gsl_linalg_SV_solve483904
Ref: 54e483904
Ref: linalg c gsl_linalg_SV_leverage484757
Ref: 54f484757
Node: Cholesky Decomposition485224
Ref: linalg cholesky-decomposition485374
Ref: 550485374
Ref: linalg sec-cholesky-decomposition485374
Ref: 3f5485374
Ref: linalg c gsl_linalg_cholesky_decomp1486343
Ref: 551486343
Ref: linalg c gsl_linalg_complex_cholesky_decomp486405
Ref: 552486405
Ref: linalg c gsl_linalg_cholesky_decomp487385
Ref: 553487385
Ref: linalg c gsl_linalg_cholesky_solve487539
Ref: 554487539
Ref: linalg c gsl_linalg_complex_cholesky_solve487658
Ref: 555487658
Ref: linalg c gsl_linalg_cholesky_svx488102
Ref: 556488102
Ref: linalg c gsl_linalg_complex_cholesky_svx488198
Ref: 557488198
Ref: linalg c gsl_linalg_cholesky_invert488727
Ref: 558488727
Ref: linalg c gsl_linalg_complex_cholesky_invert488795
Ref: 559488795
Ref: linalg c gsl_linalg_cholesky_decomp2489221
Ref: 55a489221
Ref: linalg c gsl_linalg_cholesky_solve2490247
Ref: 55b490247
Ref: linalg c gsl_linalg_cholesky_svx2490645
Ref: 55c490645
Ref: linalg c gsl_linalg_cholesky_scale491136
Ref: 55d491136
Ref: linalg c gsl_linalg_cholesky_scale_apply491686
Ref: 55e491686
Ref: linalg c gsl_linalg_cholesky_rcond491939
Ref: 55f491939
Node: Pivoted Cholesky Decomposition492458
Ref: linalg pivoted-cholesky-decomposition492611
Ref: 560492611
Ref: linalg c gsl_linalg_pcholesky_decomp493362
Ref: 561493362
Ref: linalg c gsl_linalg_pcholesky_solve494156
Ref: 562494156
Ref: linalg c gsl_linalg_pcholesky_svx494558
Ref: 563494558
Ref: linalg c gsl_linalg_pcholesky_decomp2495071
Ref: 564495071
Ref: linalg c gsl_linalg_pcholesky_solve2496276
Ref: 565496276
Ref: linalg c gsl_linalg_pcholesky_svx2496758
Ref: 566496758
Ref: linalg c gsl_linalg_pcholesky_invert497341
Ref: 567497341
Ref: linalg c gsl_linalg_pcholesky_rcond497678
Ref: 568497678
Node: Modified Cholesky Decomposition498224
Ref: linalg modified-cholesky-decomposition498373
Ref: 569498373
Ref: linalg c gsl_linalg_mcholesky_decomp499458
Ref: 56a499458
Ref: linalg c gsl_linalg_mcholesky_solve500415
Ref: 56b500415
Ref: linalg c gsl_linalg_mcholesky_svx500829
Ref: 56c500829
Ref: linalg c gsl_linalg_mcholesky_rcond501353
Ref: 56d501353
Node: LDLT Decomposition501895
Ref: linalg ldlt-decomposition502066
Ref: 56e502066
Ref: linalg sec-ldlt-decomposition502066
Ref: 56f502066
Ref: linalg c gsl_linalg_ldlt_decomp502863
Ref: 570502863
Ref: linalg c gsl_linalg_ldlt_solve503640
Ref: 572503640
Ref: linalg c gsl_linalg_ldlt_svx503967
Ref: 573503967
Ref: linalg c gsl_linalg_ldlt_rcond504392
Ref: 571504392
Node: Tridiagonal Decomposition of Real Symmetric Matrices504892
Ref: linalg tridiagonal-decomposition-of-real-symmetric-matrices505079
Ref: 574505079
Ref: linalg c gsl_linalg_symmtd_decomp505375
Ref: 575505375
Ref: linalg c gsl_linalg_symmtd_unpack506041
Ref: 576506041
Ref: linalg c gsl_linalg_symmtd_unpack_T506534
Ref: 577506534
Node: Tridiagonal Decomposition of Hermitian Matrices506911
Ref: linalg tridiagonal-decomposition-of-hermitian-matrices507121
Ref: 578507121
Ref: linalg c gsl_linalg_hermtd_decomp507408
Ref: 579507408
Ref: linalg c gsl_linalg_hermtd_unpack508135
Ref: 57a508135
Ref: linalg c gsl_linalg_hermtd_unpack_T508649
Ref: 57b508649
Node: Hessenberg Decomposition of Real Matrices509033
Ref: linalg hessenberg-decomposition-of-real-matrices509243
Ref: 57c509243
Ref: linalg c gsl_linalg_hessenberg_decomp509728
Ref: 57d509728
Ref: linalg c gsl_linalg_hessenberg_unpack510433
Ref: 57e510433
Ref: linalg c gsl_linalg_hessenberg_unpack_accum510809
Ref: 57f510809
Ref: linalg c gsl_linalg_hessenberg_set_zero511418
Ref: 580511418
Node: Hessenberg-Triangular Decomposition of Real Matrices511714
Ref: linalg hessenberg-triangular-decomposition-of-real-matrices511894
Ref: 581511894
Ref: linalg c gsl_linalg_hesstri_decomp512381
Ref: 582512381
Node: Bidiagonalization512917
Ref: linalg bidiagonalization513072
Ref: 583513072
Ref: linalg c gsl_linalg_bidiag_decomp513424
Ref: 584513424
Ref: linalg c gsl_linalg_bidiag_unpack514164
Ref: 585514164
Ref: linalg c gsl_linalg_bidiag_unpack2514835
Ref: 586514835
Ref: linalg c gsl_linalg_bidiag_unpack_B515348
Ref: 587515348
Node: Givens Rotations515723
Ref: linalg givens-rotations515853
Ref: 588515853
Ref: linalg c gsl_linalg_givens516495
Ref: 589516495
Ref: linalg c gsl_linalg_givens_gv516782
Ref: 58a516782
Node: Householder Transformations517148
Ref: linalg householder-transformations517298
Ref: 58b517298
Ref: linalg c gsl_linalg_householder_transform517801
Ref: 58c517801
Ref: linalg c gsl_linalg_complex_householder_transform517871
Ref: 58d517871
Ref: linalg c gsl_linalg_householder_hm518558
Ref: 58e518558
Ref: linalg c gsl_linalg_complex_householder_hm518661
Ref: 58f518661
Ref: linalg c gsl_linalg_householder_mh519032
Ref: 590519032
Ref: linalg c gsl_linalg_complex_householder_mh519135
Ref: 591519135
Ref: linalg c gsl_linalg_householder_hv519507
Ref: 592519507
Ref: linalg c gsl_linalg_complex_householder_hv519610
Ref: 593519610
Node: Householder solver for linear systems519967
Ref: linalg householder-solver-for-linear-systems520120
Ref: 594520120
Ref: linalg c gsl_linalg_HH_solve520209
Ref: 595520209
Ref: linalg c gsl_linalg_HH_svx520576
Ref: 596520576
Node: Tridiagonal Systems520935
Ref: linalg tridiagonal-systems521079
Ref: 597521079
Ref: linalg c gsl_linalg_solve_tridiag521511
Ref: 598521511
Ref: linalg c gsl_linalg_solve_symm_tridiag522141
Ref: 599522141
Ref: linalg c gsl_linalg_solve_cyc_tridiag522712
Ref: 59a522712
Ref: linalg c gsl_linalg_solve_symm_cyc_tridiag523367
Ref: 59b523367
Node: Triangular Systems523979
Ref: linalg triangular-systems524100
Ref: 59c524100
Ref: linalg c gsl_linalg_tri_invert524151
Ref: 59d524151
Ref: linalg c gsl_linalg_complex_tri_invert524255
Ref: 59e524255
Ref: linalg c gsl_linalg_tri_LTL524736
Ref: 59f524736
Ref: linalg c gsl_linalg_complex_tri_LHL524789
Ref: 5a0524789
Ref: linalg c gsl_linalg_tri_UL525009
Ref: 5a1525009
Ref: linalg c gsl_linalg_complex_tri_UL525062
Ref: 5a2525062
Ref: linalg c gsl_linalg_tri_rcond525433
Ref: 5a3525433
Node: Banded Systems526051
Ref: linalg banded-systems526162
Ref: 5a4526162
Node: General Banded Format526807
Ref: linalg general-banded-format526912
Ref: 5a5526912
Node: Symmetric Banded Format528429
Ref: linalg sec-symmetric-banded528566
Ref: 5a6528566
Ref: linalg symmetric-banded-format528566
Ref: 5a7528566
Node: Banded LU Decomposition530197
Ref: linalg banded-lu-decomposition530342
Ref: 5a8530342
Ref: linalg c gsl_linalg_LU_band_decomp532274
Ref: 5a9532274
Ref: linalg c gsl_linalg_LU_band_solve533097
Ref: 5aa533097
Ref: linalg c gsl_linalg_LU_band_svx533691
Ref: 5ab533691
Ref: linalg c gsl_linalg_LU_band_unpack534285
Ref: 5ac534285
Node: Banded Cholesky Decomposition534930
Ref: linalg banded-cholesky-decomposition535077
Ref: 5ad535077
Ref: linalg c gsl_linalg_cholesky_band_decomp535627
Ref: 5ae535627
Ref: linalg c gsl_linalg_cholesky_band_solve536587
Ref: 5b0536587
Ref: linalg c gsl_linalg_cholesky_band_solvem536706
Ref: 5b1536706
Ref: linalg c gsl_linalg_cholesky_band_svx537081
Ref: 5b2537081
Ref: linalg c gsl_linalg_cholesky_band_svxm537177
Ref: 5b3537177
Ref: linalg c gsl_linalg_cholesky_band_invert537675
Ref: 5b4537675
Ref: linalg c gsl_linalg_cholesky_band_unpack538101
Ref: 5b5538101
Ref: linalg c gsl_linalg_cholesky_band_scale538449
Ref: 5b6538449
Ref: linalg c gsl_linalg_cholesky_band_scale_apply539011
Ref: 5b7539011
Ref: linalg c gsl_linalg_cholesky_band_rcond539304
Ref: 5af539304
Node: Banded LDLT Decomposition539825
Ref: linalg banded-ldlt-decomposition539940
Ref: 5b8539940
Ref: linalg c gsl_linalg_ldlt_band_decomp540479
Ref: 5b9540479
Ref: linalg c gsl_linalg_ldlt_band_solve541077
Ref: 5ba541077
Ref: linalg c gsl_linalg_ldlt_band_svx541431
Ref: 5bb541431
Ref: linalg c gsl_linalg_ldlt_band_unpack541888
Ref: 5bc541888
Ref: linalg c gsl_linalg_ldlt_band_rcond542313
Ref: 5bd542313
Node: Balancing542825
Ref: linalg balancing542929
Ref: 5be542929
Ref: linalg id1542929
Ref: 5bf542929
Ref: linalg c gsl_linalg_balance_matrix543354
Ref: 5c0543354
Node: Examples<9>543630
Ref: linalg examples543753
Ref: 5c1543753
Node: References and Further Reading<9>545600
Ref: linalg references-and-further-reading545705
Ref: 5c2545705
Node: Eigensystems548095
Ref: eigen doc548208
Ref: 5c3548208
Ref: eigen eigensystems548208
Ref: 5c4548208
Node: Real Symmetric Matrices549331
Ref: eigen real-symmetric-matrices549439
Ref: 5c5549439
Ref: eigen c gsl_eigen_symm_workspace549780
Ref: 5c6549780
Ref: eigen c gsl_eigen_symm_alloc549919
Ref: 5c7549919
Ref: eigen c gsl_eigen_symm_free550194
Ref: 5c8550194
Ref: eigen c gsl_eigen_symm550350
Ref: 5c9550350
Ref: eigen c gsl_eigen_symmv_workspace550879
Ref: 5ca550879
Ref: eigen c gsl_eigen_symmv_alloc551035
Ref: 5cb551035
Ref: eigen c gsl_eigen_symmv_free551329
Ref: 5cc551329
Ref: eigen c gsl_eigen_symmv551487
Ref: 5cd551487
Node: Complex Hermitian Matrices552347
Ref: eigen complex-hermitian-matrices552490
Ref: 5ce552490
Ref: eigen c gsl_eigen_herm_workspace552674
Ref: 5cf552674
Ref: eigen c gsl_eigen_herm_alloc552813
Ref: 5d0552813
Ref: eigen c gsl_eigen_herm_free553091
Ref: 5d1553091
Ref: eigen c gsl_eigen_herm553247
Ref: 5d2553247
Ref: eigen c gsl_eigen_hermv_workspace553881
Ref: 5d3553881
Ref: eigen c gsl_eigen_hermv_alloc554037
Ref: 5d4554037
Ref: eigen c gsl_eigen_hermv_free554334
Ref: 5d5554334
Ref: eigen c gsl_eigen_hermv554492
Ref: 5d6554492
Node: Real Nonsymmetric Matrices555473
Ref: eigen real-nonsymmetric-matrices555641
Ref: 5d7555641
Ref: eigen c gsl_eigen_nonsymm_workspace556141
Ref: 5d8556141
Ref: eigen c gsl_eigen_nonsymm_alloc556286
Ref: 5d9556286
Ref: eigen c gsl_eigen_nonsymm_free556570
Ref: 5da556570
Ref: eigen c gsl_eigen_nonsymm_params556742
Ref: 5db556742
Ref: eigen c gsl_eigen_nonsymm558452
Ref: 5dc558452
Ref: eigen c gsl_eigen_nonsymm_Z559281
Ref: 5dd559281
Ref: eigen c gsl_eigen_nonsymmv_workspace559579
Ref: 5de559579
Ref: eigen c gsl_eigen_nonsymmv_alloc559741
Ref: 5df559741
Ref: eigen c gsl_eigen_nonsymmv_free560044
Ref: 5e0560044
Ref: eigen c gsl_eigen_nonsymmv_params560218
Ref: 5e1560218
Ref: eigen c gsl_eigen_nonsymmv560760
Ref: 5e2560760
Ref: eigen c gsl_eigen_nonsymmv_Z561642
Ref: 5e3561642
Node: Real Generalized Symmetric-Definite Eigensystems561955
Ref: eigen real-generalized-symmetric-definite-eigensystems562148
Ref: 5e4562148
Ref: eigen c gsl_eigen_gensymm_workspace563043
Ref: 5e5563043
Ref: eigen c gsl_eigen_gensymm_alloc563197
Ref: 5e6563197
Ref: eigen c gsl_eigen_gensymm_free563503
Ref: 5e7563503
Ref: eigen c gsl_eigen_gensymm563675
Ref: 5e8563675
Ref: eigen c gsl_eigen_gensymmv_workspace564116
Ref: 5e9564116
Ref: eigen c gsl_eigen_gensymmv_alloc564287
Ref: 5ea564287
Ref: eigen c gsl_eigen_gensymmv_free564617
Ref: 5eb564617
Ref: eigen c gsl_eigen_gensymmv564791
Ref: 5ec564791
Node: Complex Generalized Hermitian-Definite Eigensystems565353
Ref: eigen complex-generalized-hermitian-definite-eigensystems565562
Ref: 5ed565562
Ref: eigen c gsl_eigen_genherm_workspace566272
Ref: 5ee566272
Ref: eigen c gsl_eigen_genherm_alloc566426
Ref: 5ef566426
Ref: eigen c gsl_eigen_genherm_free566740
Ref: 5f0566740
Ref: eigen c gsl_eigen_genherm566912
Ref: 5f1566912
Ref: eigen c gsl_eigen_genhermv_workspace567382
Ref: 5f2567382
Ref: eigen c gsl_eigen_genhermv_alloc567553
Ref: 5f3567553
Ref: eigen c gsl_eigen_genhermv_free567886
Ref: 5f4567886
Ref: eigen c gsl_eigen_genhermv568060
Ref: 5f5568060
Node: Real Generalized Nonsymmetric Eigensystems568649
Ref: eigen real-generalized-nonsymmetric-eigensystems568846
Ref: 5f6568846
Ref: eigen c gsl_eigen_gen_workspace570638
Ref: 5f7570638
Ref: eigen c gsl_eigen_gen_alloc570778
Ref: 5f8570778
Ref: eigen c gsl_eigen_gen_free571069
Ref: 5f9571069
Ref: eigen c gsl_eigen_gen_params571223
Ref: 5fa571223
Ref: eigen c gsl_eigen_gen572393
Ref: 5fb572393
Ref: eigen c gsl_eigen_gen_QZ573505
Ref: 5fc573505
Ref: eigen c gsl_eigen_genv_workspace573898
Ref: 5fd573898
Ref: eigen c gsl_eigen_genv_alloc574055
Ref: 5fe574055
Ref: eigen c gsl_eigen_genv_free574366
Ref: 5ff574366
Ref: eigen c gsl_eigen_genv574522
Ref: 600574522
Ref: eigen c gsl_eigen_genv_QZ575608
Ref: 601575608
Node: Sorting Eigenvalues and Eigenvectors576040
Ref: eigen sorting-eigenvalues-and-eigenvectors576198
Ref: 602576198
Ref: eigen c gsl_eigen_symmv_sort576283
Ref: 603576283
Ref: eigen c gsl_eigen_symmv_sort gsl_eigen_sort_t576703
Ref: 604576703
Ref: eigen c gsl_eigen_hermv_sort577418
Ref: 605577418
Ref: eigen c gsl_eigen_nonsymmv_sort577865
Ref: 606577865
Ref: eigen c gsl_eigen_gensymmv_sort578454
Ref: 607578454
Ref: eigen c gsl_eigen_genhermv_sort578893
Ref: 608578893
Ref: eigen c gsl_eigen_genv_sort579343
Ref: 609579343
Node: Examples<10>579978
Ref: eigen examples580128
Ref: 60a580128
Node: References and Further Reading<10>585337
Ref: eigen references-and-further-reading585442
Ref: 60b585442
Node: Fast Fourier Transforms FFTs586337
Ref: fft doc586457
Ref: 60c586457
Ref: fft fast-fourier-transforms-ffts586457
Ref: 60d586457
Node: Mathematical Definitions587589
Ref: fft mathematical-definitions587717
Ref: 60f587717
Node: Overview of complex data FFTs590031
Ref: fft overview-of-complex-data-ffts590205
Ref: 612590205
Node: Radix-2 FFT routines for complex data592992
Ref: fft radix-2-fft-routines-for-complex-data593183
Ref: 613593183
Ref: fft c gsl_fft_complex_radix2_forward593765
Ref: 35593765
Ref: fft c gsl_fft_complex_radix2_transform593881
Ref: 614593881
Ref: fft c gsl_fft_complex_radix2_backward594033
Ref: 615594033
Ref: fft c gsl_fft_complex_radix2_inverse594150
Ref: 616594150
Ref: fft c gsl_fft_complex_radix2_dif_forward594903
Ref: 617594903
Ref: fft c gsl_fft_complex_radix2_dif_transform595023
Ref: 618595023
Ref: fft c gsl_fft_complex_radix2_dif_backward595179
Ref: 619595179
Ref: fft c gsl_fft_complex_radix2_dif_inverse595300
Ref: 61a595300
Ref: fft fig-fft-complex-radix2597449
Ref: 61b597449
Node: Mixed-radix FFT routines for complex data597574
Ref: fft mixed-radix-fft-routines-for-complex-data597762
Ref: 61c597762
Ref: fft c gsl_fft_complex_wavetable_alloc600025
Ref: 61d600025
Ref: fft c gsl_fft_complex_wavetable_free601211
Ref: 61f601211
Ref: fft c gsl_fft_complex_wavetable601946
Ref: 61e601946
Ref: fft c gsl_fft_complex_workspace603152
Ref: 620603152
Ref: fft c gsl_fft_complex_workspace_alloc603309
Ref: 621603309
Ref: fft c gsl_fft_complex_workspace_free603511
Ref: 622603511
Ref: fft c gsl_fft_complex_forward603840
Ref: 610603840
Ref: fft c gsl_fft_complex_transform604036
Ref: 623604036
Ref: fft c gsl_fft_complex_backward604268
Ref: 624604268
Ref: fft c gsl_fft_complex_inverse604465
Ref: 611604465
Node: Overview of real data FFTs607887
Ref: fft overview-of-real-data-ffts608072
Ref: 625608072
Node: Radix-2 FFT routines for real data610344
Ref: fft radix-2-fft-routines-for-real-data610526
Ref: 626610526
Ref: fft c gsl_fft_real_radix2_transform610862
Ref: 627610862
Ref: fft c gsl_fft_halfcomplex_radix2_inverse612963
Ref: 629612963
Ref: fft c gsl_fft_halfcomplex_radix2_backward613067
Ref: 62a613067
Ref: fft c gsl_fft_halfcomplex_radix2_unpack613494
Ref: 628613494
Node: Mixed-radix FFT routines for real data614846
Ref: fft mixed-radix-fft-routines-for-real-data615036
Ref: 62b615036
Ref: fft c gsl_fft_real_wavetable618265
Ref: 62d618265
Ref: fft c gsl_fft_halfcomplex_wavetable618298
Ref: 62e618298
Ref: fft c gsl_fft_real_wavetable_alloc618422
Ref: 62f618422
Ref: fft c gsl_fft_halfcomplex_wavetable_alloc618522
Ref: 630618522
Ref: fft c gsl_fft_real_wavetable_free619677
Ref: 631619677
Ref: fft c gsl_fft_halfcomplex_wavetable_free619770
Ref: 632619770
Ref: fft c gsl_fft_real_workspace620170
Ref: 633620170
Ref: fft c gsl_fft_real_workspace_alloc620275
Ref: 634620275
Ref: fft c gsl_fft_real_workspace_free620563
Ref: 635620563
Ref: fft c gsl_fft_real_transform620917
Ref: 62c620917
Ref: fft c gsl_fft_halfcomplex_transform621090
Ref: 636621090
Ref: fft c gsl_fft_real_unpack622062
Ref: 637622062
Ref: fft c gsl_fft_halfcomplex_unpack622714
Ref: 638622714
Ref: fft fig-fft-real-mixedradix626090
Ref: 639626090
Node: References and Further Reading<11>626215
Ref: fft fft-references626362
Ref: 60e626362
Ref: fft references-and-further-reading626362
Ref: 63a626362
Node: Numerical Integration629045
Ref: integration doc629177
Ref: 63b629177
Ref: integration numerical-integration629177
Ref: 63c629177
Node: Introduction<2>630902
Ref: integration introduction631027
Ref: 63d631027
Node: Integrands without weight functions633127
Ref: integration integrands-without-weight-functions633256
Ref: 63e633256
Node: Integrands with weight functions633934
Ref: integration integrands-with-weight-functions634113
Ref: 63f634113
Node: Integrands with singular weight functions634623
Ref: integration integrands-with-singular-weight-functions634758
Ref: 640634758
Node: QNG non-adaptive Gauss-Kronrod integration635520
Ref: integration qng-non-adaptive-gauss-kronrod-integration635678
Ref: 641635678
Ref: integration c gsl_integration_qng635982
Ref: 642635982
Node: QAG adaptive integration636840
Ref: integration qag-adaptive-integration637027
Ref: 643637027
Ref: integration c gsl_integration_workspace637471
Ref: 644637471
Ref: integration c gsl_integration_workspace_alloc637610
Ref: 645637610
Ref: integration c gsl_integration_workspace_free638019
Ref: 646638019
Ref: integration c gsl_integration_qag638197
Ref: 647638197
Node: QAGS adaptive integration with singularities640374
Ref: integration qags-adaptive-integration-with-singularities640571
Ref: 648640571
Ref: integration c gsl_integration_qags641172
Ref: 649641172
Node: QAGP adaptive integration with known singular points642242
Ref: integration qagp-adaptive-integration-with-known-singular-points642462
Ref: 64a642462
Ref: integration c gsl_integration_qagp642579
Ref: 64b642579
Node: QAGI adaptive integration on infinite intervals643637
Ref: integration qagi-adaptive-integration-on-infinite-intervals643866
Ref: 64c643866
Ref: integration c gsl_integration_qagi643973
Ref: 64d643973
Ref: integration c gsl_integration_qagiu644766
Ref: 64e644766
Ref: integration c gsl_integration_qagil645334
Ref: 64f645334
Node: QAWC adaptive integration for Cauchy principal values645902
Ref: integration qawc-adaptive-integration-for-cauchy-principal-values646127
Ref: 650646127
Ref: integration c gsl_integration_qawc646246
Ref: 651646246
Node: QAWS adaptive integration for singular functions647079
Ref: integration qaws-adaptive-integration-for-singular-functions647308
Ref: 652647308
Ref: integration c gsl_integration_qaws_table647644
Ref: 653647644
Ref: integration c gsl_integration_qaws_table_alloc647764
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Ref: integration c gsl_integration_qaws_table_set649796
Ref: 655649796
Ref: integration c gsl_integration_qaws_table_free650096
Ref: 656650096
Ref: integration c gsl_integration_qaws650316
Ref: 657650316
Node: QAWO adaptive integration for oscillatory functions651354
Ref: integration qawo-adaptive-integration-for-oscillatory-functions651577
Ref: 658651577
Ref: integration c gsl_integration_qawo_table_alloc651945
Ref: 659651945
Ref: integration c gsl_integration_qawo_table_alloc GSL_INTEG_COSINE652661
Ref: 65a652661
Ref: integration c gsl_integration_qawo_table_alloc GSL_INTEG_SINE652695
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Ref: integration c gsl_integration_qawo_table_set653293
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Ref: integration c gsl_integration_qawo_table_set_length653597
Ref: 65e653597
Ref: integration c gsl_integration_qawo_table_free653819
Ref: 65f653819
Ref: integration c gsl_integration_qawo654003
Ref: 65c654003
Node: QAWF adaptive integration for Fourier integrals655380
Ref: integration qawf-adaptive-integration-for-fourier-integrals655588
Ref: 660655588
Ref: integration c gsl_integration_qawf655697
Ref: 661655697
Node: CQUAD doubly-adaptive integration658102
Ref: integration cquad-doubly-adaptive-integration658278
Ref: 662658278
Ref: integration c gsl_integration_cquad_workspace_alloc659149
Ref: 663659149
Ref: integration c gsl_integration_cquad_workspace_free659655
Ref: 664659655
Ref: integration c gsl_integration_cquad659845
Ref: 665659845
Node: Romberg integration661483
Ref: integration romberg-integration661638
Ref: 666661638
Ref: integration c gsl_integration_romberg_alloc662220
Ref: 667662220
Ref: integration c gsl_integration_romberg_free662670
Ref: 668662670
Ref: integration c gsl_integration_romberg662854
Ref: 669662854
Node: Gauss-Legendre integration663861
Ref: integration gauss-legendre-integration664006
Ref: 66a664006
Ref: integration c gsl_integration_glfixed_table_alloc664538
Ref: 66b664538
Ref: integration c gsl_integration_glfixed664951
Ref: 66c664951
Ref: integration c gsl_integration_glfixed_point665217
Ref: 66d665217
Ref: integration c gsl_integration_glfixed_table_free665773
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Node: Fixed point quadratures665955
Ref: integration fixed-point-quadratures666092
Ref: 66f666092
Ref: integration tab-fixed-quadratures667035
Ref: 670667035
Ref: integration c gsl_integration_fixed_workspace670397
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Ref: integration c gsl_integration_fixed_alloc670954
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Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type671644
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Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_legendre671809
Ref: 674671809
Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_chebyshev672101
Ref: 675672101
Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_gegenbauer672401
Ref: 676672401
Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_jacobi672658
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Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_laguerre672834
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Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_hermite673087
Ref: 679673087
Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_exponential673338
Ref: 67a673338
Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_rational673597
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Ref: integration c gsl_integration_fixed_alloc gsl_integration_fixed_type gsl_integration_fixed_chebyshev2673777
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Ref: integration c gsl_integration_fixed_free674078
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Ref: integration c gsl_integration_fixed_n674257
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Ref: integration c gsl_integration_fixed_nodes674428
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Ref: integration c gsl_integration_fixed_weights674640
Ref: 680674640
Ref: integration c gsl_integration_fixed674856
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Node: Error codes675401
Ref: integration error-codes675524
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Node: Examples<11>676444
Ref: integration examples676578
Ref: 683676578
Node: Adaptive integration example676687
Ref: integration adaptive-integration-example676804
Ref: 684676804
Node: Fixed-point quadrature example678711
Ref: integration fixed-point-quadrature-example678828
Ref: 685678828
Node: References and Further Reading<12>681502
Ref: integration references-and-further-reading681616
Ref: 686681616
Node: Random Number Generation682943
Ref: rng doc683069
Ref: 687683069
Ref: rng random-number-generation683069
Ref: 688683069
Node: General comments on random numbers684418
Ref: rng general-comments-on-random-numbers684560
Ref: 689684560
Node: The Random Number Generator Interface686387
Ref: rng the-random-number-generator-interface686576
Ref: 68a686576
Ref: rng c gsl_rng_type687429
Ref: 68b687429
Ref: rng c gsl_rng687452
Ref: 68c687452
Node: Random number generator initialization687842
Ref: rng random-number-generator-initialization688036
Ref: 68d688036
Ref: rng c gsl_rng_alloc688125
Ref: 68e688125
Ref: rng c gsl_rng_set688988
Ref: 691688988
Ref: rng c gsl_rng_free690158
Ref: 693690158
Node: Sampling from a random number generator690294
Ref: rng sampling-from-a-random-number-generator690494
Ref: 694690494
Ref: rng c gsl_rng_get690880
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Ref: rng c gsl_rng_uniform691319
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Ref: rng c gsl_rng_uniform_pos691926
Ref: 69a691926
Ref: rng c gsl_rng_uniform_int692368
Ref: 69b692368
Node: Auxiliary random number generator functions693727
Ref: rng auxiliary-random-number-generator-functions693924
Ref: 69f693924
Ref: rng c gsl_rng_name694191
Ref: 6a0694191
Ref: rng c gsl_rng_max694471
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Ref: rng c gsl_rng_min694629
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Ref: rng c gsl_rng_state694944
Ref: 6a1694944
Ref: rng c gsl_rng_size694997
Ref: 6a2694997
Ref: rng c gsl_rng_types_setup695425
Ref: 6a3695425
Node: Random number environment variables696093
Ref: rng random-number-environment-variables696288
Ref: 6a4696288
Ref: rng c GSL_RNG_TYPE696669
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Ref: rng c GSL_RNG_SEED696859
Ref: 690696859
Ref: rng c gsl_rng_default696980
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Ref: rng c gsl_rng_default_seed697301
Ref: 68f697301
Ref: rng c gsl_rng_env_setup697670
Ref: 6a6697670
Node: Copying random number generator state699429
Ref: rng copying-random-number-generator-state699630
Ref: 6a8699630
Ref: rng c gsl_rng_memcpy699966
Ref: 6a9699966
Ref: rng c gsl_rng_clone700279
Ref: 6aa700279
Node: Reading and writing random number generator state700474
Ref: rng reading-and-writing-random-number-generator-state700674
Ref: 6ab700674
Ref: rng c gsl_rng_fwrite700891
Ref: 6ac700891
Ref: rng c gsl_rng_fread701342
Ref: 6ad701342
Node: Random number generator algorithms701939
Ref: rng random-number-generator-algorithms702131
Ref: 6ae702131
Ref: rng c gsl_rng_mt19937702903
Ref: 69d702903
Ref: rng c gsl_rng_ranlxs0704415
Ref: 6af704415
Ref: rng c gsl_rng_ranlxs1704471
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Ref: rng c gsl_rng_ranlxs2704527
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Ref: rng c gsl_rng_ranlxd1705534
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Ref: rng c gsl_rng_ranlxd2705590
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Ref: rng c gsl_rng_ranlux705855
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Ref: rng c gsl_rng_ranlux389705910
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Ref: rng c gsl_rng_cmrg707398
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Ref: rng c gsl_rng_mrg708281
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Ref: rng c gsl_rng_taus708957
Ref: 69e708957
Ref: rng c gsl_rng_taus2709010
Ref: 6b6709010
Ref: rng c gsl_rng_gfsr4710473
Ref: 6b7710473
Node: Unix random number generators712472
Ref: rng unix-random-number-generators712645
Ref: 6b8712645
Ref: rng c gsl_rng_rand713386
Ref: 6b9713386
Ref: rng c gsl_rng_random_bsd713734
Ref: 6ba713734
Ref: rng c gsl_rng_random_libc5713793
Ref: 6bb713793
Ref: rng c gsl_rng_random_glibc2713854
Ref: 6bc713854
Ref: rng c gsl_rng_rand48715403
Ref: 6bd715403
Node: Other random number generators716449
Ref: rng other-random-number-generators716599
Ref: 6be716599
Ref: rng c gsl_rng_ranf717628
Ref: 6bf717628
Ref: rng c gsl_rng_ranmar719105
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Ref: rng c gsl_rng_r250719417
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Ref: rng c gsl_rng_tt800720011
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Ref: rng c gsl_rng_vax720603
Ref: 6c3720603
Ref: rng c gsl_rng_transputer720947
Ref: 6c4720947
Ref: rng c gsl_rng_randu721242
Ref: 6c5721242
Ref: rng c gsl_rng_minstd721588
Ref: 6c6721588
Ref: rng c gsl_rng_uni722465
Ref: 6c7722465
Ref: rng c gsl_rng_uni32722517
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Ref: rng c gsl_rng_slatec722810
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Ref: rng c gsl_rng_zuf722992
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Ref: rng c gsl_rng_knuthran2723458
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Ref: rng c gsl_rng_knuthran2002723785
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Ref: rng c gsl_rng_knuthran723846
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Ref: rng c gsl_rng_borosh13724278
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Ref: rng c gsl_rng_fishman18724335
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Ref: rng c gsl_rng_fishman20724393
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Ref: rng c gsl_rng_lecuyer21724451
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Ref: rng c gsl_rng_waterman14724509
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Ref: rng c gsl_rng_fishman2x725066
Ref: 6d3725066
Ref: rng c gsl_rng_coveyou725482
Ref: 6d4725482
Node: Performance725807
Ref: rng performance725940
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Node: Examples<12>726896
Ref: rng examples727033
Ref: 6d6727033
Node: References and Further Reading<13>728363
Ref: rng references-and-further-reading728505
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Node: Acknowledgements729644
Ref: rng acknowledgements729765
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Node: Quasi-Random Sequences730105
Ref: qrng doc730237
Ref: 6d9730237
Ref: qrng quasi-random-sequences730237
Ref: 6da730237
Node: Quasi-random number generator initialization731176
Ref: qrng quasi-random-number-generator-initialization731334
Ref: 6db731334
Ref: qrng c gsl_qrng731435
Ref: 6dc731435
Ref: qrng c gsl_qrng_alloc731519
Ref: 6dd731519
Ref: qrng c gsl_qrng_free731970
Ref: 6de731970
Ref: qrng c gsl_qrng_init732108
Ref: 6df732108
Node: Sampling from a quasi-random number generator732346
Ref: qrng sampling-from-a-quasi-random-number-generator732562
Ref: 6e0732562
Ref: qrng c gsl_qrng_get732665
Ref: 6e1732665
Node: Auxiliary quasi-random number generator functions733099
Ref: qrng auxiliary-quasi-random-number-generator-functions733327
Ref: 6e2733327
Ref: qrng c gsl_qrng_name733438
Ref: 6e3733438
Ref: qrng c gsl_qrng_size733567
Ref: 6e4733567
Ref: qrng c gsl_qrng_state733622
Ref: 6e5733622
Node: Saving and restoring quasi-random number generator state734048
Ref: qrng saving-and-restoring-quasi-random-number-generator-state734271
Ref: 6e6734271
Ref: qrng c gsl_qrng_memcpy734396
Ref: 6e7734396
Ref: qrng c gsl_qrng_clone734720
Ref: 6e8734720
Node: Quasi-random number generator algorithms734918
Ref: qrng quasi-random-number-generator-algorithms735104
Ref: 6e9735104
Ref: qrng c gsl_qrng_type735260
Ref: 6ea735260
Ref: qrng c gsl_qrng_type gsl_qrng_niederreiter_2735285
Ref: 6eb735285
Ref: qrng c gsl_qrng_type gsl_qrng_sobol735543
Ref: 6ec735543
Ref: qrng c gsl_qrng_type gsl_qrng_halton735790
Ref: 6ed735790
Ref: qrng c gsl_qrng_type gsl_qrng_reversehalton735852
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Node: Examples<13>736237
Ref: qrng examples736377
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Ref: qrng fig-qrng737274
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Node: References737383
Ref: qrng references737474
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Node: Random Number Distributions737872
Ref: randist doc737990
Ref: 6f2737990
Ref: randist chap-random-number-distributions737990
Ref: 695737990
Ref: randist random-number-distributions737990
Ref: 6f3737990
Node: Introduction<3>741162
Ref: randist introduction741276
Ref: 6f4741276
Node: The Gaussian Distribution742934
Ref: randist the-gaussian-distribution743087
Ref: 6f5743087
Ref: randist c gsl_ran_gaussian743150
Ref: 6f6743150
Ref: randist c gsl_ran_gaussian_pdf743809
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Ref: randist c gsl_ran_gaussian_ziggurat744095
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Ref: randist c gsl_ran_gaussian_ratio_method744185
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Ref: randist c gsl_ran_ugaussian744513
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Ref: randist c gsl_ran_ugaussian_pdf744571
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Ref: randist c gsl_ran_ugaussian_ratio_method744625
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Ref: randist c gsl_cdf_gaussian_P744875
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Ref: randist c gsl_cdf_gaussian_Q744940
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Ref: randist c gsl_cdf_gaussian_Pinv745005
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Ref: randist c gsl_cdf_gaussian_Qinv745073
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Ref: randist c gsl_cdf_ugaussian_P745322
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Ref: randist c gsl_cdf_ugaussian_Q745374
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Ref: randist c gsl_cdf_ugaussian_Pinv745426
Ref: 703745426
Ref: randist c gsl_cdf_ugaussian_Qinv745481
Ref: 704745481
Node: The Gaussian Tail Distribution745676
Ref: randist the-gaussian-tail-distribution745849
Ref: 705745849
Ref: randist c gsl_ran_gaussian_tail745922
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Ref: randist c gsl_ran_gaussian_tail_pdf746755
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Ref: randist c gsl_ran_ugaussian_tail747111
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Ref: randist c gsl_ran_ugaussian_tail_pdf747184
Ref: 709747184
Node: The Bivariate Gaussian Distribution747443
Ref: randist the-bivariate-gaussian-distribution747629
Ref: 70a747629
Ref: randist c gsl_ran_bivariate_gaussian747712
Ref: 70b747712
Ref: randist c gsl_ran_bivariate_gaussian_pdf748462
Ref: 70c748462
Node: The Multivariate Gaussian Distribution748923
Ref: randist the-multivariate-gaussian-distribution749107
Ref: 70d749107
Ref: randist c gsl_ran_multivariate_gaussian749196
Ref: 70e749196
Ref: randist c gsl_ran_multivariate_gaussian_pdf750031
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Ref: randist c gsl_ran_multivariate_gaussian_log_pdf750202
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Ref: randist c gsl_ran_multivariate_gaussian_mean750676
Ref: 711750676
Ref: randist c gsl_ran_multivariate_gaussian_vcov751215
Ref: 712751215
Node: The Exponential Distribution751851
Ref: randist the-exponential-distribution752024
Ref: 713752024
Ref: randist c gsl_ran_exponential752093
Ref: 714752093
Ref: randist c gsl_ran_exponential_pdf752366
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Ref: randist c gsl_cdf_exponential_P752642
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Ref: randist c gsl_cdf_exponential_Q752707
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Ref: randist c gsl_cdf_exponential_Pinv752772
Ref: 718752772
Ref: randist c gsl_cdf_exponential_Qinv752840
Ref: 719752840
Node: The Laplace Distribution753075
Ref: randist the-laplace-distribution753244
Ref: 71a753244
Ref: randist c gsl_ran_laplace753305
Ref: 71b753305
Ref: randist c gsl_ran_laplace_pdf753582
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Ref: randist c gsl_cdf_laplace_P753844
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Ref: randist c gsl_cdf_laplace_Q753904
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Ref: randist c gsl_cdf_laplace_Pinv753964
Ref: 71f753964
Ref: randist c gsl_cdf_laplace_Qinv754027
Ref: 720754027
Node: The Exponential Power Distribution754253
Ref: randist the-exponential-power-distribution754417
Ref: 721754417
Ref: randist c gsl_ran_exppow754498
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Ref: randist c gsl_ran_exppow_pdf755001
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Ref: randist c gsl_cdf_exppow_P755324
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Ref: randist c gsl_cdf_exppow_Q755393
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Node: The Cauchy Distribution755639
Ref: randist the-cauchy-distribution755804
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Ref: randist c gsl_ran_cauchy755863
Ref: 727755863
Ref: randist c gsl_ran_cauchy_pdf756281
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Ref: randist c gsl_cdf_cauchy_P756550
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Ref: randist c gsl_cdf_cauchy_Q756609
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Ref: randist c gsl_cdf_cauchy_Pinv756668
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Ref: randist c gsl_cdf_cauchy_Qinv756730
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Node: The Rayleigh Distribution756964
Ref: randist the-rayleigh-distribution757125
Ref: 72d757125
Ref: randist c gsl_ran_rayleigh757190
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Ref: randist c gsl_ran_rayleigh_pdf757494
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Ref: randist c gsl_cdf_rayleigh_P757777
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Ref: randist c gsl_cdf_rayleigh_Q757842
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Ref: randist c gsl_cdf_rayleigh_Pinv757907
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Ref: randist c gsl_cdf_rayleigh_Qinv757975
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Node: The Rayleigh Tail Distribution758221
Ref: randist the-rayleigh-tail-distribution758382
Ref: 734758382
Ref: randist c gsl_ran_rayleigh_tail758457
Ref: 735758457
Ref: randist c gsl_ran_rayleigh_tail_pdf758841
Ref: 736758841
Node: The Landau Distribution759194
Ref: randist the-landau-distribution759365
Ref: 737759365
Ref: randist c gsl_ran_landau759426
Ref: 738759426
Ref: randist c gsl_ran_landau_pdf759946
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Node: The Levy alpha-Stable Distributions760192
Ref: randist the-levy-alpha-stable-distributions760372
Ref: 73a760372
Ref: randist c gsl_ran_levy760457
Ref: 73b760457
Node: The Levy skew alpha-Stable Distribution761307
Ref: randist the-levy-skew-alpha-stable-distribution761486
Ref: 73c761486
Ref: randist c gsl_ran_levy_skew761579
Ref: 73d761579
Node: The Gamma Distribution762983
Ref: randist the-gamma-distribution763156
Ref: 73e763156
Ref: randist c gsl_ran_gamma763215
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Ref: randist c gsl_ran_gamma_knuth763810
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Ref: randist c gsl_ran_gamma_pdf763991
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Ref: randist c gsl_cdf_gamma_P764280
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Ref: randist c gsl_cdf_gamma_Q764348
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Ref: randist c gsl_cdf_gamma_Pinv764416
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Ref: randist c gsl_cdf_gamma_Qinv764487
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Node: The Flat Uniform Distribution764742
Ref: randist the-flat-uniform-distribution764902
Ref: 746764902
Ref: randist c gsl_ran_flat764979
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Ref: randist c gsl_ran_flat_pdf765281
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Ref: randist c gsl_cdf_flat_P765558
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Ref: randist c gsl_cdf_flat_Q765625
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Ref: randist c gsl_cdf_flat_Pinv765692
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Ref: randist c gsl_cdf_flat_Qinv765762
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Node: The Lognormal Distribution766004
Ref: randist the-lognormal-distribution766170
Ref: 74d766170
Ref: randist c gsl_ran_lognormal766237
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Ref: randist c gsl_ran_lognormal_pdf766559
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Ref: randist c gsl_cdf_lognormal_P766884
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Ref: randist c gsl_cdf_lognormal_Pinv767062
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Ref: randist c gsl_cdf_lognormal_Qinv767154
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Node: The Chi-squared Distribution767441
Ref: randist the-chi-squared-distribution767596
Ref: 754767596
Ref: randist c gsl_ran_chisq767898
Ref: 755767898
Ref: randist c gsl_ran_chisq_pdf768223
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Ref: randist c gsl_cdf_chisq_P768501
Ref: 757768501
Ref: randist c gsl_cdf_chisq_Q768560
Ref: 758768560
Ref: randist c gsl_cdf_chisq_Pinv768619
Ref: 759768619
Ref: randist c gsl_cdf_chisq_Qinv768681
Ref: 75a768681
Node: The F-distribution768925
Ref: randist the-f-distribution769072
Ref: 75b769072
Ref: randist c gsl_ran_fdist769351
Ref: 75c769351
Ref: randist c gsl_ran_fdist_pdf769861
Ref: 75d769861
Ref: randist c gsl_cdf_fdist_P770174
Ref: 75e770174
Ref: randist c gsl_cdf_fdist_Q770246
Ref: 75f770246
Ref: randist c gsl_cdf_fdist_Pinv770318
Ref: 760770318
Ref: randist c gsl_cdf_fdist_Qinv770403
Ref: 761770403
Node: The t-distribution770681
Ref: randist the-t-distribution770821
Ref: 762770821
Ref: randist c gsl_ran_tdist771132
Ref: 763771132
Ref: randist c gsl_ran_tdist_pdf771446
Ref: 764771446
Ref: randist c gsl_cdf_tdist_P771714
Ref: 765771714
Ref: randist c gsl_cdf_tdist_Q771773
Ref: 766771773
Ref: randist c gsl_cdf_tdist_Pinv771832
Ref: 767771832
Ref: randist c gsl_cdf_tdist_Qinv771894
Ref: 768771894
Node: The Beta Distribution772128
Ref: randist the-beta-distribution772275
Ref: 769772275
Ref: randist c gsl_ran_beta772332
Ref: 76a772332
Ref: randist c gsl_ran_beta_pdf772623
Ref: 76b772623
Ref: randist c gsl_cdf_beta_P772909
Ref: 76c772909
Ref: randist c gsl_cdf_beta_Q772976
Ref: 76d772976
Ref: randist c gsl_cdf_beta_Pinv773043
Ref: 76e773043
Ref: randist c gsl_cdf_beta_Qinv773113
Ref: 76f773113
Node: The Logistic Distribution773366
Ref: randist the-logistic-distribution773518
Ref: 770773518
Ref: randist c gsl_ran_logistic773583
Ref: 771773583
Ref: randist c gsl_ran_logistic_pdf773863
Ref: 772773863
Ref: randist c gsl_cdf_logistic_P774138
Ref: 773774138
Ref: randist c gsl_cdf_logistic_Q774199
Ref: 774774199
Ref: randist c gsl_cdf_logistic_Pinv774260
Ref: 775774260
Ref: randist c gsl_cdf_logistic_Qinv774324
Ref: 776774324
Node: The Pareto Distribution774562
Ref: randist the-pareto-distribution774723
Ref: 777774723
Ref: randist c gsl_ran_pareto774784
Ref: 778774784
Ref: randist c gsl_ran_pareto_pdf775066
Ref: 779775066
Ref: randist c gsl_cdf_pareto_P775362
Ref: 77a775362
Ref: randist c gsl_cdf_pareto_Q775431
Ref: 77b775431
Ref: randist c gsl_cdf_pareto_Pinv775500
Ref: 77c775500
Ref: randist c gsl_cdf_pareto_Qinv775572
Ref: 77d775572
Node: Spherical Vector Distributions775833
Ref: randist spherical-vector-distributions775993
Ref: 77e775993
Ref: randist c gsl_ran_dir_2d776237
Ref: 77f776237
Ref: randist c gsl_ran_dir_2d_trig_method776322
Ref: 780776322
Ref: randist c gsl_ran_dir_3d777611
Ref: 781777611
Ref: randist c gsl_ran_dir_nd778171
Ref: 782778171
Node: The Weibull Distribution778808
Ref: randist the-weibull-distribution778975
Ref: 783778975
Ref: randist c gsl_ran_weibull779038
Ref: 784779038
Ref: randist c gsl_ran_weibull_pdf779317
Ref: 785779317
Ref: randist c gsl_cdf_weibull_P779616
Ref: 786779616
Ref: randist c gsl_cdf_weibull_Q779686
Ref: 787779686
Ref: randist c gsl_cdf_weibull_Pinv779756
Ref: 788779756
Ref: randist c gsl_cdf_weibull_Qinv779829
Ref: 789779829
Node: The Type-1 Gumbel Distribution780092
Ref: randist the-type-1-gumbel-distribution780259
Ref: 78a780259
Ref: randist c gsl_ran_gumbel1780334
Ref: 78b780334
Ref: randist c gsl_ran_gumbel1_pdf780639
Ref: 78c780639
Ref: randist c gsl_cdf_gumbel1_P780940
Ref: 78d780940
Ref: randist c gsl_cdf_gumbel1_Q781010
Ref: 78e781010
Ref: randist c gsl_cdf_gumbel1_Pinv781080
Ref: 78f781080
Ref: randist c gsl_cdf_gumbel1_Qinv781153
Ref: 790781153
Node: The Type-2 Gumbel Distribution781418
Ref: randist the-type-2-gumbel-distribution781587
Ref: 791781587
Ref: randist c gsl_ran_gumbel2781662
Ref: 792781662
Ref: randist c gsl_ran_gumbel2_pdf781960
Ref: 793781960
Ref: randist c gsl_cdf_gumbel2_P782261
Ref: 794782261
Ref: randist c gsl_cdf_gumbel2_Q782331
Ref: 795782331
Ref: randist c gsl_cdf_gumbel2_Pinv782401
Ref: 796782401
Ref: randist c gsl_cdf_gumbel2_Qinv782474
Ref: 797782474
Node: The Dirichlet Distribution782739
Ref: randist the-dirichlet-distribution782908
Ref: 798782908
Ref: randist c gsl_ran_dirichlet782975
Ref: 799782975
Ref: randist c gsl_ran_dirichlet_pdf783887
Ref: 79a783887
Ref: randist c gsl_ran_dirichlet_lnpdf784200
Ref: 79b784200
Node: General Discrete Distributions784483
Ref: randist general-discrete-distributions784646
Ref: 79c784646
Ref: randist c gsl_ran_discrete_t786995
Ref: 79d786995
Ref: randist c gsl_ran_discrete_preproc787119
Ref: 79e787119
Ref: randist c gsl_ran_discrete787747
Ref: 79f787747
Ref: randist c gsl_ran_discrete_pdf787961
Ref: 7a0787961
Ref: randist c gsl_ran_discrete_free788414
Ref: 7a1788414
Node: The Poisson Distribution788544
Ref: randist the-poisson-distribution788707
Ref: 7a2788707
Ref: randist c gsl_ran_poisson788770
Ref: 7a3788770
Ref: randist c gsl_ran_poisson_pdf789074
Ref: 7a4789074
Ref: randist c gsl_cdf_poisson_P789346
Ref: 7a5789346
Ref: randist c gsl_cdf_poisson_Q789413
Ref: 7a6789413
Node: The Bernoulli Distribution789624
Ref: randist the-bernoulli-distribution789782
Ref: 7a7789782
Ref: randist c gsl_ran_bernoulli789849
Ref: 7a8789849
Ref: randist c gsl_ran_bernoulli_pdf790149
Ref: 7a9790149
Node: The Binomial Distribution790442
Ref: randist the-binomial-distribution790604
Ref: 7aa790604
Ref: randist c gsl_ran_binomial790669
Ref: 7ab790669
Ref: randist c gsl_ran_binomial_pdf791088
Ref: 7ac791088
Ref: randist c gsl_cdf_binomial_P791411
Ref: 7ad791411
Ref: randist c gsl_cdf_binomial_Q791504
Ref: 7ae791504
Node: The Multinomial Distribution791765
Ref: randist the-multinomial-distribution791935
Ref: 7af791935
Ref: randist c gsl_ran_multinomial792006
Ref: 7b0792006
Ref: randist c gsl_ran_multinomial_pdf793084
Ref: 7b1793084
Ref: randist c gsl_ran_multinomial_lnpdf793387
Ref: 7b2793387
Node: The Negative Binomial Distribution793655
Ref: randist the-negative-binomial-distribution793823
Ref: 7b3793823
Ref: randist c gsl_ran_negative_binomial793906
Ref: 7b4793906
Ref: randist c gsl_ran_negative_binomial_pdf794436
Ref: 7b5794436
Ref: randist c gsl_cdf_negative_binomial_P794772
Ref: 7b6794772
Ref: randist c gsl_cdf_negative_binomial_Q794868
Ref: 7b7794868
Node: The Pascal Distribution795141
Ref: randist the-pascal-distribution795307
Ref: 7b8795307
Ref: randist c gsl_ran_pascal795368
Ref: 7b9795368
Ref: randist c gsl_ran_pascal_pdf795730
Ref: 7ba795730
Ref: randist c gsl_cdf_pascal_P796047
Ref: 7bb796047
Ref: randist c gsl_cdf_pascal_Q796138
Ref: 7bc796138
Node: The Geometric Distribution796395
Ref: randist the-geometric-distribution796558
Ref: 7bd796558
Ref: randist c gsl_ran_geometric796625
Ref: 7be796625
Ref: randist c gsl_ran_geometric_pdf797153
Ref: 7bf797153
Ref: randist c gsl_cdf_geometric_P797446
Ref: 7c0797446
Ref: randist c gsl_cdf_geometric_Q797514
Ref: 7c1797514
Node: The Hypergeometric Distribution797727
Ref: randist the-hypergeometric-distribution797895
Ref: 7c2797895
Ref: randist c gsl_ran_hypergeometric797972
Ref: 7c3797972
Ref: randist c gsl_ran_hypergeometric_pdf798706
Ref: 7c4798706
Ref: randist c gsl_cdf_hypergeometric_P799083
Ref: 7c5799083
Ref: randist c gsl_cdf_hypergeometric_Q799206
Ref: 7c6799206
Node: The Logarithmic Distribution799519
Ref: randist the-logarithmic-distribution799685
Ref: 7c7799685
Ref: randist c gsl_ran_logarithmic799756
Ref: 7c8799756
Ref: randist c gsl_ran_logarithmic_pdf800073
Ref: 7c9800073
Node: The Wishart Distribution800372
Ref: randist the-wishart-distribution800529
Ref: 7ca800529
Ref: randist c gsl_ran_wishart800592
Ref: 7cb800592
Ref: randist c gsl_ran_wishart_pdf801307
Ref: 7cc801307
Ref: randist c gsl_ran_wishart_log_pdf801481
Ref: 7cd801481
Node: Shuffling and Sampling802049
Ref: randist shuffling-and-sampling802190
Ref: 7ce802190
Ref: randist c gsl_ran_shuffle802642
Ref: 7cf802642
Ref: randist c gsl_ran_choose803361
Ref: 7d0803361
Ref: randist c gsl_ran_sample804610
Ref: 7d1804610
Node: Examples<14>805095
Ref: randist examples805246
Ref: 7d2805246
Ref: randist fig-rand-walk807423
Ref: 7d3807423
Node: References and Further Reading<14>808292
Ref: randist references-and-further-reading808412
Ref: 7d4808412
Node: Statistics810409
Ref: statistics doc810523
Ref: 7d5810523
Ref: statistics statistics810523
Ref: 7d6810523
Node: Mean Standard Deviation and Variance812001
Ref: statistics mean-standard-deviation-and-variance812112
Ref: 7d7812112
Ref: statistics c gsl_stats_mean812199
Ref: 1d812199
Ref: statistics c gsl_stats_variance812721
Ref: 7d8812721
Ref: statistics c gsl_stats_variance_m813651
Ref: 7d9813651
Ref: statistics c gsl_stats_sd814049
Ref: 7da814049
Ref: statistics c gsl_stats_sd_m814140
Ref: 7db814140
Ref: statistics c gsl_stats_tss814420
Ref: 7dc814420
Ref: statistics c gsl_stats_tss_m814512
Ref: 7dd814512
Ref: statistics c gsl_stats_variance_with_fixed_mean814947
Ref: 7de814947
Ref: statistics c gsl_stats_sd_with_fixed_mean815464
Ref: 7df815464
Node: Absolute deviation815785
Ref: statistics absolute-deviation815941
Ref: 7e0815941
Ref: statistics c gsl_stats_absdev815990
Ref: 7e1815990
Ref: statistics c gsl_stats_absdev_m816652
Ref: 7e2816652
Node: Higher moments skewness and kurtosis817185
Ref: statistics higher-moments-skewness-and-kurtosis817320
Ref: 7e3817320
Ref: statistics c gsl_stats_skew817409
Ref: 7e4817409
Ref: statistics c gsl_stats_skew_m_sd818040
Ref: 7e5818040
Ref: statistics c gsl_stats_kurtosis818546
Ref: 7e6818546
Ref: statistics c gsl_stats_kurtosis_m_sd819045
Ref: 7e7819045
Node: Autocorrelation819562
Ref: statistics autocorrelation819689
Ref: 7e8819689
Ref: statistics c gsl_stats_lag1_autocorrelation819732
Ref: 7e9819732
Ref: statistics c gsl_stats_lag1_autocorrelation_m820113
Ref: 7ea820113
Node: Covariance820409
Ref: statistics covariance820511
Ref: 7eb820511
Ref: statistics c gsl_stats_covariance820544
Ref: 7ec820544
Ref: statistics c gsl_stats_covariance_m820952
Ref: 7ed820952
Node: Correlation821493
Ref: statistics correlation821596
Ref: 7ee821596
Ref: statistics c gsl_stats_correlation821631
Ref: 7ef821631
Ref: statistics c gsl_stats_spearman822239
Ref: 7f0822239
Node: Weighted Samples822966
Ref: statistics weighted-samples823085
Ref: 7f1823085
Ref: statistics c gsl_stats_wmean823579
Ref: 7f2823579
Ref: statistics c gsl_stats_wvariance824036
Ref: 7f3824036
Ref: statistics c gsl_stats_wvariance_m824760
Ref: 7f4824760
Ref: statistics c gsl_stats_wsd825069
Ref: 7f5825069
Ref: statistics c gsl_stats_wsd_m825406
Ref: 7f6825406
Ref: statistics c gsl_stats_wvariance_with_fixed_mean825680
Ref: 7f7825680
Ref: statistics c gsl_stats_wsd_with_fixed_mean826255
Ref: 7f8826255
Ref: statistics c gsl_stats_wtss826598
Ref: 7f9826598
Ref: statistics c gsl_stats_wtss_m826731
Ref: 7fa826731
Ref: statistics c gsl_stats_wabsdev827246
Ref: 7fb827246
Ref: statistics c gsl_stats_wabsdev_m827604
Ref: 7fc827604
Ref: statistics c gsl_stats_wskew827912
Ref: 7fd827912
Ref: statistics c gsl_stats_wskew_m_sd828202
Ref: 7fe828202
Ref: statistics c gsl_stats_wkurtosis828580
Ref: 7ff828580
Ref: statistics c gsl_stats_wkurtosis_m_sd828884
Ref: 800828884
Node: Maximum and Minimum values829266
Ref: statistics maximum-and-minimum-values829396
Ref: 801829396
Ref: statistics c gsl_stats_max829773
Ref: 802829773
Ref: statistics c gsl_stats_min830299
Ref: 803830299
Ref: statistics c gsl_stats_minmax830826
Ref: 804830826
Ref: statistics c gsl_stats_max_index831083
Ref: 805831083
Ref: statistics c gsl_stats_min_index831524
Ref: 806831524
Ref: statistics c gsl_stats_minmax_index831965
Ref: 807831965
Node: Median and Percentiles832269
Ref: statistics median-and-percentiles832399
Ref: 808832399
Ref: statistics c gsl_stats_median_from_sorted_data832811
Ref: 809832811
Ref: statistics c gsl_stats_median833696
Ref: 80a833696
Ref: statistics c gsl_stats_quantile_from_sorted_data834141
Ref: 80b834141
Node: Order Statistics835444
Ref: statistics order-statistics835573
Ref: 80c835573
Ref: statistics c gsl_stats_select836102
Ref: 80d836102
Node: Robust Location Estimates836534
Ref: statistics robust-location-estimates836663
Ref: 80e836663
Node: Trimmed Mean837342
Ref: statistics trimmed-mean837445
Ref: 80f837445
Ref: statistics c gsl_stats_trmean_from_sorted_data837956
Ref: 810837956
Node: Gastwirth Estimator838595
Ref: statistics gastwirth-estimator838698
Ref: 811838698
Ref: statistics c gsl_stats_gastwirth_from_sorted_data839053
Ref: 812839053
Node: Robust Scale Estimates839556
Ref: statistics robust-scale-estimates839681
Ref: 813839681
Node: Median Absolute Deviation MAD840170
Ref: statistics median-absolute-deviation-mad840281
Ref: 814840281
Ref: statistics sec-mad-statistic840281
Ref: 815840281
Ref: statistics c gsl_stats_mad0840857
Ref: 816840857
Ref: statistics c gsl_stats_mad840978
Ref: 817840978
Node: S_n Statistic841553
Ref: statistics s-n-statistic841686
Ref: 818841686
Ref: statistics sec-sn-statistic841686
Ref: 819841686
Ref: statistics c gsl_stats_Sn0_from_sorted_data842267
Ref: 81a842267
Ref: statistics c gsl_stats_Sn_from_sorted_data842421
Ref: 81b842421
Node: Q_n Statistic843234
Ref: statistics q-n-statistic843329
Ref: 81c843329
Ref: statistics sec-qn-statistic843329
Ref: 81d843329
Ref: statistics c gsl_stats_Qn0_from_sorted_data843796
Ref: 81e843796
Ref: statistics c gsl_stats_Qn_from_sorted_data843965
Ref: 81f843965
Node: Examples<15>844815
Ref: statistics examples844949
Ref: 820844949
Node: References and Further Reading<15>847444
Ref: statistics references-and-further-reading847547
Ref: 821847547
Node: Running Statistics849017
Ref: rstat doc849128
Ref: 822849128
Ref: rstat running-statistics849128
Ref: 823849128
Node: Initializing the Accumulator850250
Ref: rstat initializing-the-accumulator850373
Ref: 824850373
Ref: rstat c gsl_rstat_workspace850442
Ref: 825850442
Ref: rstat c gsl_rstat_alloc850628
Ref: 826850628
Ref: rstat c gsl_rstat_free850816
Ref: 827850816
Ref: rstat c gsl_rstat_reset850962
Ref: 828850962
Node: Adding Data to the Accumulator851150
Ref: rstat adding-data-to-the-accumulator851300
Ref: 829851300
Ref: rstat c gsl_rstat_add851373
Ref: 82a851373
Ref: rstat c gsl_rstat_n851640
Ref: 82b851640
Node: Current Statistics851790
Ref: rstat current-statistics851921
Ref: 82c851921
Ref: rstat c gsl_rstat_min851970
Ref: 82d851970
Ref: rstat c gsl_rstat_max852109
Ref: 82e852109
Ref: rstat c gsl_rstat_mean852248
Ref: 82f852248
Ref: rstat c gsl_rstat_variance852443
Ref: 830852443
Ref: rstat c gsl_rstat_sd852669
Ref: 831852669
Ref: rstat c gsl_rstat_sd_mean852890
Ref: 832852890
Ref: rstat c gsl_rstat_rms853084
Ref: 833853084
Ref: rstat c gsl_rstat_skew853301
Ref: 834853301
Ref: rstat c gsl_rstat_kurtosis853524
Ref: 835853524
Ref: rstat c gsl_rstat_median853762
Ref: 836853762
Node: Quantiles853923
Ref: rstat quantiles854036
Ref: 837854036
Ref: rstat c gsl_rstat_quantile_workspace854458
Ref: 838854458
Ref: rstat c gsl_rstat_quantile_alloc854592
Ref: 839854592
Ref: rstat c gsl_rstat_quantile_free854924
Ref: 83a854924
Ref: rstat c gsl_rstat_quantile_reset855098
Ref: 83b855098
Ref: rstat c gsl_rstat_quantile_add855314
Ref: 83c855314
Ref: rstat c gsl_rstat_quantile_get855518
Ref: 83d855518
Node: Examples<16>855675
Ref: rstat examples855804
Ref: 83e855804
Node: References and Further Reading<16>861162
Ref: rstat references-and-further-reading861273
Ref: 83f861273
Node: Moving Window Statistics861658
Ref: movstat doc861776
Ref: 840861776
Ref: movstat moving-window-statistics861776
Ref: 841861776
Node: Introduction<4>862676
Ref: movstat introduction862780
Ref: 842862780
Node: Handling Endpoints863689
Ref: movstat handling-endpoints863841
Ref: 843863841
Ref: movstat c gsl_movstat_end_t864191
Ref: 844864191
Ref: movstat c gsl_movstat_end_t GSL_MOVSTAT_END_PADZERO864345
Ref: 845864345
Ref: movstat c gsl_movstat_end_t GSL_MOVSTAT_END_PADVALUE864719
Ref: 846864719
Ref: movstat c gsl_movstat_end_t GSL_MOVSTAT_END_TRUNCATE865067
Ref: 847865067
Node: Allocation for Moving Window Statistics865244
Ref: movstat allocation-for-moving-window-statistics865392
Ref: 848865392
Ref: movstat c gsl_movstat_workspace865483
Ref: 849865483
Ref: movstat c gsl_movstat_alloc865585
Ref: 84a865585
Ref: movstat c gsl_movstat_alloc2865916
Ref: 84b865916
Ref: movstat c gsl_movstat_free866294
Ref: 84c866294
Node: Moving Mean866426
Ref: movstat moving-mean866594
Ref: 84d866594
Ref: movstat c gsl_movstat_mean867023
Ref: 84e867023
Node: Moving Variance and Standard Deviation867492
Ref: movstat moving-variance-and-standard-deviation867647
Ref: 84f867647
Ref: movstat c gsl_movstat_variance868076
Ref: 850868076
Ref: movstat c gsl_movstat_sd868557
Ref: 851868557
Node: Moving Minimum and Maximum869052
Ref: movstat moving-minimum-and-maximum869206
Ref: 852869206
Ref: movstat c gsl_movstat_min869432
Ref: 853869432
Ref: movstat c gsl_movstat_max869899
Ref: 854869899
Ref: movstat c gsl_movstat_minmax870366
Ref: 855870366
Node: Moving Sum870854
Ref: movstat moving-sum870983
Ref: 856870983
Ref: movstat c gsl_movstat_sum871139
Ref: 857871139
Node: Moving Median871605
Ref: movstat moving-median871731
Ref: 858871731
Ref: movstat c gsl_movstat_median871886
Ref: 859871886
Node: Robust Scale Estimation872357
Ref: movstat robust-scale-estimation872503
Ref: 85a872503
Node: Moving MAD873179
Ref: movstat moving-mad873269
Ref: 85b873269
Ref: movstat c gsl_movstat_mad0873698
Ref: 85c873698
Ref: movstat c gsl_movstat_mad873873
Ref: 85d873873
Node: Moving QQR874609
Ref: movstat moving-qqr874718
Ref: 85e874718
Ref: movstat c gsl_movstat_qqr875355
Ref: 85f875355
Node: Moving S_n875993
Ref: movstat moving-s-n876102
Ref: 860876102
Ref: movstat c gsl_movstat_Sn876366
Ref: 861876366
Node: Moving Q_n876924
Ref: movstat moving-q-n877014
Ref: 862877014
Ref: movstat c gsl_movstat_Qn877250
Ref: 863877250
Node: User-defined Moving Statistics877808
Ref: movstat user-defined-moving-statistics877953
Ref: 864877953
Ref: movstat c gsl_movstat_function878605
Ref: 866878605
Ref: movstat c gsl_movstat_function function879024
Ref: 868879024
Ref: movstat c gsl_movstat_function params879347
Ref: 869879347
Ref: movstat c gsl_movstat_apply879447
Ref: 867879447
Ref: movstat c gsl_movstat_fill880017
Ref: 865880017
Node: Accumulators880732
Ref: movstat accumulators880866
Ref: 86a880866
Ref: movstat c gsl_movstat_accum881161
Ref: 86b881161
Ref: movstat c gsl_movstat_accum size881731
Ref: 86c881731
Ref: movstat c gsl_movstat_accum init881926
Ref: 86d881926
Ref: movstat c gsl_movstat_accum insert882100
Ref: 86e882100
Ref: movstat c gsl_movstat_accum delete882469
Ref: 86f882469
Ref: movstat c gsl_movstat_accum get882656
Ref: 870882656
Ref: movstat c gsl_movstat_accum_min883085
Ref: 871883085
Ref: movstat c gsl_movstat_accum_max883152
Ref: 872883152
Ref: movstat c gsl_movstat_accum_minmax883219
Ref: 873883219
Ref: movstat c gsl_movstat_accum_mean883408
Ref: 874883408
Ref: movstat c gsl_movstat_accum_sd883476
Ref: 875883476
Ref: movstat c gsl_movstat_accum_variance883542
Ref: 876883542
Ref: movstat c gsl_movstat_accum_median883751
Ref: 877883751
Ref: movstat c gsl_movstat_accum_Sn883943
Ref: 878883943
Ref: movstat c gsl_movstat_accum_Qn884009
Ref: 879884009
Ref: movstat c gsl_movstat_accum_sum884191
Ref: 87a884191
Ref: movstat c gsl_movstat_accum_qqr884316
Ref: 87b884316
Node: Examples<17>884454
Ref: movstat examples884592
Ref: 87c884592
Node: Example 1884777
Ref: movstat example-1884867
Ref: 87d884867
Ref: movstat fig-movstat1885071
Ref: 87e885071
Node: Example 2 Robust Scale886781
Ref: movstat example-2-robust-scale886916
Ref: 87f886916
Ref: movstat fig-movstat2887866
Ref: 880887866
Node: Example 3 User-defined Moving Window891083
Ref: movstat example-3-user-defined-moving-window891200
Ref: 881891200
Ref: movstat fig-movstat3891767
Ref: 882891767
Node: References and Further Reading<17>893771
Ref: movstat references-and-further-reading893888
Ref: 883893888
Node: Digital Filtering894468
Ref: filter doc894578
Ref: 884894578
Ref: filter digital-filtering894578
Ref: 885894578
Node: Introduction<5>894862
Ref: filter introduction894962
Ref: 886894962
Node: Handling Endpoints<2>895343
Ref: filter handling-endpoints895474
Ref: 887895474
Ref: filter c gsl_filter_end_t895819
Ref: 888895819
Ref: filter c gsl_filter_end_t GSL_FILTER_END_PADZERO895972
Ref: 889895972
Ref: filter c gsl_filter_end_t GSL_FILTER_END_PADVALUE896347
Ref: 88a896347
Ref: filter c gsl_filter_end_t GSL_FILTER_END_TRUNCATE896696
Ref: 88b896696
Node: Linear Digital Filters896872
Ref: filter linear-digital-filters897013
Ref: 88c897013
Node: Gaussian Filter897100
Ref: filter gaussian-filter897175
Ref: 88d897175
Ref: filter c gsl_filter_gaussian_alloc898617
Ref: 88e898617
Ref: filter c gsl_filter_gaussian_free898974
Ref: 88f898974
Ref: filter c gsl_filter_gaussian899131
Ref: 890899131
Ref: filter c gsl_filter_gaussian_kernel899834
Ref: 891899834
Node: Nonlinear Digital Filters900487
Ref: filter nonlinear-digital-filters900619
Ref: 892900619
Node: Standard Median Filter901143
Ref: filter standard-median-filter901260
Ref: 893901260
Ref: filter c gsl_filter_median_alloc901598
Ref: 894901598
Ref: filter c gsl_filter_median_free901990
Ref: 895901990
Ref: filter c gsl_filter_median902143
Ref: 896902143
Node: Recursive Median Filter902598
Ref: filter recursive-median-filter902748
Ref: 897902748
Ref: filter c gsl_filter_rmedian_alloc903469
Ref: 898903469
Ref: filter c gsl_filter_rmedian_free903864
Ref: 899903864
Ref: filter c gsl_filter_rmedian904019
Ref: 89a904019
Node: Impulse Detection Filter904477
Ref: filter impulse-detection-filter904596
Ref: 89b904596
Ref: filter c gsl_filter_scale_t906921
Ref: 89c906921
Ref: filter c gsl_filter_scale_t GSL_FILTER_SCALE_MAD907044
Ref: 89d907044
Ref: filter c gsl_filter_scale_t GSL_FILTER_SCALE_IQR907417
Ref: 89e907417
Ref: filter c gsl_filter_scale_t GSL_FILTER_SCALE_SN908036
Ref: 89f908036
Ref: filter c gsl_filter_scale_t GSL_FILTER_SCALE_QN908231
Ref: 8a0908231
Ref: filter c gsl_filter_impulse_alloc909295
Ref: 8a1909295
Ref: filter c gsl_filter_impulse_free909682
Ref: 8a2909682
Ref: filter c gsl_filter_impulse909837
Ref: 8a3909837
Node: Examples<18>910815
Ref: filter examples910959
Ref: 8a4910959
Node: Gaussian Example 1911105
Ref: filter gaussian-example-1911200
Ref: 8a5911200
Ref: filter fig-filt-gaussian911487
Ref: 8a6911487
Node: Gaussian Example 2914708
Ref: filter gaussian-example-2914838
Ref: 8a7914838
Ref: filter fig-filt-gaussian2915561
Ref: 8a8915561
Node: Square Wave Signal Example918865
Ref: filter square-wave-signal-example919002
Ref: 8a9919002
Ref: filter fig-filt-edge919504
Ref: 8aa919504
Node: Impulse Detection Example922152
Ref: filter impulse-detection-example922262
Ref: 8ab922262
Ref: filter fig-impulse922744
Ref: 8ac922744
Node: References and Further Reading<18>925461
Ref: filter references-and-further-reading925571
Ref: 8ad925571
Node: Histograms926186
Ref: histogram doc926280
Ref: 8ae926280
Ref: histogram histograms926280
Ref: 8af926280
Node: The histogram struct927955
Ref: histogram the-histogram-struct928052
Ref: 8b0928052
Ref: histogram c gsl_histogram928154
Ref: 8b1928154
Node: Histogram allocation930237
Ref: histogram histogram-allocation930361
Ref: 8b2930361
Ref: histogram c gsl_histogram_alloc930903
Ref: 8b3930903
Ref: histogram c gsl_histogram_set_ranges931458
Ref: 8b4931458
Ref: histogram c gsl_histogram_set_ranges_uniform932612
Ref: 8b5932612
Ref: histogram c gsl_histogram_free933218
Ref: 8b6933218
Node: Copying Histograms933377
Ref: histogram copying-histograms933522
Ref: 8b7933522
Ref: histogram c gsl_histogram_memcpy933571
Ref: 8b8933571
Ref: histogram c gsl_histogram_clone933898
Ref: 8b9933898
Node: Updating and accessing histogram elements934125
Ref: histogram updating-and-accessing-histogram-elements934276
Ref: 8ba934276
Ref: histogram c gsl_histogram_increment934631
Ref: 8bb934631
Ref: histogram c gsl_histogram_accumulate935513
Ref: 8bc935513
Ref: histogram c gsl_histogram_get935819
Ref: 8bd935819
Ref: histogram c gsl_histogram_get_range936203
Ref: 8be936203
Ref: histogram c gsl_histogram_max937076
Ref: 8bf937076
Ref: histogram c gsl_histogram_min937140
Ref: 8c0937140
Ref: histogram c gsl_histogram_bins937204
Ref: 8c1937204
Ref: histogram c gsl_histogram_reset937530
Ref: 8c2937530
Node: Searching histogram ranges937674
Ref: histogram searching-histogram-ranges937827
Ref: 8c3937827
Ref: histogram c gsl_histogram_find938021
Ref: 8c4938021
Node: Histogram Statistics938762
Ref: histogram histogram-statistics938894
Ref: 8c5938894
Ref: histogram c gsl_histogram_max_val938947
Ref: 8c6938947
Ref: histogram c gsl_histogram_max_bin939100
Ref: 8c7939100
Ref: histogram c gsl_histogram_min_val939353
Ref: 8c8939353
Ref: histogram c gsl_histogram_min_bin939506
Ref: 8c9939506
Ref: histogram c gsl_histogram_mean939759
Ref: 8ca939759
Ref: histogram c gsl_histogram_sigma940098
Ref: 8cb940098
Ref: histogram c gsl_histogram_sum940457
Ref: 8cc940457
Node: Histogram Operations940628
Ref: histogram histogram-operations940764
Ref: 8cd940764
Ref: histogram c gsl_histogram_equal_bins_p940817
Ref: 8ce940817
Ref: histogram c gsl_histogram_add941053
Ref: 8cf941053
Ref: histogram c gsl_histogram_sub941376
Ref: 8d0941376
Ref: histogram c gsl_histogram_mul941707
Ref: 8d1941707
Ref: histogram c gsl_histogram_div942053
Ref: 8d2942053
Ref: histogram c gsl_histogram_scale942396
Ref: 8d3942396
Ref: histogram c gsl_histogram_shift942631
Ref: 8d4942631
Node: Reading and writing histograms942865
Ref: histogram reading-and-writing-histograms943007
Ref: 8d5943007
Ref: histogram c gsl_histogram_fwrite943191
Ref: 8d6943191
Ref: histogram c gsl_histogram_fread943646
Ref: 8d7943646
Ref: histogram c gsl_histogram_fprintf944238
Ref: 8d8944238
Ref: histogram c gsl_histogram_fscanf945557
Ref: 8d9945557
Node: Resampling from histograms946138
Ref: histogram resampling-from-histograms946305
Ref: 8da946305
Node: The histogram probability distribution struct946938
Ref: histogram the-histogram-probability-distribution-struct947106
Ref: 8db947106
Ref: histogram c gsl_histogram_pdf947945
Ref: 8dc947945
Ref: histogram c gsl_histogram_pdf_alloc948906
Ref: 8dd948906
Ref: histogram c gsl_histogram_pdf_init949333
Ref: 8de949333
Ref: histogram c gsl_histogram_pdf_free949769
Ref: 8df949769
Ref: histogram c gsl_histogram_pdf_sample949960
Ref: 8e0949960
Node: Example programs for histograms950494
Ref: histogram example-programs-for-histograms950662
Ref: 8e1950662
Ref: histogram fig-histogram952473
Ref: 8e2952473
Node: Two dimensional histograms952550
Ref: histogram two-dimensional-histograms952696
Ref: 8e3952696
Node: The 2D histogram struct953270
Ref: histogram the-2d-histogram-struct953408
Ref: 8e4953408
Ref: histogram c gsl_histogram2d953534
Ref: 8e5953534
Node: 2D Histogram allocation955597
Ref: histogram d-histogram-allocation955730
Ref: 8e6955730
Ref: histogram c gsl_histogram2d_alloc956286
Ref: 8e7956286
Ref: histogram c gsl_histogram2d_set_ranges956917
Ref: 8e8956917
Ref: histogram c gsl_histogram2d_set_ranges_uniform957343
Ref: 8e9957343
Ref: histogram c gsl_histogram2d_free957733
Ref: 8ea957733
Node: Copying 2D Histograms957899
Ref: histogram copying-2d-histograms958053
Ref: 8eb958053
Ref: histogram c gsl_histogram2d_memcpy958110
Ref: 8ec958110
Ref: histogram c gsl_histogram2d_clone958443
Ref: 8ed958443
Node: Updating and accessing 2D histogram elements958676
Ref: histogram updating-and-accessing-2d-histogram-elements958836
Ref: 8ee958836
Ref: histogram c gsl_histogram2d_increment959259
Ref: 8ef959259
Ref: histogram c gsl_histogram2d_accumulate959992
Ref: 8f0959992
Ref: histogram c gsl_histogram2d_get960314
Ref: 8f1960314
Ref: histogram c gsl_histogram2d_get_xrange960744
Ref: 8f2960744
Ref: histogram c gsl_histogram2d_get_yrange960868
Ref: 8f3960868
Ref: histogram c gsl_histogram2d_xmax961766
Ref: 8f4961766
Ref: histogram c gsl_histogram2d_xmin961835
Ref: 8f5961835
Ref: histogram c gsl_histogram2d_nx961904
Ref: 8f6961904
Ref: histogram c gsl_histogram2d_ymax961971
Ref: 8f7961971
Ref: histogram c gsl_histogram2d_ymin962040
Ref: 8f8962040
Ref: histogram c gsl_histogram2d_ny962109
Ref: 8f9962109
Ref: histogram c gsl_histogram2d_reset962471
Ref: 8fa962471
Node: Searching 2D histogram ranges962619
Ref: histogram searching-2d-histogram-ranges962781
Ref: 8fb962781
Ref: histogram c gsl_histogram2d_find962987
Ref: 8fc962987
Node: 2D Histogram Statistics963721
Ref: histogram d-histogram-statistics963862
Ref: 8fd963862
Ref: histogram c gsl_histogram2d_max_val963923
Ref: 8fe963923
Ref: histogram c gsl_histogram2d_max_bin964080
Ref: 8ff964080
Ref: histogram c gsl_histogram2d_min_val964459
Ref: 900964459
Ref: histogram c gsl_histogram2d_min_bin964616
Ref: 901964616
Ref: histogram c gsl_histogram2d_xmean964995
Ref: 902964995
Ref: histogram c gsl_histogram2d_ymean965283
Ref: 903965283
Ref: histogram c gsl_histogram2d_xsigma965571
Ref: 904965571
Ref: histogram c gsl_histogram2d_ysigma965875
Ref: 905965875
Ref: histogram c gsl_histogram2d_cov966179
Ref: 906966179
Ref: histogram c gsl_histogram2d_sum966479
Ref: 907966479
Node: 2D Histogram Operations966654
Ref: histogram d-histogram-operations966799
Ref: 908966799
Ref: histogram c gsl_histogram2d_equal_bins_p966860
Ref: 909966860
Ref: histogram c gsl_histogram2d_add967095
Ref: 90a967095
Ref: histogram c gsl_histogram2d_sub967430
Ref: 90b967430
Ref: histogram c gsl_histogram2d_mul967773
Ref: 90c967773
Ref: histogram c gsl_histogram2d_div968130
Ref: 90d968130
Ref: histogram c gsl_histogram2d_scale968485
Ref: 90e968485
Ref: histogram c gsl_histogram2d_shift968736
Ref: 90f968736
Node: Reading and writing 2D histograms968988
Ref: histogram reading-and-writing-2d-histograms969139
Ref: 910969139
Ref: histogram c gsl_histogram2d_fwrite969347
Ref: 911969347
Ref: histogram c gsl_histogram2d_fread969806
Ref: 912969806
Ref: histogram c gsl_histogram2d_fprintf970415
Ref: 913970415
Ref: histogram c gsl_histogram2d_fscanf972237
Ref: 914972237
Node: Resampling from 2D histograms972836
Ref: histogram resampling-from-2d-histograms972998
Ref: 915972998
Ref: histogram c gsl_histogram2d_pdf973711
Ref: 916973711
Ref: histogram c gsl_histogram2d_pdf_alloc975002
Ref: 917975002
Ref: histogram c gsl_histogram2d_pdf_init975485
Ref: 918975485
Ref: histogram c gsl_histogram2d_pdf_free975938
Ref: 919975938
Ref: histogram c gsl_histogram2d_pdf_sample976149
Ref: 91a976149
Node: Example programs for 2D histograms976496
Ref: histogram example-programs-for-2d-histograms976616
Ref: 91b976616
Node: N-tuples978683
Ref: ntuple doc978783
Ref: 91c978783
Ref: ntuple n-tuples978783
Ref: 91d978783
Node: The ntuple struct979963
Ref: ntuple the-ntuple-struct980051
Ref: 91e980051
Ref: ntuple c gsl_ntuple980098
Ref: 91f980098
Node: Creating ntuples980532
Ref: ntuple creating-ntuples980660
Ref: 920980660
Ref: ntuple c gsl_ntuple_create980705
Ref: 921980705
Node: Opening an existing ntuple file981246
Ref: ntuple opening-an-existing-ntuple-file981372
Ref: 922981372
Ref: ntuple c gsl_ntuple_open981447
Ref: 923981447
Node: Writing ntuples981920
Ref: ntuple writing-ntuples982045
Ref: 924982045
Ref: ntuple c gsl_ntuple_write982088
Ref: 925982088
Ref: ntuple c gsl_ntuple_bookdata982276
Ref: 926982276
Node: Reading ntuples982404
Ref: ntuple reading-ntuples982520
Ref: 927982520
Ref: ntuple c gsl_ntuple_read982563
Ref: 928982563
Node: Closing an ntuple file982753
Ref: ntuple closing-an-ntuple-file982881
Ref: 929982881
Ref: ntuple c gsl_ntuple_close982938
Ref: 92a982938
Node: Histogramming ntuple values983105
Ref: ntuple histogramming-ntuple-values983230
Ref: 92b983230
Ref: ntuple c gsl_ntuple_select_fn983683
Ref: 92d983683
Ref: ntuple c gsl_ntuple_value_fn984164
Ref: 92e984164
Ref: ntuple c gsl_ntuple_project984616
Ref: 92c984616
Node: Examples<19>985385
Ref: ntuple examples985522
Ref: 92f985522
Ref: ntuple fig-ntuples988758
Ref: 930988758
Node: References and Further Reading<19>988826
Ref: ntuple references-and-further-reading988927
Ref: 931988927
Node: Monte Carlo Integration989133
Ref: montecarlo doc989242
Ref: 932989242
Ref: montecarlo monte-carlo-integration989242
Ref: 933989242
Node: Interface990399
Ref: montecarlo interface990495
Ref: 934990495
Ref: montecarlo c gsl_monte_function991503
Ref: 935991503
Node: PLAIN Monte Carlo993744
Ref: montecarlo plain-monte-carlo993854
Ref: 936993854
Ref: montecarlo c gsl_monte_plain_state994868
Ref: 937994868
Ref: montecarlo c gsl_monte_plain_alloc994961
Ref: 938994961
Ref: montecarlo c gsl_monte_plain_init995178
Ref: 939995178
Ref: montecarlo c gsl_monte_plain_integrate995404
Ref: 93a995404
Ref: montecarlo c gsl_monte_plain_free996288
Ref: 93b996288
Node: MISER996449
Ref: montecarlo miser996555
Ref: 93c996555
Ref: montecarlo c gsl_monte_miser_state998475
Ref: 93d998475
Ref: montecarlo c gsl_monte_miser_alloc998571
Ref: 93e998571
Ref: montecarlo c gsl_monte_miser_init998858
Ref: 93f998858
Ref: montecarlo c gsl_monte_miser_integrate999084
Ref: 940999084
Ref: montecarlo c gsl_monte_miser_free999968
Ref: 941999968
Ref: montecarlo c gsl_monte_miser_params_get1000246
Ref: 9421000246
Ref: montecarlo c gsl_monte_miser_params_set1000492
Ref: 9431000492
Ref: montecarlo c gsl_monte_miser_params1001119
Ref: 9441001119
Ref: montecarlo c gsl_monte_miser_params estimate_frac1001153
Ref: 9451001153
Ref: montecarlo c gsl_monte_miser_params min_calls1001423
Ref: 9461001423
Ref: montecarlo c gsl_monte_miser_params min_calls_per_bisection1001945
Ref: 9471001945
Ref: montecarlo c gsl_monte_miser_params alpha1002435
Ref: 9481002435
Ref: montecarlo c gsl_monte_miser_params dither1003311
Ref: 9491003311
Ref: MISER-Footnote-11003819
Node: VEGAS1003958
Ref: montecarlo vegas1004059
Ref: 94a1004059
Ref: montecarlo c gsl_monte_vegas_state1006334
Ref: 94b1006334
Ref: montecarlo c gsl_monte_vegas_alloc1006430
Ref: 94c1006430
Ref: montecarlo c gsl_monte_vegas_init1006717
Ref: 94d1006717
Ref: montecarlo c gsl_monte_vegas_integrate1006943
Ref: 94e1006943
Ref: montecarlo c gsl_monte_vegas_free1008111
Ref: 94f1008111
Ref: montecarlo c gsl_monte_vegas_chisq1009788
Ref: 9501009788
Ref: montecarlo c gsl_monte_vegas_runval1010302
Ref: 9511010302
Ref: montecarlo c gsl_monte_vegas_params_get1010716
Ref: 9521010716
Ref: montecarlo c gsl_monte_vegas_params_set1010962
Ref: 9531010962
Ref: montecarlo c gsl_monte_vegas_params1011589
Ref: 9541011589
Ref: montecarlo c gsl_monte_vegas_params alpha1011623
Ref: 9551011623
Ref: montecarlo c gsl_monte_vegas_params iterations1011897
Ref: 9561011897
Ref: montecarlo c gsl_monte_vegas_params stage1012058
Ref: 9571012058
Ref: montecarlo c gsl_monte_vegas_params mode1012979
Ref: 9581012979
Ref: montecarlo c gsl_monte_vegas_params verbose1013471
Ref: 9591013471
Ref: montecarlo c gsl_monte_vegas_params ostream1013502
Ref: 95a1013502
Node: Examples<20>1014231
Ref: montecarlo examples1014361
Ref: 95b1014361
Node: References and Further Reading<20>1020282
Ref: montecarlo references-and-further-reading1020398
Ref: 95c1020398
Node: Simulated Annealing1021040
Ref: siman doc1021172
Ref: 95d1021172
Ref: siman simulated-annealing1021172
Ref: 95e1021172
Node: Simulated Annealing algorithm1022251
Ref: siman simulated-annealing-algorithm1022375
Ref: 95f1022375
Node: Simulated Annealing functions1023405
Ref: siman simulated-annealing-functions1023550
Ref: 9601023550
Ref: siman c gsl_siman_solve1023621
Ref: 9611023621
Ref: siman c gsl_siman_Efunc_t1026203
Ref: 9621026203
Ref: siman c gsl_siman_step_t1026366
Ref: 9631026366
Ref: siman c gsl_siman_metric_t1026688
Ref: 9641026688
Ref: siman c gsl_siman_print_t1026886
Ref: 9651026886
Ref: siman c gsl_siman_copy_t1027050
Ref: 9661027050
Ref: siman c gsl_siman_copy_construct_t1027231
Ref: 9671027231
Ref: siman c gsl_siman_destroy_t1027414
Ref: 9681027414
Ref: siman c gsl_siman_params_t1027588
Ref: 9691027588
Node: Examples<21>1028694
Ref: siman examples1028844
Ref: 96a1028844
Node: Trivial example1029211
Ref: siman trivial-example1029311
Ref: 96b1029311
Ref: siman fig-siman-test1031873
Ref: 96c1031873
Ref: siman fig-siman-energy1032074
Ref: 96d1032074
Node: Traveling Salesman Problem1032155
Ref: siman traveling-salesman-problem1032255
Ref: 96e1032255
Node: References and Further Reading<21>1046679
Ref: siman references-and-further-reading1046791
Ref: 96f1046791
Node: Ordinary Differential Equations1047053
Ref: ode-initval doc1047175
Ref: 9701047175
Ref: ode-initval ordinary-differential-equations1047175
Ref: 9711047175
Node: Defining the ODE System1048292
Ref: ode-initval defining-the-ode-system1048411
Ref: 9721048411
Ref: ode-initval c gsl_odeiv2_system1048811
Ref: 9731048811
Node: Stepping Functions1051578
Ref: ode-initval stepping-functions1051732
Ref: 9771051732
Ref: ode-initval c gsl_odeiv2_step1051947
Ref: 9781051947
Ref: ode-initval c gsl_odeiv2_step_alloc1052039
Ref: 9791052039
Ref: ode-initval c gsl_odeiv2_step_reset1052520
Ref: 9761052520
Ref: ode-initval c gsl_odeiv2_step_free1052760
Ref: 97a1052760
Ref: ode-initval c gsl_odeiv2_step_name1052920
Ref: 97b1052920
Ref: ode-initval c gsl_odeiv2_step_order1053234
Ref: 97c1053234
Ref: ode-initval c gsl_odeiv2_step_set_driver1053481
Ref: 97d1053481
Ref: ode-initval c gsl_odeiv2_step_apply1053967
Ref: 97e1053967
Ref: ode-initval c gsl_odeiv2_step_type1056409
Ref: 97f1056409
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rk21056441
Ref: 9801056441
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rk41056571
Ref: 9811056571
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rkf451056877
Ref: 9821056877
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rkck1057094
Ref: 9831057094
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rk8pd1057250
Ref: 9841057250
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rk1imp1057412
Ref: 9851057412
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rk2imp1057831
Ref: 9861057831
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_rk4imp1058233
Ref: 9871058233
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_bsimp1058584
Ref: 9881058584
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_msadams1058852
Ref: 9891058852
Ref: ode-initval c gsl_odeiv2_step_type gsl_odeiv2_step_msbdf1059380
Ref: 98a1059380
Node: Adaptive Step-size Control1060072
Ref: ode-initval adaptive-step-size-control1060212
Ref: 98b1060212
Ref: ode-initval c gsl_odeiv2_control1060461
Ref: 98c1060461
Ref: ode-initval c gsl_odeiv2_control_type1060544
Ref: 98d1060544
Ref: ode-initval c gsl_odeiv2_control_standard_new1060618
Ref: 98e1060618
Ref: ode-initval c gsl_odeiv2_control_y_new1062231
Ref: 98f1062231
Ref: ode-initval c gsl_odeiv2_control_yp_new1062679
Ref: 9901062679
Ref: ode-initval c gsl_odeiv2_control_scaled_new1063148
Ref: 9911063148
Ref: ode-initval c gsl_odeiv2_control_alloc1063869
Ref: 9921063869
Ref: ode-initval c gsl_odeiv2_control_init1064281
Ref: 9931064281
Ref: ode-initval c gsl_odeiv2_control_free1064691
Ref: 9941064691
Ref: ode-initval c gsl_odeiv2_control_hadjust1064856
Ref: 9951064856
Ref: ode-initval c gsl_odeiv2_control_name1065856
Ref: 9961065856
Ref: ode-initval c gsl_odeiv2_control_errlevel1066186
Ref: 9971066186
Ref: ode-initval c gsl_odeiv2_control_set_driver1066634
Ref: 9981066634
Node: Evolution1066853
Ref: ode-initval evolution1066981
Ref: 9991066981
Ref: ode-initval c gsl_odeiv2_evolve1067182
Ref: 99a1067182
Ref: ode-initval c gsl_odeiv2_evolve_alloc1067296
Ref: 99b1067296
Ref: ode-initval c gsl_odeiv2_evolve_apply1067531
Ref: 99c1067531
Ref: ode-initval c gsl_odeiv2_evolve_apply_fixed_step1069862
Ref: 99d1069862
Ref: ode-initval c gsl_odeiv2_evolve_reset1070547
Ref: 9751070547
Ref: ode-initval c gsl_odeiv2_evolve_free1070792
Ref: 99e1070792
Ref: ode-initval c gsl_odeiv2_evolve_set_driver1070957
Ref: 99f1070957
Node: Driver1071572
Ref: ode-initval driver1071686
Ref: 9a01071686
Ref: ode-initval c gsl_odeiv2_driver_alloc_y_new1071825
Ref: 9a11071825
Ref: ode-initval c gsl_odeiv2_driver_alloc_yp_new1072033
Ref: 9a21072033
Ref: ode-initval c gsl_odeiv2_driver_alloc_standard_new1072242
Ref: 9a31072242
Ref: ode-initval c gsl_odeiv2_driver_alloc_scaled_new1072506
Ref: 9a41072506
Ref: ode-initval c gsl_odeiv2_driver_set_hmin1073268
Ref: 9a51073268
Ref: ode-initval c gsl_odeiv2_driver_set_hmax1073490
Ref: 9a61073490
Ref: ode-initval c gsl_odeiv2_driver_set_nmax1073728
Ref: 9a71073728
Ref: ode-initval c gsl_odeiv2_driver_apply1073996
Ref: 9a81073996
Ref: ode-initval c gsl_odeiv2_driver_apply_fixed_step1075017
Ref: 9a91075017
Ref: ode-initval c gsl_odeiv2_driver_reset1075550
Ref: 9741075550
Ref: ode-initval c gsl_odeiv2_driver_reset_hstart1075678
Ref: 9aa1075678
Ref: ode-initval c gsl_odeiv2_driver_free1075977
Ref: 9ab1075977
Node: Examples<22>1076148
Ref: ode-initval examples1076287
Ref: 9ac1076287
Node: References and Further Reading<22>1081538
Ref: ode-initval references-and-further-reading1081662
Ref: 9ad1081662
Node: Interpolation1083329
Ref: interp doc1083457
Ref: 9ae1083457
Ref: interp interpolation1083457
Ref: 9af1083457
Ref: interp sec-interpolation1083457
Ref: 9b01083457
Node: Introduction to 1D Interpolation1084865
Ref: interp introduction-to-1d-interpolation1084983
Ref: 9b21084983
Node: 1D Interpolation Functions1085352
Ref: interp d-interpolation-functions1085501
Ref: 9b31085501
Ref: interp c gsl_interp1085707
Ref: 9b41085707
Ref: interp c gsl_interp_alloc1085766
Ref: 9b51085766
Ref: interp c gsl_interp_init1086013
Ref: 9b61086013
Ref: interp c gsl_interp_free1086685
Ref: 9b71086685
Node: 1D Interpolation Types1086812
Ref: interp d-interpolation-types1086962
Ref: 9b81086962
Ref: interp c gsl_interp_type1087090
Ref: 9b91087090
Ref: interp c gsl_interp_type gsl_interp_linear1087117
Ref: 9ba1087117
Ref: interp c gsl_interp_type gsl_interp_polynomial1087294
Ref: 9bb1087294
Ref: interp c gsl_interp_type gsl_interp_cspline1087700
Ref: 9bc1087700
Ref: interp c gsl_interp_type gsl_interp_cspline_periodic1088075
Ref: 9bd1088075
Ref: interp c gsl_interp_type gsl_interp_akima1088654
Ref: 9be1088654
Ref: interp c gsl_interp_type gsl_interp_akima_periodic1088862
Ref: 9bf1088862
Ref: interp c gsl_interp_type gsl_interp_steffen1089095
Ref: 9c01089095
Ref: interp c gsl_interp_name1089729
Ref: 9c11089729
Ref: interp c gsl_interp_min_size1090073
Ref: 9c21090073
Ref: interp c gsl_interp_type_min_size1090157
Ref: 9c31090157
Node: 1D Index Look-up and Acceleration1090478
Ref: interp d-index-look-up-and-acceleration1090642
Ref: 9c41090642
Ref: interp c gsl_interp_accel1090855
Ref: 9c51090855
Ref: interp c gsl_interp_bsearch1091131
Ref: 9c61091131
Ref: interp c gsl_interp_accel_alloc1091556
Ref: 9c71091556
Ref: interp c gsl_interp_accel_find1092105
Ref: 9c81092105
Ref: interp c gsl_interp_accel_reset1092637
Ref: 9c91092637
Ref: interp c gsl_interp_accel_free1092905
Ref: 9ca1092905
Node: 1D Evaluation of Interpolating Functions1093036
Ref: interp d-evaluation-of-interpolating-functions1093203
Ref: 9cb1093203
Ref: interp c gsl_interp_eval1093296
Ref: 9cc1093296
Ref: interp c gsl_interp_eval_e1093451
Ref: 9cd1093451
Ref: interp c gsl_interp_eval_deriv1094034
Ref: 9ce1094034
Ref: interp c gsl_interp_eval_deriv_e1094195
Ref: 9cf1094195
Ref: interp c gsl_interp_eval_deriv21094632
Ref: 9d01094632
Ref: interp c gsl_interp_eval_deriv2_e1094794
Ref: 9d11094794
Ref: interp c gsl_interp_eval_integ1095241
Ref: 9d21095241
Ref: interp c gsl_interp_eval_integ_e1095412
Ref: 9d31095412
Node: 1D Higher-level Interface1095896
Ref: interp d-higher-level-interface1096063
Ref: 9d41096063
Ref: interp c gsl_spline1096573
Ref: 9d51096573
Ref: interp c gsl_spline_alloc1096693
Ref: 9d61096693
Ref: interp c gsl_spline_init1096799
Ref: 9d71096799
Ref: interp c gsl_spline_free1096916
Ref: 9d81096916
Ref: interp c gsl_spline_name1096973
Ref: 9d91096973
Ref: interp c gsl_spline_min_size1097043
Ref: 9da1097043
Ref: interp c gsl_spline_eval1097128
Ref: 9db1097128
Ref: interp c gsl_spline_eval_e1097235
Ref: 9dc1097235
Ref: interp c gsl_spline_eval_deriv1097353
Ref: 9dd1097353
Ref: interp c gsl_spline_eval_deriv_e1097466
Ref: 9de1097466
Ref: interp c gsl_spline_eval_deriv21097590
Ref: 9df1097590
Ref: interp c gsl_spline_eval_deriv2_e1097704
Ref: 9e01097704
Ref: interp c gsl_spline_eval_integ1097830
Ref: 9e11097830
Ref: interp c gsl_spline_eval_integ_e1097953
Ref: 9e21097953
Node: 1D Interpolation Example Programs1098092
Ref: interp d-interpolation-example-programs1098251
Ref: 9e31098251
Ref: interp fig-interp1099655
Ref: 9e41099655
Ref: interp fig-interpp1101245
Ref: 9e51101245
Ref: interp fig-interp-compare1103409
Ref: 9e61103409
Node: Introduction to 2D Interpolation1103907
Ref: interp introduction-to-2d-interpolation1104067
Ref: 9e71104067
Node: 2D Interpolation Functions1104441
Ref: interp id11104590
Ref: 9e81104590
Ref: interp c gsl_interp2d1104798
Ref: 9e91104798
Ref: interp c gsl_interp2d_alloc1104859
Ref: 9ea1104859
Ref: interp c gsl_interp2d_init1105222
Ref: 9eb1105222
Ref: interp c gsl_interp2d_free1106189
Ref: 9ec1106189
Node: 2D Interpolation Grids1106320
Ref: interp d-interpolation-grids1106459
Ref: 9ed1106459
Ref: interp c gsl_interp2d_set1106860
Ref: 9ee1106860
Ref: interp c gsl_interp2d_get1107133
Ref: 9ef1107133
Ref: interp c gsl_interp2d_idx1107392
Ref: 9f01107392
Node: 2D Interpolation Types1107644
Ref: interp id21107797
Ref: 9f11107797
Ref: interp c gsl_interp2d_type1107856
Ref: 9f21107856
Ref: interp c gsl_interp2d_type gsl_interp2d_bilinear1107969
Ref: 9f31107969
Ref: interp c gsl_interp2d_type gsl_interp2d_bicubic1108154
Ref: 9f41108154
Ref: interp c gsl_interp2d_name1108260
Ref: 9f51108260
Ref: interp c gsl_interp2d_min_size1108611
Ref: 9f61108611
Ref: interp c gsl_interp2d_type_min_size1108699
Ref: 9f71108699
Node: 2D Evaluation of Interpolating Functions1109014
Ref: interp id31109170
Ref: 9f81109170
Ref: interp c gsl_interp2d_eval1109265
Ref: 9f91109265
Ref: interp c gsl_interp2d_eval_e1109500
Ref: 9fa1109500
Ref: interp c gsl_interp2d_eval_extrap1110227
Ref: 9fb1110227
Ref: interp c gsl_interp2d_eval_extrap_e1110469
Ref: 9fc1110469
Ref: interp c gsl_interp2d_eval_deriv_x1111233
Ref: 9fd1111233
Ref: interp c gsl_interp2d_eval_deriv_x_e1111476
Ref: 9fe1111476
Ref: interp c gsl_interp2d_eval_deriv_y1112239
Ref: 9ff1112239
Ref: interp c gsl_interp2d_eval_deriv_y_e1112482
Ref: a001112482
Ref: interp c gsl_interp2d_eval_deriv_xx1113245
Ref: a011113245
Ref: interp c gsl_interp2d_eval_deriv_xx_e1113489
Ref: a021113489
Ref: interp c gsl_interp2d_eval_deriv_yy1114257
Ref: a031114257
Ref: interp c gsl_interp2d_eval_deriv_yy_e1114501
Ref: a041114501
Ref: interp c gsl_interp2d_eval_deriv_xy1115269
Ref: a051115269
Ref: interp c gsl_interp2d_eval_deriv_xy_e1115513
Ref: a061115513
Node: 2D Higher-level Interface1116290
Ref: interp id41116457
Ref: a071116457
Ref: interp c gsl_spline2d1116978
Ref: a081116978
Ref: interp c gsl_spline2d_alloc1117102
Ref: a091117102
Ref: interp c gsl_spline2d_init1117229
Ref: a0a1117229
Ref: interp c gsl_spline2d_free1117394
Ref: a0b1117394
Ref: interp c gsl_spline2d_name1117455
Ref: a0c1117455
Ref: interp c gsl_spline2d_min_size1117529
Ref: a0d1117529
Ref: interp c gsl_spline2d_eval1117618
Ref: a0e1117618
Ref: interp c gsl_spline2d_eval_e1117786
Ref: a0f1117786
Ref: interp c gsl_spline2d_eval_extrap1117965
Ref: a101117965
Ref: interp c gsl_spline2d_eval_extrap_e1118140
Ref: a111118140
Ref: interp c gsl_spline2d_eval_deriv_x1118326
Ref: a121118326
Ref: interp c gsl_spline2d_eval_deriv_x_e1118502
Ref: a131118502
Ref: interp c gsl_spline2d_eval_deriv_y1118689
Ref: a141118689
Ref: interp c gsl_spline2d_eval_deriv_y_e1118865
Ref: a151118865
Ref: interp c gsl_spline2d_eval_deriv_xx1119052
Ref: a161119052
Ref: interp c gsl_spline2d_eval_deriv_xx_e1119229
Ref: a171119229
Ref: interp c gsl_spline2d_eval_deriv_yy1119417
Ref: a181119417
Ref: interp c gsl_spline2d_eval_deriv_yy_e1119594
Ref: a191119594
Ref: interp c gsl_spline2d_eval_deriv_xy1119782
Ref: a1a1119782
Ref: interp c gsl_spline2d_eval_deriv_xy_e1119959
Ref: a1b1119959
Ref: interp c gsl_spline2d_set1120147
Ref: a1c1120147
Ref: interp c gsl_spline2d_get1120283
Ref: a1d1120283
Node: 2D Interpolation Example programs1120542
Ref: interp id51120703
Ref: a1e1120703
Ref: interp fig-interp2d1122755
Ref: a1f1122755
Node: References and Further Reading<23>1122818
Ref: interp references-and-further-reading1122945
Ref: a201122945
Node: Numerical Differentiation1123525
Ref: diff doc1123646
Ref: a211123646
Ref: diff numerical-differentiation1123646
Ref: a221123646
Node: Functions1124108
Ref: diff functions1124201
Ref: a231124201
Ref: diff c gsl_deriv_central1124232
Ref: a241124232
Ref: diff c gsl_deriv_forward1125273
Ref: a251125273
Ref: diff c gsl_deriv_backward1126439
Ref: a261126439
Node: Examples<23>1127257
Ref: diff examples1127393
Ref: a271127393
Node: References and Further Reading<24>1128682
Ref: diff references-and-further-reading1128800
Ref: a281128800
Node: Chebyshev Approximations1129209
Ref: cheb doc1129336
Ref: a291129336
Ref: cheb chebyshev-approximations1129336
Ref: a2a1129336
Node: Definitions1130240
Ref: cheb definitions1130366
Ref: a2b1130366
Ref: cheb c gsl_cheb_series1130401
Ref: a2c1130401
Node: Creation and Calculation of Chebyshev Series1131099
Ref: cheb creation-and-calculation-of-chebyshev-series1131253
Ref: a2d1131253
Ref: cheb c gsl_cheb_alloc1131354
Ref: a2e1131354
Ref: cheb c gsl_cheb_free1131599
Ref: a2f1131599
Ref: cheb c gsl_cheb_init1131741
Ref: a301131741
Node: Auxiliary Functions1132147
Ref: cheb auxiliary-functions1132317
Ref: a311132317
Ref: cheb c gsl_cheb_order1132449
Ref: a321132449
Ref: cheb c gsl_cheb_size1132587
Ref: a331132587
Ref: cheb c gsl_cheb_coeffs1132650
Ref: a341132650
Node: Chebyshev Series Evaluation1132890
Ref: cheb chebyshev-series-evaluation1133041
Ref: a351133041
Ref: cheb c gsl_cheb_eval1133108
Ref: a361133108
Ref: cheb c gsl_cheb_eval_err1133284
Ref: a371133284
Ref: cheb c gsl_cheb_eval_n1133679
Ref: a381133679
Ref: cheb c gsl_cheb_eval_n_err1133933
Ref: a391133933
Node: Derivatives and Integrals1134401
Ref: cheb derivatives-and-integrals1134545
Ref: a3a1134545
Ref: cheb c gsl_cheb_calc_deriv1134889
Ref: a3b1134889
Ref: cheb c gsl_cheb_calc_integ1135259
Ref: a3c1135259
Node: Examples<24>1135721
Ref: cheb examples1135872
Ref: a3d1135872
Ref: cheb fig-cheb1137241
Ref: a3e1137241
Node: References and Further Reading<25>1137319
Ref: cheb references-and-further-reading1137436
Ref: a3f1137436
Node: Series Acceleration1137735
Ref: sum doc1137855
Ref: a401137855
Ref: sum series-acceleration1137855
Ref: a411137855
Node: Acceleration functions1138525
Ref: sum acceleration-functions1138660
Ref: a421138660
Ref: sum c gsl_sum_levin_u_workspace1139618
Ref: a431139618
Ref: sum c gsl_sum_levin_u_alloc1139696
Ref: a441139696
Ref: sum c gsl_sum_levin_u_free1139932
Ref: a451139932
Ref: sum c gsl_sum_levin_u_accel1140090
Ref: a461140090
Node: Acceleration functions without error estimation1141043
Ref: sum acceleration-functions-without-error-estimation1141199
Ref: a471141199
Ref: sum c gsl_sum_levin_utrunc_workspace1142289
Ref: a481142289
Ref: sum c gsl_sum_levin_utrunc_alloc1142396
Ref: a491142396
Ref: sum c gsl_sum_levin_utrunc_free1142666
Ref: a4a1142666
Ref: sum c gsl_sum_levin_utrunc_accel1142844
Ref: a4b1142844
Node: Examples<25>1143856
Ref: sum examples1144024
Ref: a4c1144024
Node: References and Further Reading<26>1146185
Ref: sum references-and-further-reading1146297
Ref: a4d1146297
Node: Wavelet Transforms1147069
Ref: dwt doc1147191
Ref: a4e1147191
Ref: dwt wavelet-transforms1147191
Ref: a4f1147191
Node: Definitions<2>1147681
Ref: dwt definitions1147774
Ref: a501147774
Node: Initialization1148651
Ref: dwt initialization1148772
Ref: a511148772
Ref: dwt c gsl_wavelet1148813
Ref: a521148813
Ref: dwt c gsl_wavelet_alloc1148954
Ref: a531148954
Ref: dwt c gsl_wavelet_type1149395
Ref: a541149395
Ref: dwt c gsl_wavelet_type gsl_wavelet_daubechies1149423
Ref: a551149423
Ref: dwt c gsl_wavelet_type gsl_wavelet_daubechies_centered1149495
Ref: a561149495
Ref: dwt c gsl_wavelet_type gsl_wavelet_haar1149774
Ref: a571149774
Ref: dwt c gsl_wavelet_type gsl_wavelet_haar_centered1149840
Ref: a581149840
Ref: dwt c gsl_wavelet_type gsl_wavelet_bspline1150035
Ref: a591150035
Ref: dwt c gsl_wavelet_type gsl_wavelet_bspline_centered1150104
Ref: a5a1150104
Ref: dwt c gsl_wavelet_name1150610
Ref: a5b1150610
Ref: dwt c gsl_wavelet_free1150772
Ref: a5c1150772
Ref: dwt c gsl_wavelet_workspace1150885
Ref: a5d1150885
Ref: dwt c gsl_wavelet_workspace_alloc1151065
Ref: a5e1151065
Ref: dwt c gsl_wavelet_workspace_free1151654
Ref: a5f1151654
Node: Transform Functions1151808
Ref: dwt transform-functions1151927
Ref: a601151927
Node: Wavelet transforms in one dimension1152365
Ref: dwt wavelet-transforms-in-one-dimension1152501
Ref: a611152501
Ref: dwt c gsl_wavelet_transform1152588
Ref: a621152588
Ref: dwt c gsl_wavelet_transform_forward1152766
Ref: a631152766
Ref: dwt c gsl_wavelet_transform_inverse1152925
Ref: a641152925
Node: Wavelet transforms in two dimension1154541
Ref: dwt wavelet-transforms-in-two-dimension1154677
Ref: a651154677
Ref: dwt c gsl_wavelet2d_transform1155988
Ref: a661155988
Ref: dwt c gsl_wavelet2d_transform_forward1156183
Ref: a671156183
Ref: dwt c gsl_wavelet2d_transform_inverse1156359
Ref: a681156359
Ref: dwt c gsl_wavelet2d_transform_matrix1157522
Ref: a691157522
Ref: dwt c gsl_wavelet2d_transform_matrix_forward1157685
Ref: a6a1157685
Ref: dwt c gsl_wavelet2d_transform_matrix_inverse1157819
Ref: a6b1157819
Ref: dwt c gsl_wavelet2d_nstransform1158062
Ref: a6c1158062
Ref: dwt c gsl_wavelet2d_nstransform_forward1158259
Ref: a6d1158259
Ref: dwt c gsl_wavelet2d_nstransform_inverse1158437
Ref: a6e1158437
Ref: dwt c gsl_wavelet2d_nstransform_matrix1158711
Ref: a6f1158711
Ref: dwt c gsl_wavelet2d_nstransform_matrix_forward1158876
Ref: a701158876
Ref: dwt c gsl_wavelet2d_nstransform_matrix_inverse1159012
Ref: a711159012
Node: Examples<26>1159287
Ref: dwt examples1159426
Ref: a721159426
Ref: dwt fig-dwt1161502
Ref: a731161502
Node: References and Further Reading<27>1161677
Ref: dwt references-and-further-reading1161788
Ref: a741161788
Node: Discrete Hankel Transforms1163874
Ref: dht doc1164005
Ref: a751164005
Ref: dht discrete-hankel-transforms1164005
Ref: a761164005
Node: Definitions<3>1164352
Ref: dht definitions1164451
Ref: a771164451
Node: Functions<2>1167662
Ref: dht functions1167804
Ref: a7c1167804
Ref: dht c gsl_dht1167835
Ref: a781167835
Ref: dht c gsl_dht_alloc1167911
Ref: a791167911
Ref: dht c gsl_dht_init1168070
Ref: a7a1168070
Ref: dht c gsl_dht_new1168264
Ref: a7d1168264
Ref: dht c gsl_dht_free1168539
Ref: a7e1168539
Ref: dht c gsl_dht_apply1168639
Ref: a7b1168639
Ref: dht c gsl_dht_x_sample1169096
Ref: a7f1169096
Ref: dht c gsl_dht_k_sample1169379
Ref: a801169379
Node: References and Further Reading<28>1169563
Ref: dht references-and-further-reading1169682
Ref: a811169682
Node: One Dimensional Root-Finding1169963
Ref: roots doc1170104
Ref: a821170104
Ref: roots one-dimensional-root-finding1170104
Ref: a831170104
Node: Overview1171207
Ref: roots overview1171297
Ref: a841171297
Node: Caveats1173091
Ref: roots caveats1173213
Ref: a871173213
Node: Initializing the Solver1174827
Ref: roots initializing-the-solver1174972
Ref: a881174972
Ref: roots c gsl_root_fsolver1175031
Ref: a851175031
Ref: roots c gsl_root_fdfsolver1175156
Ref: a861175156
Ref: roots c gsl_root_fsolver_alloc1175276
Ref: a891175276
Ref: roots c gsl_root_fdfsolver_alloc1175896
Ref: a8a1175896
Ref: roots c gsl_root_fsolver_set1176549
Ref: a8b1176549
Ref: roots c gsl_root_fdfsolver_set1176873
Ref: a8c1176873
Ref: roots c gsl_root_fsolver_free1177169
Ref: a8d1177169
Ref: roots c gsl_root_fdfsolver_free1177232
Ref: a8e1177232
Ref: roots c gsl_root_fsolver_name1177388
Ref: a8f1177388
Ref: roots c gsl_root_fdfsolver_name1177474
Ref: a901177474
Node: Providing the function to solve1177787
Ref: roots providing-function-to-solve1177950
Ref: a911177950
Ref: roots providing-the-function-to-solve1177950
Ref: a921177950
Ref: roots c gsl_function1178252
Ref: a931178252
Ref: roots c gsl_function_fdf1179390
Ref: a941179390
Node: Search Bounds and Guesses1181484
Ref: roots search-bounds-and-guesses1181633
Ref: a951181633
Node: Iteration1182331
Ref: roots iteration1182475
Ref: a961182475
Ref: roots c gsl_root_fsolver_iterate1182801
Ref: a971182801
Ref: roots c gsl_root_fdfsolver_iterate1182866
Ref: a981182866
Ref: roots c gsl_root_fsolver_root1183696
Ref: a991183696
Ref: roots c gsl_root_fdfsolver_root1183767
Ref: a9a1183767
Ref: roots c gsl_root_fsolver_x_lower1183949
Ref: a9b1183949
Ref: roots c gsl_root_fsolver_x_upper1184033
Ref: a9c1184033
Node: Search Stopping Parameters1184213
Ref: roots search-stopping-parameters1184358
Ref: a9d1184358
Ref: roots c gsl_root_test_interval1184837
Ref: a9e1184837
Ref: roots c gsl_root_test_delta1185868
Ref: a9f1185868
Ref: roots c gsl_root_test_residual1186325
Ref: aa01186325
Node: Root Bracketing Algorithms1186853
Ref: roots root-bracketing-algorithms1187030
Ref: aa11187030
Ref: roots c gsl_root_fsolver_type1187597
Ref: aa21187597
Ref: roots c gsl_root_fsolver_type gsl_root_fsolver_bisection1187630
Ref: aa31187630
Ref: roots c gsl_root_fsolver_type gsl_root_fsolver_falsepos1188461
Ref: aa41188461
Ref: roots c gsl_root_fsolver_type gsl_root_fsolver_brent1189421
Ref: aa51189421
Node: Root Finding Algorithms using Derivatives1190582
Ref: roots root-finding-algorithms-using-derivatives1190745
Ref: aa61190745
Ref: roots c gsl_root_fdfsolver_type1191263
Ref: aa71191263
Ref: roots c gsl_root_fdfsolver_type gsl_root_fdfsolver_newton1191298
Ref: aa81191298
Ref: roots c gsl_root_fdfsolver_type gsl_root_fdfsolver_secant1191954
Ref: aa91191954
Ref: roots c gsl_root_fdfsolver_type gsl_root_fdfsolver_steffenson1193410
Ref: aaa1193410
Ref: Root Finding Algorithms using Derivatives-Footnote-11194541
Node: Examples<27>1194703
Ref: roots examples1194874
Ref: aab1194874
Node: References and Further Reading<29>1201984
Ref: roots references-and-further-reading1202105
Ref: aac1202105
Node: One Dimensional Minimization1202624
Ref: min doc1202768
Ref: aad1202768
Ref: min one-dimensional-minimization1202768
Ref: aae1202768
Node: Overview<2>1203933
Ref: min overview1204029
Ref: aaf1204029
Ref: min fig-min-interval1204280
Ref: ab01204280
Node: Caveats<2>1205789
Ref: min caveats1205920
Ref: ab21205920
Node: Initializing the Minimizer1207104
Ref: min initializing-the-minimizer1207258
Ref: ab31207258
Ref: min c gsl_min_fminimizer1207323
Ref: ab11207323
Ref: min c gsl_min_fminimizer_alloc1207405
Ref: ab41207405
Ref: min c gsl_min_fminimizer_set1208057
Ref: ab51208057
Ref: min c gsl_min_fminimizer_set_with_values1208600
Ref: ab61208600
Ref: min c gsl_min_fminimizer_free1209080
Ref: ab71209080
Ref: min c gsl_min_fminimizer_name1209238
Ref: ab81209238
Node: Providing the function to minimize1209557
Ref: min providing-the-function-to-minimize1209713
Ref: ab91209713
Node: Iteration<2>1210039
Ref: min iteration1210188
Ref: aba1210188
Ref: min c gsl_min_fminimizer_iterate1210520
Ref: abb1210520
Ref: min c gsl_min_fminimizer_x_minimum1211256
Ref: abc1211256
Ref: min c gsl_min_fminimizer_x_upper1211464
Ref: abd1211464
Ref: min c gsl_min_fminimizer_x_lower1211552
Ref: abe1211552
Ref: min c gsl_min_fminimizer_f_minimum1211757
Ref: abf1211757
Ref: min c gsl_min_fminimizer_f_upper1211847
Ref: ac01211847
Ref: min c gsl_min_fminimizer_f_lower1211935
Ref: ac11211935
Node: Stopping Parameters1212210
Ref: min stopping-parameters1212348
Ref: ac21212348
Ref: min c gsl_min_test_interval1212791
Ref: ac31212791
Node: Minimization Algorithms1213848
Ref: min minimization-algorithms1213986
Ref: ac41213986
Ref: min c gsl_min_fminimizer_type1214471
Ref: ac51214471
Ref: min c gsl_min_fminimizer_type gsl_min_fminimizer_goldensection1214506
Ref: ac61214506
Ref: min c gsl_min_fminimizer_type gsl_min_fminimizer_brent1215577
Ref: ac71215577
Ref: min c gsl_min_fminimizer_type gsl_min_fminimizer_quad_golden1216554
Ref: ac81216554
Node: Examples<28>1216788
Ref: min examples1216941
Ref: ac91216941
Node: References and Further Reading<30>1219506
Ref: min references-and-further-reading1219627
Ref: aca1219627
Node: Multidimensional Root-Finding1219946
Ref: multiroots doc1220091
Ref: acb1220091
Ref: multiroots multidimensional-root-finding1220091
Ref: acc1220091
Node: Overview<3>1221385
Ref: multiroots overview1221498
Ref: acd1221498
Node: Initializing the Solver<2>1223584
Ref: multiroots initializing-the-solver1223740
Ref: ad01223740
Ref: multiroots c gsl_multiroot_fsolver1224022
Ref: acf1224022
Ref: multiroots c gsl_multiroot_fdfsolver1224141
Ref: ace1224141
Ref: multiroots c gsl_multiroot_fsolver_alloc1224259
Ref: ad11224259
Ref: multiroots c gsl_multiroot_fdfsolver_alloc1225024
Ref: ad21225024
Ref: multiroots c gsl_multiroot_fsolver_set1225822
Ref: ad31225822
Ref: multiroots c gsl_multiroot_fdfsolver_set1225951
Ref: ad41225951
Ref: multiroots c gsl_multiroot_fsolver_free1226405
Ref: ad51226405
Ref: multiroots c gsl_multiroot_fdfsolver_free1226478
Ref: ad61226478
Ref: multiroots c gsl_multiroot_fsolver_name1226654
Ref: ad71226654
Ref: multiroots c gsl_multiroot_fdfsolver_name1226750
Ref: ad81226750
Node: Providing the function to solve<2>1227077
Ref: multiroots providing-the-function-to-solve1227234
Ref: ad91227234
Ref: multiroots c gsl_multiroot_function1227487
Ref: ada1227487
Ref: multiroots c gsl_multiroot_function_fdf1229021
Ref: adb1229021
Node: Iteration<3>1232103
Ref: multiroots iteration1232263
Ref: adc1232263
Ref: multiroots c gsl_multiroot_fsolver_iterate1232589
Ref: add1232589
Ref: multiroots c gsl_multiroot_fdfsolver_iterate1232674
Ref: ade1232674
Ref: multiroots c gsl_multiroot_fsolver_root1233422
Ref: adf1233422
Ref: multiroots c gsl_multiroot_fdfsolver_root1233530
Ref: ae01233530
Ref: multiroots c gsl_multiroot_fsolver_f1233760
Ref: ae11233760
Ref: multiroots c gsl_multiroot_fdfsolver_f1233865
Ref: ae21233865
Ref: multiroots c gsl_multiroot_fsolver_dx1234119
Ref: ae31234119
Ref: multiroots c gsl_multiroot_fdfsolver_dx1234225
Ref: ae41234225
Node: Search Stopping Parameters<2>1234443
Ref: multiroots search-stopping-parameters1234597
Ref: ae51234597
Ref: multiroots c gsl_multiroot_test_delta1235098
Ref: ae61235098
Ref: multiroots c gsl_multiroot_test_residual1235676
Ref: ae71235676
Node: Algorithms using Derivatives1236230
Ref: multiroots algorithms-using-derivatives1236402
Ref: ae81236402
Ref: multiroots c gsl_multiroot_fdfsolver_type1236880
Ref: ae91236880
Ref: multiroots c gsl_multiroot_fdfsolver_type gsl_multiroot_fdfsolver_hybridsj1237014
Ref: aea1237014
Ref: multiroots c gsl_multiroot_fdfsolver_type gsl_multiroot_fdfsolver_hybridj1239874
Ref: aeb1239874
Ref: multiroots c gsl_multiroot_fdfsolver_type gsl_multiroot_fdfsolver_newton1240266
Ref: aec1240266
Ref: multiroots c gsl_multiroot_fdfsolver_type gsl_multiroot_fdfsolver_gnewton1241148
Ref: aed1241148
Node: Algorithms without Derivatives1241780
Ref: multiroots algorithms-without-derivatives1241935
Ref: aee1241935
Ref: multiroots c gsl_multiroot_fsolver_type1242396
Ref: aef1242396
Ref: multiroots c gsl_multiroot_fsolver_type gsl_multiroot_fsolver_hybrids1242530
Ref: af01242530
Ref: multiroots c gsl_multiroot_fsolver_type gsl_multiroot_fsolver_hybrid1243030
Ref: af11243030
Ref: multiroots c gsl_multiroot_fsolver_type gsl_multiroot_fsolver_dnewton1243241
Ref: af21243241
Ref: multiroots c gsl_multiroot_fsolver_type gsl_multiroot_fsolver_broyden1244254
Ref: af31244254
Node: Examples<29>1245441
Ref: multiroots examples1245602
Ref: af41245602
Node: References and Further Reading<31>1254142
Ref: multiroots references-and-further-reading1254264
Ref: af51254264
Node: Multidimensional Minimization1255343
Ref: multimin doc1255488
Ref: af61255488
Ref: multimin multidimensional-minimization1255488
Ref: af71255488
Node: Overview<4>1256751
Ref: multimin overview1256848
Ref: af81256848
Node: Caveats<3>1258862
Ref: multimin caveats1259011
Ref: afb1259011
Node: Initializing the Multidimensional Minimizer1259596
Ref: multimin initializing-the-multidimensional-minimizer1259766
Ref: afc1259766
Ref: multimin c gsl_multimin_fdfminimizer1260058
Ref: af91260058
Ref: multimin c gsl_multimin_fminimizer1260165
Ref: afa1260165
Ref: multimin c gsl_multimin_fdfminimizer_alloc1260272
Ref: afd1260272
Ref: multimin c gsl_multimin_fminimizer_alloc1260429
Ref: afe1260429
Ref: multimin c gsl_multimin_fdfminimizer_set1260922
Ref: aff1260922
Ref: multimin c gsl_multimin_fminimizer_set1261104
Ref: b001261104
Ref: multimin c gsl_multimin_fdfminimizer_free1262521
Ref: b011262521
Ref: multimin c gsl_multimin_fminimizer_free1262612
Ref: b021262612
Ref: multimin c gsl_multimin_fdfminimizer_name1262790
Ref: b031262790
Ref: multimin c gsl_multimin_fminimizer_name1262894
Ref: b041262894
Node: Providing a function to minimize1263237
Ref: multimin providing-a-function-to-minimize1263409
Ref: b051263409
Ref: multimin c gsl_multimin_function_fdf1263846
Ref: b061263846
Ref: multimin c gsl_multimin_function1265334
Ref: b071265334
Ref: multimin multimin-paraboloid1265958
Ref: b081265958
Node: Iteration<4>1267469
Ref: multimin iteration1267615
Ref: b091267615
Ref: multimin c gsl_multimin_fdfminimizer_iterate1267919
Ref: b0a1267919
Ref: multimin c gsl_multimin_fminimizer_iterate1268012
Ref: b0b1268012
Ref: multimin c gsl_multimin_fdfminimizer_x1268644
Ref: b0c1268644
Ref: multimin c gsl_multimin_fminimizer_x1268757
Ref: b0d1268757
Ref: multimin c gsl_multimin_fdfminimizer_minimum1268866
Ref: b0e1268866
Ref: multimin c gsl_multimin_fminimizer_minimum1268968
Ref: b0f1268968
Ref: multimin c gsl_multimin_fdfminimizer_gradient1269066
Ref: b101269066
Ref: multimin c gsl_multimin_fdfminimizer_dx1269186
Ref: b111269186
Ref: multimin c gsl_multimin_fminimizer_size1269300
Ref: b121269300
Ref: multimin c gsl_multimin_fdfminimizer_restart1269669
Ref: b131269669
Node: Stopping Criteria1269873
Ref: multimin stopping-criteria1270014
Ref: b141270014
Ref: multimin c gsl_multimin_test_gradient1270453
Ref: b151270453
Ref: multimin c gsl_multimin_test_size1271114
Ref: b161271114
Node: Algorithms with Derivatives1271486
Ref: multimin algorithms-with-derivatives1271648
Ref: b171271648
Ref: multimin c gsl_multimin_fdfminimizer_type1271935
Ref: b181271935
Ref: multimin c gsl_multimin_fdfminimizer_type gsl_multimin_fdfminimizer_conjugate_fr1272045
Ref: b191272045
Ref: multimin c gsl_multimin_fdfminimizer_type gsl_multimin_fdfminimizer_conjugate_pr1273166
Ref: b1a1273166
Ref: multimin c gsl_multimin_fdfminimizer_type gsl_multimin_fdfminimizer_vector_bfgs21273667
Ref: b1b1273667
Ref: multimin c gsl_multimin_fdfminimizer_type gsl_multimin_fdfminimizer_vector_bfgs1273784
Ref: b1c1273784
Ref: multimin c gsl_multimin_fdfminimizer_type gsl_multimin_fdfminimizer_steepest_descent1275040
Ref: b1d1275040
Node: Algorithms without Derivatives<2>1275718
Ref: multimin algorithms-without-derivatives1275875
Ref: b1e1275875
Ref: multimin c gsl_multimin_fminimizer_type1276051
Ref: b1f1276051
Ref: multimin c gsl_multimin_fminimizer_type gsl_multimin_fminimizer_nmsimplex21276174
Ref: b201276174
Ref: multimin c gsl_multimin_fminimizer_type gsl_multimin_fminimizer_nmsimplex1276285
Ref: b211276285
Ref: multimin c gsl_multimin_fminimizer_type gsl_multimin_fminimizer_nmsimplex2rand1278672
Ref: b221278672
Node: Examples<30>1279417
Ref: multimin examples1279581
Ref: b231279581
Ref: multimin fig-multimin1282331
Ref: b241282331
Node: References and Further Reading<32>1286087
Ref: multimin references-and-further-reading1286209
Ref: b251286209
Node: Linear Least-Squares Fitting1286948
Ref: lls doc1287095
Ref: b261287095
Ref: lls linear-least-squares-fitting1287095
Ref: b271287095
Node: Overview<5>1288009
Ref: lls overview1288112
Ref: b281288112
Ref: lls sec-lls-overview1288112
Ref: b291288112
Node: Linear regression1290366
Ref: lls linear-regression1290504
Ref: b2a1290504
Node: Linear regression with a constant term1290815
Ref: lls linear-regression-with-a-constant-term1290958
Ref: b2b1290958
Ref: lls c gsl_fit_linear1291182
Ref: b2c1291182
Ref: lls c gsl_fit_wlinear1292268
Ref: b2d1292268
Ref: lls c gsl_fit_linear_est1293394
Ref: b2e1293394
Node: Linear regression without a constant term1293894
Ref: lls linear-regression-without-a-constant-term1294037
Ref: b2f1294037
Ref: lls c gsl_fit_mul1294280
Ref: b301294280
Ref: lls c gsl_fit_wmul1295060
Ref: b311295060
Ref: lls c gsl_fit_mul_est1296041
Ref: b321296041
Node: Multi-parameter regression1296424
Ref: lls multi-parameter-regression1296573
Ref: b331296573
Ref: lls c gsl_multifit_linear_workspace1297969
Ref: b341297969
Ref: lls c gsl_multifit_linear_alloc1298100
Ref: b351298100
Ref: lls c gsl_multifit_linear_free1298505
Ref: b361298505
Ref: lls c gsl_multifit_linear_svd1298674
Ref: b371298674
Ref: lls c gsl_multifit_linear_bsvd1298942
Ref: b381298942
Ref: lls c gsl_multifit_linear1299343
Ref: b391299343
Ref: lls c gsl_multifit_linear_tsvd1300645
Ref: b3a1300645
Ref: lls c gsl_multifit_wlinear1301877
Ref: b3b1301877
Ref: lls c gsl_multifit_wlinear_tsvd1302859
Ref: b3c1302859
Ref: lls c gsl_multifit_linear_est1304116
Ref: b3d1304116
Ref: lls c gsl_multifit_linear_residuals1304568
Ref: b3e1304568
Ref: lls c gsl_multifit_linear_rank1304891
Ref: b3f1304891
Node: Regularized regression1305313
Ref: lls regularized-regression1305469
Ref: b401305469
Ref: lls sec-regularized-regression1305469
Ref: b411305469
Ref: lls c gsl_multifit_linear_stdform11310776
Ref: b421310776
Ref: lls c gsl_multifit_linear_wstdform11310976
Ref: b461310976
Ref: lls c gsl_multifit_linear_L_decomp1312382
Ref: b481312382
Ref: lls c gsl_multifit_linear_stdform21313115
Ref: b431313115
Ref: lls c gsl_multifit_linear_wstdform21313366
Ref: b491313366
Ref: lls c gsl_multifit_linear_solve1314674
Ref: b4a1314674
Ref: lls c gsl_multifit_linear_genform11315864
Ref: b441315864
Ref: lls c gsl_multifit_linear_genform21316450
Ref: b451316450
Ref: lls c gsl_multifit_linear_wgenform21316712
Ref: b4b1316712
Ref: lls c gsl_multifit_linear_applyW1317638
Ref: b471317638
Ref: lls c gsl_multifit_linear_lcurve1318273
Ref: b4c1318273
Ref: lls c gsl_multifit_linear_lcurvature1319375
Ref: b4d1319375
Ref: lls c gsl_multifit_linear_lcorner1320455
Ref: b4e1320455
Ref: lls c gsl_multifit_linear_lcorner21321507
Ref: b4f1321507
Ref: lls c gsl_multifit_linear_gcv_init1322840
Ref: b501322840
Ref: lls c gsl_multifit_linear_gcv_curve1323489
Ref: b511323489
Ref: lls c gsl_multifit_linear_gcv_min1323986
Ref: b521323986
Ref: lls c gsl_multifit_linear_gcv_calc1324605
Ref: b531324605
Ref: lls c gsl_multifit_linear_gcv1324896
Ref: b541324896
Ref: lls c gsl_multifit_linear_Lk1325629
Ref: b551325629
Ref: lls c gsl_multifit_linear_Lsobolev1325976
Ref: b561325976
Ref: lls c gsl_multifit_linear_rcond1326934
Ref: b571326934
Node: Robust linear regression1327345
Ref: lls robust-linear-regression1327501
Ref: b581327501
Ref: lls c gsl_multifit_robust_workspace1330630
Ref: b591330630
Ref: lls c gsl_multifit_robust_alloc1330734
Ref: b5a1330734
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type1331188
Ref: b5b1331188
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type gsl_multifit_robust_default1331229
Ref: b5c1331229
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type gsl_multifit_robust_bisquare1331523
Ref: b5d1331523
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type gsl_multifit_robust_cauchy1331985
Ref: b5e1331985
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type gsl_multifit_robust_fair1332591
Ref: b5f1332591
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type gsl_multifit_robust_huber1332996
Ref: b601332996
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type gsl_multifit_robust_ols1333682
Ref: b611333682
Ref: lls c gsl_multifit_robust_alloc gsl_multifit_robust_type gsl_multifit_robust_welsch1334125
Ref: b621334125
Ref: lls c gsl_multifit_robust_free1334538
Ref: b631334538
Ref: lls c gsl_multifit_robust_name1334714
Ref: b641334714
Ref: lls c gsl_multifit_robust_tune1334939
Ref: b651334939
Ref: lls c gsl_multifit_robust_maxiter1335358
Ref: b661335358
Ref: lls c gsl_multifit_robust_weights1335703
Ref: b671335703
Ref: lls c gsl_multifit_robust1336471
Ref: b681336471
Ref: lls c gsl_multifit_robust_est1337715
Ref: b6a1337715
Ref: lls c gsl_multifit_robust_residuals1338168
Ref: b6b1338168
Ref: lls c gsl_multifit_robust_statistics1338790
Ref: b691338790
Ref: lls c gsl_multifit_robust_statistics gsl_multifit_robust_stats1339277
Ref: b6c1339277
Node: Large dense linear systems1341211
Ref: lls large-dense-linear-systems1341360
Ref: b6d1341360
Node: Normal Equations Approach1343469
Ref: lls normal-equations-approach1343595
Ref: b6e1343595
Node: Tall Skinny QR TSQR Approach1344904
Ref: lls tall-skinny-qr-tsqr-approach1345080
Ref: b6f1345080
Node: Large Dense Linear Systems Solution Steps1346524
Ref: lls large-dense-linear-systems-solution-steps1346716
Ref: b701346716
Node: Large Dense Linear Least Squares Routines1347671
Ref: lls large-dense-linear-least-squares-routines1347826
Ref: b711347826
Ref: lls c gsl_multilarge_linear_workspace1347925
Ref: b721347925
Ref: lls c gsl_multilarge_linear_alloc1348063
Ref: b731348063
Ref: lls c gsl_multilarge_linear_alloc gsl_multilarge_linear_type1348413
Ref: b741348413
Ref: lls c gsl_multilarge_linear_alloc gsl_multilarge_linear_type gsl_multilarge_linear_normal1348637
Ref: b751348637
Ref: lls c gsl_multilarge_linear_alloc gsl_multilarge_linear_type gsl_multilarge_linear_tsqr1349082
Ref: b761349082
Ref: lls c gsl_multilarge_linear_free1349575
Ref: b771349575
Ref: lls c gsl_multilarge_linear_name1349755
Ref: b781349755
Ref: lls c gsl_multilarge_linear_reset1349944
Ref: b791349944
Ref: lls c gsl_multilarge_linear_stdform11350156
Ref: b7a1350156
Ref: lls c gsl_multilarge_linear_wstdform11350360
Ref: b7b1350360
Ref: lls c gsl_multilarge_linear_L_decomp1351527
Ref: b7d1351527
Ref: lls c gsl_multilarge_linear_stdform21351995
Ref: b7f1351995
Ref: lls c gsl_multilarge_linear_wstdform21352235
Ref: b7e1352235
Ref: lls c gsl_multilarge_linear_accumulate1353330
Ref: b811353330
Ref: lls c gsl_multilarge_linear_solve1353846
Ref: b821353846
Ref: lls c gsl_multilarge_linear_genform11354424
Ref: b7c1354424
Ref: lls c gsl_multilarge_linear_genform21355014
Ref: b801355014
Ref: lls c gsl_multilarge_linear_lcurve1355531
Ref: b831355531
Ref: lls c gsl_multilarge_linear_matrix_ptr1356555
Ref: b841356555
Ref: lls c gsl_multilarge_linear_rhs_ptr1356889
Ref: b851356889
Ref: lls c gsl_multilarge_linear_rcond1357239
Ref: b861357239
Node: Troubleshooting1357846
Ref: lls troubleshooting1357983
Ref: b871357983
Node: Examples<31>1358473
Ref: lls examples1358618
Ref: b881358618
Node: Simple Linear Regression Example1358996
Ref: lls simple-linear-regression-example1359128
Ref: b891359128
Ref: lls fig-fit-wlinear1361221
Ref: b8a1361221
Node: Multi-parameter Linear Regression Example1361305
Ref: lls multi-parameter-linear-regression-example1361485
Ref: b8b1361485
Ref: lls fig-fit-wlinear21366355
Ref: b8c1366355
Node: Regularized Linear Regression Example 11366434
Ref: lls regularized-linear-regression-example-11366621
Ref: b8d1366621
Ref: lls fig-regularized1368319
Ref: b8e1368319
Node: Regularized Linear Regression Example 21374090
Ref: lls regularized-linear-regression-example-21374268
Ref: b8f1374268
Ref: lls fig-regularized21375744
Ref: b901375744
Node: Robust Linear Regression Example1381081
Ref: lls robust-linear-regression-example1381257
Ref: b911381257
Ref: lls fig-robust1385004
Ref: b921385004
Node: Large Dense Linear Regression Example1385077
Ref: lls large-dense-linear-regression-example1385205
Ref: b931385205
Ref: lls fig-multilarge1387127
Ref: b941387127
Node: References and Further Reading<33>1392728
Ref: lls references-and-further-reading1392849
Ref: b951392849
Node: Nonlinear Least-Squares Fitting1395023
Ref: nls doc1395154
Ref: b961395154
Ref: nls nonlinear-least-squares-fitting1395154
Ref: b971395154
Node: Overview<6>1396931
Ref: nls overview1397059
Ref: b981397059
Node: Solving the Trust Region Subproblem TRS1401311
Ref: nls solving-the-trust-region-subproblem-trs1401480
Ref: b991401480
Node: Levenberg-Marquardt1403271
Ref: nls levenberg-marquardt1403422
Ref: b9a1403422
Node: Levenberg-Marquardt with Geodesic Acceleration1404822
Ref: nls levenberg-marquardt-with-geodesic-acceleration1404988
Ref: b9b1404988
Node: Dogleg1406499
Ref: nls dogleg1406659
Ref: b9c1406659
Node: Double Dogleg1407893
Ref: nls double-dogleg1408031
Ref: b9d1408031
Node: Two Dimensional Subspace1408623
Ref: nls two-dimensional-subspace1408788
Ref: b9e1408788
Node: Steihaug-Toint Conjugate Gradient1409527
Ref: nls steihaug-toint-conjugate-gradient1409670
Ref: b9f1409670
Node: Weighted Nonlinear Least-Squares1410231
Ref: nls weighted-nonlinear-least-squares1410407
Ref: ba01410407
Node: Tunable Parameters1411430
Ref: nls sec-tunable-parameters1411593
Ref: ba21411593
Ref: nls tunable-parameters1411593
Ref: ba31411593
Ref: nls c gsl_multifit_nlinear_parameters1411872
Ref: ba41411872
Ref: nls c gsl_multilarge_nlinear_parameters1412967
Ref: ba51412967
Ref: nls c gsl_multifit_nlinear_trs1414166
Ref: ba61414166
Ref: nls c gsl_multilarge_nlinear_trs1414201
Ref: ba71414201
Ref: nls c gsl_multilarge_nlinear_trs gsl_multifit_nlinear_trs_lm1414387
Ref: ba81414387
Ref: nls c gsl_multilarge_nlinear_trs gsl_multilarge_nlinear_trs_lm1414487
Ref: ba91414487
Ref: nls c gsl_multilarge_nlinear_trs gsl_multifit_nlinear_trs_lmaccel1414651
Ref: baa1414651
Ref: nls c gsl_multilarge_nlinear_trs gsl_multilarge_nlinear_trs_lmaccel1414756
Ref: bab1414756
Ref: nls c gsl_multilarge_nlinear_trs gsl_multifit_nlinear_trs_dogleg1414962
Ref: bac1414962
Ref: nls c gsl_multilarge_nlinear_trs gsl_multilarge_nlinear_trs_dogleg1415066
Ref: bad1415066
Ref: nls c gsl_multilarge_nlinear_trs gsl_multifit_nlinear_trs_ddogleg1415221
Ref: bae1415221
Ref: nls c gsl_multilarge_nlinear_trs gsl_multilarge_nlinear_trs_ddogleg1415326
Ref: baf1415326
Ref: nls c gsl_multilarge_nlinear_trs gsl_multifit_nlinear_trs_subspace2D1415489
Ref: bb01415489
Ref: nls c gsl_multilarge_nlinear_trs gsl_multilarge_nlinear_trs_subspace2D1415597
Ref: bb11415597
Ref: nls c gsl_multilarge_nlinear_trs gsl_multilarge_nlinear_trs_cgst1415761
Ref: bb21415761
Ref: nls c gsl_multifit_nlinear_scale1416000
Ref: bb31416000
Ref: nls c gsl_multilarge_nlinear_scale1416037
Ref: bb41416037
Ref: nls c gsl_multilarge_nlinear_scale gsl_multifit_nlinear_scale_more1416203
Ref: bb51416203
Ref: nls c gsl_multilarge_nlinear_scale gsl_multilarge_nlinear_scale_more1416309
Ref: bb61416309
Ref: nls c gsl_multilarge_nlinear_scale gsl_multifit_nlinear_scale_levenberg1417326
Ref: bb71417326
Ref: nls c gsl_multilarge_nlinear_scale gsl_multilarge_nlinear_scale_levenberg1417437
Ref: bb81417437
Ref: nls c gsl_multilarge_nlinear_scale gsl_multifit_nlinear_scale_marquardt1417993
Ref: bb91417993
Ref: nls c gsl_multilarge_nlinear_scale gsl_multilarge_nlinear_scale_marquardt1418104
Ref: bba1418104
Ref: nls c gsl_multifit_nlinear_solver1418531
Ref: bbb1418531
Ref: nls c gsl_multilarge_nlinear_solver1418569
Ref: bbc1418569
Ref: nls c gsl_multilarge_nlinear_solver gsl_multifit_nlinear_solver_qr1418923
Ref: bbd1418923
Ref: nls c gsl_multilarge_nlinear_solver gsl_multifit_nlinear_solver_cholesky1419365
Ref: bbe1419365
Ref: nls c gsl_multilarge_nlinear_solver gsl_multilarge_nlinear_solver_cholesky1419477
Ref: bbf1419477
Ref: nls c gsl_multilarge_nlinear_solver gsl_multifit_nlinear_solver_mcholesky1420341
Ref: bc01420341
Ref: nls c gsl_multilarge_nlinear_solver gsl_multilarge_nlinear_solver_mcholesky1420454
Ref: bc11420454
Ref: nls c gsl_multilarge_nlinear_solver gsl_multifit_nlinear_solver_svd1421158
Ref: bc21421158
Ref: nls c gsl_multifit_nlinear_fdtype1421515
Ref: bc31421515
Ref: nls c gsl_multifit_nlinear_fdtype GSL_MULTIFIT_NLINEAR_FWDIFF1421828
Ref: bc41421828
Ref: nls c gsl_multifit_nlinear_fdtype GSL_MULTIFIT_NLINEAR_CTRDIFF1422724
Ref: bc51422724
Node: Initializing the Solver<3>1424977
Ref: nls initializing-the-solver1425146
Ref: bc61425146
Ref: nls c gsl_multifit_nlinear_type1425205
Ref: bc71425205
Ref: nls c gsl_multifit_nlinear_type gsl_multifit_nlinear_trust1425416
Ref: bc81425416
Ref: nls c gsl_multifit_nlinear_alloc1425646
Ref: bc91425646
Ref: nls c gsl_multilarge_nlinear_alloc1425865
Ref: bca1425865
Ref: nls c gsl_multifit_nlinear_default_parameters1427546
Ref: bcb1427546
Ref: nls c gsl_multilarge_nlinear_default_parameters1427661
Ref: bcc1427661
Ref: nls c gsl_multifit_nlinear_init1428043
Ref: bcd1428043
Ref: nls c gsl_multifit_nlinear_winit1428195
Ref: ba11428195
Ref: nls c gsl_multilarge_nlinear_init1428371
Ref: bce1428371
Ref: nls c gsl_multifit_nlinear_free1428980
Ref: bd01428980
Ref: nls c gsl_multilarge_nlinear_free1429071
Ref: bd11429071
Ref: nls c gsl_multifit_nlinear_name1429258
Ref: bd21429258
Ref: nls c gsl_multilarge_nlinear_name1429362
Ref: bd31429362
Ref: nls c gsl_multifit_nlinear_trs_name1429700
Ref: bd41429700
Ref: nls c gsl_multilarge_nlinear_trs_name1429808
Ref: bd51429808
Node: Providing the Function to be Minimized1430185
Ref: nls providing-the-function-to-be-minimized1430348
Ref: bd61430348
Ref: nls sec-providing-function-minimized1430348
Ref: bcf1430348
Ref: nls c gsl_multifit_nlinear_fdf1430632
Ref: bd71430632
Ref: nls c gsl_multilarge_nlinear_fdf1433759
Ref: bd81433759
Node: Iteration<5>1437617
Ref: nls iteration1437777
Ref: bd91437777
Ref: nls c gsl_multifit_nlinear_iterate1437974
Ref: bda1437974
Ref: nls c gsl_multilarge_nlinear_iterate1438067
Ref: bdb1438067
Ref: nls c gsl_multifit_nlinear_position1439088
Ref: bdc1439088
Ref: nls c gsl_multilarge_nlinear_position1439208
Ref: bdd1439208
Ref: nls c gsl_multifit_nlinear_residual1439446
Ref: bde1439446
Ref: nls c gsl_multilarge_nlinear_residual1439566
Ref: bdf1439566
Ref: nls c gsl_multifit_nlinear_jac1439874
Ref: be01439874
Ref: nls c gsl_multifit_nlinear_niter1440201
Ref: be11440201
Ref: nls c gsl_multilarge_nlinear_niter1440301
Ref: be21440301
Ref: nls c gsl_multifit_nlinear_rcond1440618
Ref: be31440618
Ref: nls c gsl_multilarge_nlinear_rcond1440730
Ref: be41440730
Ref: nls c gsl_multifit_nlinear_avratio1442285
Ref: be51442285
Ref: nls c gsl_multilarge_nlinear_avratio1442387
Ref: be61442387
Node: Testing for Convergence1442828
Ref: nls testing-for-convergence1442967
Ref: be71442967
Ref: nls c gsl_multifit_nlinear_test1443455
Ref: be81443455
Ref: nls c gsl_multilarge_nlinear_test1443629
Ref: be91443629
Node: High Level Driver1445732
Ref: nls high-level-driver1445899
Ref: bea1445899
Ref: nls c gsl_multifit_nlinear_driver1446057
Ref: beb1446057
Ref: nls c gsl_multilarge_nlinear_driver1446383
Ref: bec1446383
Node: Covariance matrix of best fit parameters1448053
Ref: nls covariance-matrix-of-best-fit-parameters1448215
Ref: bee1448215
Ref: nls c gsl_multifit_nlinear_covar1448310
Ref: bef1448310
Ref: nls c gsl_multilarge_nlinear_covar1448427
Ref: bf01448427
Node: Troubleshooting<2>1450259
Ref: nls sec-nlinear-troubleshooting1450416
Ref: bed1450416
Ref: nls troubleshooting1450416
Ref: bf11450416
Node: Examples<32>1452241
Ref: nls examples1452392
Ref: bf21452392
Node: Exponential Fitting Example1452707
Ref: nls exponential-fitting-example1452824
Ref: bf31452824
Ref: nls fig-fit-exp1456770
Ref: bf41456770
Node: Geodesic Acceleration Example 11463120
Ref: nls geodesic-acceleration-example-11463277
Ref: bf51463277
Ref: nls fig-nlfit21465048
Ref: bf61465048
Node: Geodesic Acceleration Example 21470463
Ref: nls geodesic-acceleration-example-21470622
Ref: bf71470622
Ref: nls fig-nlfit2b1474413
Ref: bf81474413
Node: Comparing TRS Methods Example1482136
Ref: nls comparing-trs-methods-example1482301
Ref: bf91482301
Ref: nls fig-nlfit31484610
Ref: bfa1484610
Node: Large Nonlinear Least Squares Example1490949
Ref: nls large-nonlinear-least-squares-example1491074
Ref: bfb1491074
Node: References and Further Reading<34>1499873
Ref: nls references-and-further-reading1499997
Ref: bfc1499997
Node: Basis Splines1501447
Ref: bspline doc1501565
Ref: bfd1501565
Ref: bspline basis-splines1501565
Ref: bfe1501565
Ref: bspline chap-basis-splines1501565
Ref: 9b11501565
Node: Overview<7>1502326
Ref: bspline overview1502430
Ref: bff1502430
Node: Initializing the B-splines solver1503652
Ref: bspline initializing-the-b-splines-solver1503794
Ref: c001503794
Ref: bspline c gsl_bspline_workspace1503873
Ref: c011503873
Ref: bspline c gsl_bspline_alloc1503990
Ref: c021503990
Ref: bspline c gsl_bspline_free1504417
Ref: c031504417
Node: Constructing the knots vector1504567
Ref: bspline constructing-the-knots-vector1504721
Ref: c041504721
Ref: bspline c gsl_bspline_knots1504792
Ref: c051504792
Ref: bspline c gsl_bspline_knots_uniform1505022
Ref: c061505022
Node: Evaluation of B-splines1505353
Ref: bspline evaluation-of-b-splines1505508
Ref: c071505508
Ref: bspline c gsl_bspline_eval1505567
Ref: c081505567
Ref: bspline c gsl_bspline_eval_nonzero1506167
Ref: c0a1506167
Ref: bspline c gsl_bspline_ncoeffs1506905
Ref: c091506905
Node: Evaluation of B-spline derivatives1507072
Ref: bspline evaluation-of-b-spline-derivatives1507233
Ref: c0b1507233
Ref: bspline c gsl_bspline_deriv_eval1507314
Ref: c0c1507314
Ref: bspline c gsl_bspline_deriv_eval_nonzero1508161
Ref: c0d1508161
Node: Working with the Greville abscissae1509091
Ref: bspline working-with-the-greville-abscissae1509241
Ref: c0e1509241
Ref: bspline c gsl_bspline_greville_abscissa1509774
Ref: c0f1509774
Node: Examples<33>1510076
Ref: bspline examples1510226
Ref: c101510226
Ref: bspline fig-bspline1513975
Ref: c111513975
Node: References and Further Reading<35>1514048
Ref: bspline references-and-further-reading1514154
Ref: c121514154
Node: Sparse Matrices1514844
Ref: spmatrix doc1514950
Ref: c131514950
Ref: spmatrix sparse-matrices1514950
Ref: c141514950
Node: Data types<2>1516137
Ref: spmatrix data-types1516241
Ref: c151516241
Node: Sparse Matrix Storage Formats1518441
Ref: spmatrix sparse-matrix-storage-formats1518565
Ref: c161518565
Node: Coordinate Storage COO1519403
Ref: spmatrix coordinate-storage-coo1519529
Ref: c171519529
Ref: spmatrix sec-spmatrix-coo1519529
Ref: c181519529
Node: Compressed Sparse Column CSC1521634
Ref: spmatrix compressed-sparse-column-csc1521794
Ref: c191521794
Ref: spmatrix sec-spmatrix-csc1521794
Ref: c1a1521794
Node: Compressed Sparse Row CSR1523151
Ref: spmatrix compressed-sparse-row-csr1523280
Ref: c1b1523280
Ref: spmatrix sec-spmatrix-csr1523280
Ref: c1c1523280
Node: Overview<8>1524567
Ref: spmatrix overview1524688
Ref: c1d1524688
Ref: spmatrix c gsl_spmatrix1524910
Ref: c1e1524910
Node: Allocation1527387
Ref: spmatrix allocation1527504
Ref: c1f1527504
Ref: spmatrix c gsl_spmatrix_alloc1527886
Ref: c201527886
Ref: spmatrix c gsl_spmatrix_alloc_nzmax1528789
Ref: c211528789
Ref: spmatrix c gsl_spmatrix_alloc_nzmax GSL_SPMATRIX_COO1529809
Ref: c231529809
Ref: spmatrix c gsl_spmatrix_alloc_nzmax GSL_SPMATRIX_CSC1529904
Ref: c241529904
Ref: spmatrix c gsl_spmatrix_alloc_nzmax GSL_SPMATRIX_CSR1530003
Ref: c251530003
Ref: spmatrix c gsl_spmatrix_realloc1530180
Ref: c221530180
Ref: spmatrix c gsl_spmatrix_free1530675
Ref: c271530675
Node: Accessing Matrix Elements1530912
Ref: spmatrix accessing-matrix-elements1531046
Ref: c281531046
Ref: spmatrix c gsl_spmatrix_get1531109
Ref: c291531109
Ref: spmatrix c gsl_spmatrix_set1531404
Ref: c261531404
Ref: spmatrix c gsl_spmatrix_ptr1531693
Ref: c2a1531693
Node: Initializing Matrix Elements1532124
Ref: spmatrix initializing-matrix-elements1532276
Ref: c2b1532276
Ref: spmatrix c gsl_spmatrix_set_zero1532603
Ref: c2c1532603
Node: Reading and Writing Matrices1533009
Ref: spmatrix reading-and-writing-matrices1533152
Ref: c2d1533152
Ref: spmatrix c gsl_spmatrix_fwrite1533221
Ref: c2e1533221
Ref: spmatrix c gsl_spmatrix_fread1533750
Ref: c2f1533750
Ref: spmatrix c gsl_spmatrix_fprintf1534474
Ref: c301534474
Ref: spmatrix c gsl_spmatrix_fscanf1535181
Ref: c311535181
Node: Copying Matrices1535661
Ref: spmatrix copying-matrices1535803
Ref: c321535803
Ref: spmatrix c gsl_spmatrix_memcpy1535848
Ref: c331535848
Node: Exchanging Rows and Columns1536226
Ref: spmatrix exchanging-rows-and-columns1536357
Ref: c341536357
Ref: spmatrix c gsl_spmatrix_transpose_memcpy1536424
Ref: c351536424
Ref: spmatrix c gsl_spmatrix_transpose1536888
Ref: c361536888
Node: Matrix Operations1537589
Ref: spmatrix matrix-operations1537721
Ref: c371537721
Ref: spmatrix c gsl_spmatrix_scale1537770
Ref: c381537770
Ref: spmatrix c gsl_spmatrix_scale_columns1538114
Ref: c391538114
Ref: spmatrix c gsl_spmatrix_scale_rows1538609
Ref: c3a1538609
Ref: spmatrix c gsl_spmatrix_add1539095
Ref: c3b1539095
Ref: spmatrix c gsl_spmatrix_dense_add1539379
Ref: c3c1539379
Ref: spmatrix c gsl_spmatrix_dense_sub1539863
Ref: c3d1539863
Node: Matrix Properties1540354
Ref: spmatrix matrix-properties1540495
Ref: c3e1540495
Ref: spmatrix c gsl_spmatrix_type1540544
Ref: c3f1540544
Ref: spmatrix c gsl_spmatrix_nnz1540965
Ref: c401540965
Ref: spmatrix c gsl_spmatrix_equal1541201
Ref: c411541201
Ref: spmatrix c gsl_spmatrix_norm11541644
Ref: c421541644
Node: Finding Maximum and Minimum Elements1541986
Ref: spmatrix finding-maximum-and-minimum-elements1542127
Ref: c431542127
Ref: spmatrix c gsl_spmatrix_minmax1542214
Ref: c441542214
Ref: spmatrix c gsl_spmatrix_min_index1542611
Ref: c451542611
Node: Compressed Format1543098
Ref: spmatrix compressed-format1543266
Ref: c461543266
Ref: spmatrix c gsl_spmatrix_csc1543395
Ref: c471543395
Ref: spmatrix c gsl_spmatrix_csr1543774
Ref: c481543774
Ref: spmatrix c gsl_spmatrix_compress1544150
Ref: c491544150
Node: Conversion Between Sparse and Dense Matrices1544655
Ref: spmatrix conversion-between-sparse-and-dense-matrices1544799
Ref: c4a1544799
Ref: spmatrix c gsl_spmatrix_d2sp1545048
Ref: c4b1545048
Ref: spmatrix c gsl_spmatrix_sp2d1545315
Ref: c4c1545315
Node: Examples<34>1545617
Ref: spmatrix examples1545778
Ref: c4d1545778
Node: References and Further Reading<36>1549365
Ref: spmatrix references-and-further-reading1549473
Ref: c4e1549473
Node: Sparse BLAS Support1549794
Ref: spblas doc1549908
Ref: c4f1549908
Ref: spblas sparse-blas-support1549908
Ref: c501549908
Node: Sparse BLAS operations1550452
Ref: spblas sparse-blas-operations1550574
Ref: c511550574
Ref: spblas c gsl_spblas_dgemv1550631
Ref: c521550631
Ref: spblas c gsl_spblas_dgemm1551198
Ref: c531551198
Node: References and Further Reading<37>1551454
Ref: spblas references-and-further-reading1551576
Ref: c541551576
Node: Sparse Linear Algebra1551895
Ref: splinalg doc1552012
Ref: c551552012
Ref: splinalg sparse-linear-algebra1552012
Ref: c561552012
Node: Overview<9>1552537
Ref: splinalg overview1552640
Ref: c571552640
Node: Sparse Iterative Solvers1553266
Ref: splinalg sparse-iterative-solvers1553390
Ref: c581553390
Node: Overview<10>1553564
Ref: splinalg id11553680
Ref: c591553680
Node: Types of Sparse Iterative Solvers1554408
Ref: splinalg types-of-sparse-iterative-solvers1554567
Ref: c5a1554567
Ref: splinalg c gsl_splinalg_itersolve_type1554736
Ref: c5b1554736
Ref: splinalg c gsl_splinalg_itersolve_type gsl_splinalg_itersolve_gmres1554775
Ref: c5c1554775
Node: Iterating the Sparse Linear System1556803
Ref: splinalg iterating-the-sparse-linear-system1556941
Ref: c5e1556941
Ref: splinalg c gsl_splinalg_itersolve_alloc1557152
Ref: c5f1557152
Ref: splinalg c gsl_splinalg_itersolve_free1557708
Ref: c601557708
Ref: splinalg c gsl_splinalg_itersolve_name1557880
Ref: c611557880
Ref: splinalg c gsl_splinalg_itersolve_iterate1558051
Ref: c5d1558051
Ref: splinalg c gsl_splinalg_itersolve_normr1559219
Ref: c621559219
Node: Examples<35>1559480
Ref: splinalg examples1559627
Ref: c631559627
Ref: splinalg fig-splinalg-poisson1560666
Ref: c641560666
Node: References and Further Reading<38>1563957
Ref: splinalg references-and-further-reading1564071
Ref: c651564071
Node: Physical Constants1564454
Ref: const doc1564582
Ref: c661564582
Ref: const physical-constants1564582
Ref: c671564582
Node: Fundamental Constants1565911
Ref: const fundamental-constants1566023
Ref: c681566023
Ref: const c GSL_CONST_MKSA_SPEED_OF_LIGHT1566078
Ref: c691566078
Ref: const c GSL_CONST_MKSA_VACUUM_PERMEABILITY1566159
Ref: c6a1566159
Ref: const c GSL_CONST_MKSA_VACUUM_PERMITTIVITY1566307
Ref: c6b1566307
Ref: const c GSL_CONST_MKSA_PLANCKS_CONSTANT_H1566460
Ref: c6c1566460
Ref: const c GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR1566536
Ref: c6d1566536
Ref: const c GSL_CONST_NUM_AVOGADRO1566635
Ref: c6e1566635
Ref: const c GSL_CONST_MKSA_FARADAY1566702
Ref: c6f1566702
Ref: const c GSL_CONST_MKSA_BOLTZMANN1566774
Ref: c701566774
Ref: const c GSL_CONST_MKSA_MOLAR_GAS1566844
Ref: c711566844
Ref: const c GSL_CONST_MKSA_STANDARD_GAS_VOLUME1566916
Ref: c721566916
Ref: const c GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT1566999
Ref: c731566999
Ref: const c GSL_CONST_MKSA_GAUSS1567107
Ref: c741567107
Node: Astronomy and Astrophysics1567177
Ref: const astronomy-and-astrophysics1567324
Ref: c751567324
Ref: const c GSL_CONST_MKSA_ASTRONOMICAL_UNIT1567389
Ref: c761567389
Ref: const c GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT1567505
Ref: c771567505
Ref: const c GSL_CONST_MKSA_LIGHT_YEAR1567592
Ref: c781567592
Ref: const c GSL_CONST_MKSA_PARSEC1567670
Ref: c791567670
Ref: const c GSL_CONST_MKSA_GRAV_ACCEL1567740
Ref: c7a1567740
Ref: const c GSL_CONST_MKSA_SOLAR_MASS1567837
Ref: c7b1567837
Node: Atomic and Nuclear Physics1567902
Ref: const atomic-and-nuclear-physics1568047
Ref: c7c1568047
Ref: const c GSL_CONST_MKSA_ELECTRON_CHARGE1568112
Ref: c7d1568112
Ref: const c GSL_CONST_MKSA_ELECTRON_VOLT1568192
Ref: c7e1568192
Ref: const c GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS1568274
Ref: c7f1568274
Ref: const c GSL_CONST_MKSA_MASS_ELECTRON1568357
Ref: c801568357
Ref: const c GSL_CONST_MKSA_MASS_MUON1568435
Ref: c811568435
Ref: const c GSL_CONST_MKSA_MASS_PROTON1568507
Ref: c821568507
Ref: const c GSL_CONST_MKSA_MASS_NEUTRON1568581
Ref: c831568581
Ref: const c GSL_CONST_NUM_FINE_STRUCTURE1568657
Ref: c841568657
Ref: const c GSL_CONST_MKSA_RYDBERG1568756
Ref: c851568756
Ref: const c GSL_CONST_MKSA_BOHR_RADIUS1568930
Ref: c861568930
Ref: const c GSL_CONST_MKSA_ANGSTROM1568997
Ref: c871568997
Ref: const c GSL_CONST_MKSA_BARN1569065
Ref: c881569065
Ref: const c GSL_CONST_MKSA_BOHR_MAGNETON1569123
Ref: c891569123
Ref: const c GSL_CONST_MKSA_NUCLEAR_MAGNETON1569196
Ref: c8a1569196
Ref: const c GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT1569275
Ref: c8b1569275
Ref: const c GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT1569462
Ref: c8c1569462
Ref: const c GSL_CONST_MKSA_THOMSON_CROSS_SECTION1569560
Ref: c8d1569560
Ref: const c GSL_CONST_MKSA_DEBYE1569652
Ref: c8e1569652
Node: Measurement of Time1569733
Ref: const measurement-of-time1569866
Ref: c8f1569866
Ref: const c GSL_CONST_MKSA_MINUTE1569917
Ref: c901569917
Ref: const c GSL_CONST_MKSA_HOUR1569992
Ref: c911569992
Ref: const c GSL_CONST_MKSA_DAY1570063
Ref: c921570063
Ref: const c GSL_CONST_MKSA_WEEK1570132
Ref: c931570132
Node: Imperial Units1570203
Ref: const imperial-units1570334
Ref: c941570334
Ref: const c GSL_CONST_MKSA_INCH1570375
Ref: c951570375
Ref: const c GSL_CONST_MKSA_FOOT1570435
Ref: c961570435
Ref: const c GSL_CONST_MKSA_YARD1570495
Ref: c971570495
Ref: const c GSL_CONST_MKSA_MILE1570555
Ref: c981570555
Ref: const c GSL_CONST_MKSA_MIL1570615
Ref: c991570615
Node: Speed and Nautical Units1570695
Ref: const speed-and-nautical-units1570821
Ref: c9a1570821
Ref: const c GSL_CONST_MKSA_KILOMETERS_PER_HOUR1570882
Ref: c9b1570882
Ref: const c GSL_CONST_MKSA_MILES_PER_HOUR1570970
Ref: c9c1570970
Ref: const c GSL_CONST_MKSA_NAUTICAL_MILE1571048
Ref: c9d1571048
Ref: const c GSL_CONST_MKSA_FATHOM1571126
Ref: c9e1571126
Ref: const c GSL_CONST_MKSA_KNOT1571190
Ref: c9f1571190
Node: Printers Units1571249
Ref: const printers-units1571383
Ref: ca01571383
Ref: const c GSL_CONST_MKSA_POINT1571424
Ref: ca11571424
Ref: const c GSL_CONST_MKSA_TEXPOINT1571510
Ref: ca21571510
Node: Volume Area and Length1571594
Ref: const volume-area-and-length1571719
Ref: ca31571719
Ref: const c GSL_CONST_MKSA_MICRON1571778
Ref: ca41571778
Ref: const c GSL_CONST_MKSA_HECTARE1571842
Ref: ca51571842
Ref: const c GSL_CONST_MKSA_ACRE1571906
Ref: ca61571906
Ref: const c GSL_CONST_MKSA_LITER1571964
Ref: ca71571964
Ref: const c GSL_CONST_MKSA_US_GALLON1572026
Ref: ca81572026
Ref: const c GSL_CONST_MKSA_CANADIAN_GALLON1572096
Ref: ca91572096
Ref: const c GSL_CONST_MKSA_UK_GALLON1572178
Ref: caa1572178
Ref: const c GSL_CONST_MKSA_QUART1572248
Ref: cab1572248
Ref: const c GSL_CONST_MKSA_PINT1572310
Ref: cac1572310
Node: Mass and Weight1572370
Ref: const mass-and-weight1572505
Ref: cad1572505
Ref: const c GSL_CONST_MKSA_POUND_MASS1572548
Ref: cae1572548
Ref: const c GSL_CONST_MKSA_OUNCE_MASS1572613
Ref: caf1572613
Ref: const c GSL_CONST_MKSA_TON1572678
Ref: cb01572678
Ref: const c GSL_CONST_MKSA_METRIC_TON1572734
Ref: cb11572734
Ref: const c GSL_CONST_MKSA_UK_TON1572814
Ref: cb21572814
Ref: const c GSL_CONST_MKSA_TROY_OUNCE1572876
Ref: cb31572876
Ref: const c GSL_CONST_MKSA_CARAT1572946
Ref: cb41572946
Ref: const c GSL_CONST_MKSA_GRAM_FORCE1573006
Ref: cb51573006
Ref: const c GSL_CONST_MKSA_POUND_FORCE1573078
Ref: cb61573078
Ref: const c GSL_CONST_MKSA_KILOPOUND_FORCE1573152
Ref: cb71573152
Ref: const c GSL_CONST_MKSA_POUNDAL1573234
Ref: cb81573234
Node: Thermal Energy and Power1573299
Ref: const thermal-energy-and-power1573420
Ref: cb91573420
Ref: const c GSL_CONST_MKSA_CALORIE1573483
Ref: cba1573483
Ref: const c GSL_CONST_MKSA_BTU1573549
Ref: cbb1573549
Ref: const c GSL_CONST_MKSA_THERM1573629
Ref: cbc1573629
Ref: const c GSL_CONST_MKSA_HORSEPOWER1573691
Ref: cbd1573691
Node: Pressure1573762
Ref: const pressure1573877
Ref: cbe1573877
Ref: const c GSL_CONST_MKSA_BAR1573908
Ref: cbf1573908
Ref: const c GSL_CONST_MKSA_STD_ATMOSPHERE1573968
Ref: cc01573968
Ref: const c GSL_CONST_MKSA_TORR1574055
Ref: cc11574055
Ref: const c GSL_CONST_MKSA_METER_OF_MERCURY1574117
Ref: cc21574117
Ref: const c GSL_CONST_MKSA_INCH_OF_MERCURY1574203
Ref: cc31574203
Ref: const c GSL_CONST_MKSA_INCH_OF_WATER1574287
Ref: cc41574287
Ref: const c GSL_CONST_MKSA_PSI1574367
Ref: cc51574367
Node: Viscosity1574445
Ref: const viscosity1574558
Ref: cc61574558
Ref: const c GSL_CONST_MKSA_POISE1574591
Ref: cc71574591
Ref: const c GSL_CONST_MKSA_STOKES1574664
Ref: cc81574664
Node: Light and Illumination1574741
Ref: const light-and-illumination1574859
Ref: cc91574859
Ref: const c GSL_CONST_MKSA_STILB1574918
Ref: cca1574918
Ref: const c GSL_CONST_MKSA_LUMEN1574983
Ref: ccb1574983
Ref: const c GSL_CONST_MKSA_LUX1575052
Ref: ccc1575052
Ref: const c GSL_CONST_MKSA_PHOT1575115
Ref: ccd1575115
Ref: const c GSL_CONST_MKSA_FOOTCANDLE1575180
Ref: cce1575180
Ref: const c GSL_CONST_MKSA_LAMBERT1575257
Ref: ccf1575257
Ref: const c GSL_CONST_MKSA_FOOTLAMBERT1575326
Ref: cd01575326
Node: Radioactivity1575403
Ref: const radioactivity1575528
Ref: cd11575528
Ref: const c GSL_CONST_MKSA_CURIE1575569
Ref: cd21575569
Ref: const c GSL_CONST_MKSA_ROENTGEN1575633
Ref: cd31575633
Ref: const c GSL_CONST_MKSA_RAD1575703
Ref: cd41575703
Node: Force and Energy1575768
Ref: const force-and-energy1575879
Ref: cd51575879
Ref: const c GSL_CONST_MKSA_NEWTON1575926
Ref: cd61575926
Ref: const c GSL_CONST_MKSA_DYNE1575998
Ref: cd71575998
Ref: const c GSL_CONST_MKSA_JOULE1576074
Ref: cd81576074
Ref: const c GSL_CONST_MKSA_ERG1576145
Ref: cd91576145
Node: Prefixes1576216
Ref: const prefixes1576326
Ref: cda1576326
Ref: const c GSL_CONST_NUM_YOTTA1576409
Ref: cdb1576409
Ref: const c GSL_CONST_NUM_ZETTA1576455
Ref: cdc1576455
Ref: const c GSL_CONST_NUM_EXA1576501
Ref: cdd1576501
Ref: const c GSL_CONST_NUM_PETA1576545
Ref: cde1576545
Ref: const c GSL_CONST_NUM_TERA1576590
Ref: cdf1576590
Ref: const c GSL_CONST_NUM_GIGA1576635
Ref: ce01576635
Ref: const c GSL_CONST_NUM_MEGA1576677
Ref: ce11576677
Ref: const c GSL_CONST_NUM_KILO1576719
Ref: ce21576719
Ref: const c GSL_CONST_NUM_MILLI1576761
Ref: ce31576761
Ref: const c GSL_CONST_NUM_MICRO1576807
Ref: ce41576807
Ref: const c GSL_CONST_NUM_NANO1576853
Ref: ce51576853
Ref: const c GSL_CONST_NUM_PICO1576898
Ref: ce61576898
Ref: const c GSL_CONST_NUM_FEMTO1576944
Ref: ce71576944
Ref: const c GSL_CONST_NUM_ATTO1576991
Ref: ce81576991
Ref: const c GSL_CONST_NUM_ZEPTO1577037
Ref: ce91577037
Ref: const c GSL_CONST_NUM_YOCTO1577084
Ref: cea1577084
Node: Examples<36>1577131
Ref: const examples1577259
Ref: ceb1577259
Node: References and Further Reading<39>1578889
Ref: const references-and-further-reading1579000
Ref: cec1579000
Node: IEEE floating-point arithmetic1579607
Ref: ieee754 doc1579742
Ref: ced1579742
Ref: ieee754 chap-ieee1579742
Ref: cee1579742
Ref: ieee754 ieee-floating-point-arithmetic1579742
Ref: cef1579742
Node: Representation of floating point numbers1580227
Ref: ieee754 representation-of-floating-point-numbers1580376
Ref: cf01580376
Ref: ieee754 c gsl_ieee_fprintf_float1582405
Ref: cf11582405
Ref: ieee754 c gsl_ieee_fprintf_double1582478
Ref: cf21582478
Ref: ieee754 c gsl_ieee_printf_float1583358
Ref: cf31583358
Ref: ieee754 c gsl_ieee_printf_double1583416
Ref: cf41583416
Node: Setting up your IEEE environment1584790
Ref: ieee754 setting-up-your-ieee-environment1584982
Ref: cf51584982
Ref: ieee754 c GSL_IEEE_MODE1586051
Ref: cf61586051
Ref: ieee754 c gsl_ieee_env_setup1586131
Ref: cf71586131
Node: References and Further Reading<40>1591925
Ref: ieee754 references-and-further-reading1592068
Ref: cf81592068
Node: Debugging Numerical Programs1593095
Ref: debug doc1593231
Ref: cf91593231
Ref: debug debugging-numerical-programs1593231
Ref: cfa1593231
Node: Using gdb1593608
Ref: debug using-gdb1593726
Ref: cfb1593726
Node: Examining floating point registers1597020
Ref: debug examining-floating-point-registers1597181
Ref: cfc1597181
Node: Handling floating point exceptions1598323
Ref: debug handling-floating-point-exceptions1598517
Ref: cfd1598517
Node: GCC warning options for numerical programs1599779
Ref: debug gcc-warning-options-for-numerical-programs1599973
Ref: cfe1599973
Node: References and Further Reading<41>1603906
Ref: debug references-and-further-reading1604057
Ref: cff1604057
Node: Contributors to GSL1604699
Ref: contrib doc1604820
Ref: d001604820
Ref: contrib contributors-to-gsl1604820
Ref: d011604820
Node: Autoconf Macros1609295
Ref: autoconf doc1609405
Ref: d021609405
Ref: autoconf autoconf-macros1609405
Ref: d031609405
Ref: autoconf chap-autoconf-macros1609405
Ref: 161609405
Node: GSL CBLAS Library1613475
Ref: cblas doc1613592
Ref: d041613592
Ref: cblas chap-cblas1613592
Ref: 4761613592
Ref: cblas gsl-cblas-library1613592
Ref: d051613592
Node: Level 1<2>1613974
Ref: cblas level-11614058
Ref: d061614058
Ref: cblas c cblas_sdsdot1614085
Ref: d071614085
Ref: cblas c cblas_dsdot1614226
Ref: d081614226
Ref: cblas c cblas_sdot1614348
Ref: d091614348
Ref: cblas c cblas_ddot1614468
Ref: d0a1614468
Ref: cblas c cblas_cdotu_sub1614591
Ref: d0b1614591
Ref: cblas c cblas_cdotc_sub1614725
Ref: d0c1614725
Ref: cblas c cblas_zdotu_sub1614859
Ref: d0d1614859
Ref: cblas c cblas_zdotc_sub1614993
Ref: d0e1614993
Ref: cblas c cblas_snrm21615127
Ref: d0f1615127
Ref: cblas c cblas_sasum1615216
Ref: d101615216
Ref: cblas c cblas_dnrm21615305
Ref: d111615305
Ref: cblas c cblas_dasum1615396
Ref: d121615396
Ref: cblas c cblas_scnrm21615487
Ref: d131615487
Ref: cblas c cblas_scasum1615576
Ref: d141615576
Ref: cblas c cblas_dznrm21615665
Ref: d151615665
Ref: cblas c cblas_dzasum1615755
Ref: d161615755
Ref: cblas c cblas_isamax1615845
Ref: d171615845
Ref: cblas c cblas_idamax1615941
Ref: d181615941
Ref: cblas c cblas_icamax1616038
Ref: d191616038
Ref: cblas c cblas_izamax1616133
Ref: d1a1616133
Ref: cblas c cblas_sswap1616228
Ref: d1b1616228
Ref: cblas c cblas_scopy1616336
Ref: d1c1616336
Ref: cblas c cblas_saxpy1616450
Ref: d1d1616450
Ref: cblas c cblas_dswap1616583
Ref: d1e1616583
Ref: cblas c cblas_dcopy1616693
Ref: d1f1616693
Ref: cblas c cblas_daxpy1616809
Ref: d201616809
Ref: cblas c cblas_cswap1616945
Ref: d211616945
Ref: cblas c cblas_ccopy1617051
Ref: d221617051
Ref: cblas c cblas_caxpy1617163
Ref: d231617163
Ref: cblas c cblas_zswap1617294
Ref: d241617294
Ref: cblas c cblas_zcopy1617400
Ref: d251617400
Ref: cblas c cblas_zaxpy1617512
Ref: d261617512
Ref: cblas c cblas_srotg1617643
Ref: d271617643
Ref: cblas c cblas_srotmg1617716
Ref: d281617716
Ref: cblas c cblas_srot1617819
Ref: d291617819
Ref: cblas c cblas_srotm1617956
Ref: d2a1617956
Ref: cblas c cblas_drotg1618080
Ref: d2b1618080
Ref: cblas c cblas_drotmg1618167
Ref: d2c1618167
Ref: cblas c cblas_drot1618275
Ref: d2d1618275
Ref: cblas c cblas_drotm1618416
Ref: d2e1618416
Ref: cblas c cblas_sscal1618543
Ref: d2f1618543
Ref: cblas c cblas_dscal1618644
Ref: d301618644
Ref: cblas c cblas_cscal1618747
Ref: d311618747
Ref: cblas c cblas_zscal1618847
Ref: d321618847
Ref: cblas c cblas_csscal1618947
Ref: d331618947
Ref: cblas c cblas_zdscal1619048
Ref: d341619048
Node: Level 2<2>1619150
Ref: cblas level-21619253
Ref: d351619253
Ref: cblas c cblas_sgemv1619280
Ref: d361619280
Ref: cblas c cblas_sgbmv1619560
Ref: d371619560
Ref: cblas c cblas_strmv1619878
Ref: d381619878
Ref: cblas c cblas_stbmv1620132
Ref: d391620132
Ref: cblas c cblas_stpmv1620399
Ref: d3a1620399
Ref: cblas c cblas_strsv1620639
Ref: d3b1620639
Ref: cblas c cblas_stbsv1620893
Ref: d3c1620893
Ref: cblas c cblas_stpsv1621160
Ref: d3d1621160
Ref: cblas c cblas_dgemv1621400
Ref: d3e1621400
Ref: cblas c cblas_dgbmv1621685
Ref: d3f1621685
Ref: cblas c cblas_dtrmv1622008
Ref: d401622008
Ref: cblas c cblas_dtbmv1622264
Ref: d411622264
Ref: cblas c cblas_dtpmv1622533
Ref: d421622533
Ref: cblas c cblas_dtrsv1622775
Ref: d431622775
Ref: cblas c cblas_dtbsv1623031
Ref: d441623031
Ref: cblas c cblas_dtpsv1623300
Ref: d451623300
Ref: cblas c cblas_cgemv1623542
Ref: d461623542
Ref: cblas c cblas_cgbmv1623819
Ref: d471623819
Ref: cblas c cblas_ctrmv1624134
Ref: d481624134
Ref: cblas c cblas_ctbmv1624386
Ref: d491624386
Ref: cblas c cblas_ctpmv1624651
Ref: d4a1624651
Ref: cblas c cblas_ctrsv1624889
Ref: d4b1624889
Ref: cblas c cblas_ctbsv1625141
Ref: d4c1625141
Ref: cblas c cblas_ctpsv1625406
Ref: d4d1625406
Ref: cblas c cblas_zgemv1625644
Ref: d4e1625644
Ref: cblas c cblas_zgbmv1625921
Ref: d4f1625921
Ref: cblas c cblas_ztrmv1626236
Ref: d501626236
Ref: cblas c cblas_ztbmv1626488
Ref: d511626488
Ref: cblas c cblas_ztpmv1626753
Ref: d521626753
Ref: cblas c cblas_ztrsv1626991
Ref: d531626991
Ref: cblas c cblas_ztbsv1627243
Ref: d541627243
Ref: cblas c cblas_ztpsv1627508
Ref: d551627508
Ref: cblas c cblas_ssymv1627746
Ref: d561627746
Ref: cblas c cblas_ssbmv1628006
Ref: d571628006
Ref: cblas c cblas_sspmv1628279
Ref: d581628279
Ref: cblas c cblas_sger1628525
Ref: d591628525
Ref: cblas c cblas_ssyr1628741
Ref: d5a1628741
Ref: cblas c cblas_sspr1628940
Ref: d5b1628940
Ref: cblas c cblas_ssyr21629125
Ref: d5c1629125
Ref: cblas c cblas_sspr21629367
Ref: d5d1629367
Ref: cblas c cblas_dsymv1629584
Ref: d5e1629584
Ref: cblas c cblas_dsbmv1629849
Ref: d5f1629849
Ref: cblas c cblas_dspmv1630127
Ref: d601630127
Ref: cblas c cblas_dger1630378
Ref: d611630378
Ref: cblas c cblas_dsyr1630608
Ref: d621630608
Ref: cblas c cblas_dspr1630810
Ref: d631630810
Ref: cblas c cblas_dsyr21630998
Ref: d641630998
Ref: cblas c cblas_dspr21631244
Ref: d651631244
Ref: cblas c cblas_chemv1631475
Ref: d661631475
Ref: cblas c cblas_chbmv1631732
Ref: d671631732
Ref: cblas c cblas_chpmv1632002
Ref: d681632002
Ref: cblas c cblas_cgeru1632245
Ref: d691632245
Ref: cblas c cblas_cgerc1632459
Ref: d6a1632459
Ref: cblas c cblas_cher1632673
Ref: d6b1632673
Ref: cblas c cblas_chpr1632870
Ref: d6c1632870
Ref: cblas c cblas_cher21633052
Ref: d6d1633052
Ref: cblas c cblas_chpr21633291
Ref: d6e1633291
Ref: cblas c cblas_zhemv1633506
Ref: d6f1633506
Ref: cblas c cblas_zhbmv1633763
Ref: d701633763
Ref: cblas c cblas_zhpmv1634033
Ref: d711634033
Ref: cblas c cblas_zgeru1634276
Ref: d721634276
Ref: cblas c cblas_zgerc1634490
Ref: d731634490
Ref: cblas c cblas_zher1634704
Ref: d741634704
Ref: cblas c cblas_zhpr1634902
Ref: d751634902
Ref: cblas c cblas_zher21635085
Ref: d761635085
Ref: cblas c cblas_zhpr21635324
Ref: d771635324
Node: Level 3<2>1635539
Ref: cblas level-31635644
Ref: d781635644
Ref: cblas c cblas_sgemm1635671
Ref: d791635671
Ref: cblas c cblas_ssymm1636007
Ref: d7a1636007
Ref: cblas c cblas_ssyrk1636316
Ref: d7b1636316
Ref: cblas c cblas_ssyr2k1636590
Ref: d7c1636590
Ref: cblas c cblas_strmm1636906
Ref: d7d1636906
Ref: cblas c cblas_strsm1637229
Ref: d7e1637229
Ref: cblas c cblas_dgemm1637552
Ref: d7f1637552
Ref: cblas c cblas_dsymm1637893
Ref: d801637893
Ref: cblas c cblas_dsyrk1638207
Ref: d811638207
Ref: cblas c cblas_dsyr2k1638485
Ref: d821638485
Ref: cblas c cblas_dtrmm1638806
Ref: d831638806
Ref: cblas c cblas_dtrsm1639132
Ref: d841639132
Ref: cblas c cblas_cgemm1639458
Ref: d851639458
Ref: cblas c cblas_csymm1639791
Ref: d861639791
Ref: cblas c cblas_csyrk1640097
Ref: d871640097
Ref: cblas c cblas_csyr2k1640369
Ref: d881640369
Ref: cblas c cblas_ctrmm1640682
Ref: d891640682
Ref: cblas c cblas_ctrsm1641003
Ref: d8a1641003
Ref: cblas c cblas_zgemm1641324
Ref: d8b1641324
Ref: cblas c cblas_zsymm1641657
Ref: d8c1641657
Ref: cblas c cblas_zsyrk1641963
Ref: d8d1641963
Ref: cblas c cblas_zsyr2k1642235
Ref: d8e1642235
Ref: cblas c cblas_ztrmm1642548
Ref: d8f1642548
Ref: cblas c cblas_ztrsm1642869
Ref: d901642869
Ref: cblas c cblas_chemm1643190
Ref: d911643190
Ref: cblas c cblas_cherk1643496
Ref: d921643496
Ref: cblas c cblas_cher2k1643768
Ref: d931643768
Ref: cblas c cblas_zhemm1644081
Ref: d941644081
Ref: cblas c cblas_zherk1644387
Ref: d951644387
Ref: cblas c cblas_zher2k1644661
Ref: d961644661
Ref: cblas c cblas_xerbla1644975
Ref: d971644975
Node: Examples<37>1645067
Ref: cblas examples1645153
Ref: d981645153
Node: GNU General Public License1646618
Ref: gpl doc1646750
Ref: d991646750
Ref: gpl gnu-general-public-license1646750
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Node: GNU Free Documentation License1684724
Ref: fdl doc1684844
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Node: Index1709747

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