org.apache.commons.math3.special

Class Beta

java.lang.Object

org.apache.commons.math3.special.Beta

public class Beta
extends java.lang.Object

This is a utility class that provides computation methods related to the Beta family of functions.

Implementation of logBeta(double, double) is based on the algorithms described in

• Didonato and Morris (1986), Computation of the Incomplete Gamma Function Ratios and their Inverse, TOMS 12(4), 377-393,
• Didonato and Morris (1992), Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios, TOMS 18(3), 360-373,

and implemented in the NSWC Library of Mathematical Functions, available here. This library is "approved for public release", and the Copyright guidance indicates that unless otherwise stated in the code, all FORTRAN functions in this library are license free. Since no such notice appears in the code these functions can safely be ported to Commons-Math.

Field Summary

Fields

Modifier and Type	Field and Description
private static double	DEFAULT_EPSILON Maximum allowed numerical error.
private static double[]	DELTA The coefficients of the series expansion of the Δ function.
private static double	HALF_LOG_TWO_PI The constant value of ½log 2π.

Constructor Summary

Constructors

Modifier	Constructor and Description
private	Beta() Default constructor.

Method Summary

Methods

Modifier and Type	Method and Description
private static double	deltaMinusDeltaSum(double a, double b) Returns the value of Δ(b) - Δ(a + b), with 0 ≤ a ≤ b and b ≥ 10.
static double	logBeta(double p, double q) Returns the value of log B(p, q) for 0 ≤ x ≤ 1 and p, q > 0.
static double	logBeta(double a, double b, double epsilon, int maxIterations) Deprecated. as of version 3.1, this method is deprecated as the computation of the beta function is no longer iterative; it will be removed in version 4.0. Current implementation of this method internally calls logBeta(double, double).
private static double	logGammaMinusLogGammaSum(double a, double b) Returns the value of log[Γ(b) / Γ(a + b)] for a ≥ 0 and b ≥ 10.
private static double	logGammaSum(double a, double b) Returns the value of log Γ(a + b) for 1 ≤ a, b ≤ 2.
static double	regularizedBeta(double x, double a, double b) Returns the regularized beta function I(x, a, b).
static double	regularizedBeta(double x, double a, double b, double epsilon) Returns the regularized beta function I(x, a, b).
static double	regularizedBeta(double x, double a, double b, double epsilon, int maxIterations) Returns the regularized beta function I(x, a, b).
static double	regularizedBeta(double x, double a, double b, int maxIterations) Returns the regularized beta function I(x, a, b).
private static double	sumDeltaMinusDeltaSum(double p, double q) Returns the value of Δ(p) + Δ(q) - Δ(p + q), with p, q ≥ 10.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DEFAULT_EPSILON

private static final double DEFAULT_EPSILON

Maximum allowed numerical error.

See Also:

Constant Field Values

HALF_LOG_TWO_PI

private static final double HALF_LOG_TWO_PI

The constant value of ½log 2π.

See Also:

Constant Field Values

DELTA

private static final double[] DELTA

The coefficients of the series expansion of the Δ function. This function is defined as follows

Δ(x) = log Γ(x) - (x - 0.5) log a + a - 0.5 log 2π,

see equation (23) in Didonato and Morris (1992). The series expansion, which applies for x ≥ 10, reads

                 14
                ====
             1  \                2 n
     Δ(x) = ---  >    d  (10 / x)
             x  /      n
                ====
                n = 0
 

Constructor Detail

Beta

private Beta()

Default constructor. Prohibit instantiation.

Method Detail

regularizedBeta

public static double regularizedBeta(double x,
                     double a,
                     double b)

Returns the regularized beta function I(x, a, b).

Parameters:

x - Value.

a - Parameter a.

b - Parameter b.

Returns:

the regularized beta function I(x, a, b).

Throws:

MaxCountExceededException - if the algorithm fails to converge.

regularizedBeta

public static double regularizedBeta(double x,
                     double a,
                     double b,
                     double epsilon)

Returns the regularized beta function I(x, a, b).

Parameters:

x - Value.

a - Parameter a.

b - Parameter b.

epsilon - When the absolute value of the nth item in the series is less than epsilon the approximation ceases to calculate further elements in the series.

Returns:

the regularized beta function I(x, a, b)

Throws:

MaxCountExceededException - if the algorithm fails to converge.

regularizedBeta

public static double regularizedBeta(double x,
                     double a,
                     double b,
                     int maxIterations)

Returns the regularized beta function I(x, a, b).

Parameters:

x - the value.

a - Parameter a.

b - Parameter b.

maxIterations - Maximum number of "iterations" to complete.

Returns:

the regularized beta function I(x, a, b)

Throws:

MaxCountExceededException - if the algorithm fails to converge.

regularizedBeta

public static double regularizedBeta(double x,
                     double a,
                     double b,
                     double epsilon,
                     int maxIterations)

Returns the regularized beta function I(x, a, b). The implementation of this method is based on:

• Regularized Beta Function.
• Regularized Beta Function.

Parameters:

x - the value.

a - Parameter a.

b - Parameter b.

epsilon - When the absolute value of the nth item in the series is less than epsilon the approximation ceases to calculate further elements in the series.

maxIterations - Maximum number of "iterations" to complete.

Returns:

the regularized beta function I(x, a, b)

Throws:

MaxCountExceededException - if the algorithm fails to converge.

logBeta

@Deprecated
public static double logBeta(double a,
                        double b,
                        double epsilon,
                        int maxIterations)

Deprecated. as of version 3.1, this method is deprecated as the computation of the beta function is no longer iterative; it will be removed in version 4.0. Current implementation of this method internally calls logBeta(double, double).

Returns the natural logarithm of the beta function B(a, b). The implementation of this method is based on:

• Beta Function, equation (1).

Parameters:

a - Parameter a.

b - Parameter b.

epsilon - This parameter is ignored.

maxIterations - This parameter is ignored.

Returns:

log(B(a, b)).

logGammaSum

private static double logGammaSum(double a,
                 double b)
                           throws OutOfRangeException

Returns the value of log Γ(a + b) for 1 ≤ a, b ≤ 2. Based on the NSWC Library of Mathematics Subroutines double precision implementation, DGSMLN. In BetaTest.testLogGammaSum(), this private method is accessed through reflection.

Parameters:

a - First argument.

b - Second argument.

Returns:

the value of log(Gamma(a + b)).

Throws:

OutOfRangeException - if a or b is lower than 1.0 or greater than 2.0.

logGammaMinusLogGammaSum

private static double logGammaMinusLogGammaSum(double a,
                              double b)
                                        throws NumberIsTooSmallException

Returns the value of log[Γ(b) / Γ(a + b)] for a ≥ 0 and b ≥ 10. Based on the NSWC Library of Mathematics Subroutines double precision implementation, DLGDIV. In BetaTest.testLogGammaMinusLogGammaSum(), this private method is accessed through reflection.

Parameters:

a - First argument.

b - Second argument.

Returns:

the value of log(Gamma(b) / Gamma(a + b)).

Throws:

NumberIsTooSmallException - if a < 0.0 or b < 10.0.

deltaMinusDeltaSum

private static double deltaMinusDeltaSum(double a,
                        double b)
                                  throws OutOfRangeException,
                                         NumberIsTooSmallException

Returns the value of Δ(b) - Δ(a + b), with 0 ≤ a ≤ b and b ≥ 10. Based on equations (26), (27) and (28) in Didonato and Morris (1992).

Parameters:

a - First argument.

b - Second argument.

Returns:

the value of Delta(b) - Delta(a + b)

Throws:

OutOfRangeException - if a < 0 or a > b

NumberIsTooSmallException - if b < 10

sumDeltaMinusDeltaSum

private static double sumDeltaMinusDeltaSum(double p,
                           double q)

Returns the value of Δ(p) + Δ(q) - Δ(p + q), with p, q ≥ 10. Based on the NSWC Library of Mathematics Subroutines double precision implementation, DBCORR. In BetaTest.testSumDeltaMinusDeltaSum(), this private method is accessed through reflection.

Parameters:

p - First argument.

q - Second argument.

Returns:

the value of Delta(p) + Delta(q) - Delta(p + q).

Throws:

NumberIsTooSmallException - if p < 10.0 or q < 10.0.

logBeta

public static double logBeta(double p,
             double q)

Returns the value of log B(p, q) for 0 ≤ x ≤ 1 and p, q > 0. Based on the NSWC Library of Mathematics Subroutines implementation, DBETLN.

Parameters:

p - First argument.

q - Second argument.

Returns:

the value of log(Beta(p, q)), NaN if p <= 0 or q <= 0.