org.apache.commons.math3.analysis.polynomials

Class PolynomialFunctionLagrangeForm

java.lang.Object

org.apache.commons.math3.analysis.polynomials.PolynomialFunctionLagrangeForm

All Implemented Interfaces:

UnivariateFunction

public class PolynomialFunctionLagrangeForm
extends java.lang.Object
implements UnivariateFunction

Implements the representation of a real polynomial function in Lagrange Form. For reference, see Introduction to Numerical Analysis, ISBN 038795452X, chapter 2.

The approximated function should be smooth enough for Lagrange polynomial to work well. Otherwise, consider using splines instead.

Since:

1.2

Field Summary

Fields

Modifier and Type	Field and Description
private double[]	coefficients The coefficients of the polynomial, ordered by degree -- i.e.
private boolean	coefficientsComputed Whether the polynomial coefficients are available.
private double[]	x Interpolating points (abscissas).
private double[]	y Function values at interpolating points.

Constructor Summary

Constructors

Constructor and Description
PolynomialFunctionLagrangeForm(double[] x, double[] y) Construct a Lagrange polynomial with the given abscissas and function values.

Method Summary

Methods

Modifier and Type	Method and Description
protected void	computeCoefficients() Calculate the coefficients of Lagrange polynomial from the interpolation data.
int	degree() Returns the degree of the polynomial.
static double	evaluate(double[] x, double[] y, double z) Evaluate the Lagrange polynomial using Neville's Algorithm.
private static double	evaluateInternal(double[] x, double[] y, double z) Evaluate the Lagrange polynomial using Neville's Algorithm.
double[]	getCoefficients() Returns a copy of the coefficients array.
double[]	getInterpolatingPoints() Returns a copy of the interpolating points array.
double[]	getInterpolatingValues() Returns a copy of the interpolating values array.
double	value(double z) Calculate the function value at the given point.
static boolean	verifyInterpolationArray(double[] x, double[] y, boolean abort) Check that the interpolation arrays are valid.

Methods inherited from class java.lang.Object

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

Field Detail

coefficients

private double[] coefficients

The coefficients of the polynomial, ordered by degree -- i.e. coefficients[0] is the constant term and coefficients[n] is the coefficient of x^n where n is the degree of the polynomial.

x

private final double[] x

Interpolating points (abscissas).

y

private final double[] y

Function values at interpolating points.

coefficientsComputed

private boolean coefficientsComputed

Whether the polynomial coefficients are available.

Constructor Detail

PolynomialFunctionLagrangeForm

public PolynomialFunctionLagrangeForm(double[] x,
                              double[] y)
                               throws DimensionMismatchException,
                                      NumberIsTooSmallException,
                                      NonMonotonicSequenceException

Construct a Lagrange polynomial with the given abscissas and function values. The order of interpolating points are not important.

The constructor makes copy of the input arrays and assigns them.

Parameters:

x - interpolating points

y - function values at interpolating points

Throws:

DimensionMismatchException - if the array lengths are different.

NumberIsTooSmallException - if the number of points is less than 2.

NonMonotonicSequenceException - if two abscissae have the same value.

Method Detail

value

public double value(double z)

Calculate the function value at the given point.

Specified by:

value in interface UnivariateFunction

Parameters:

z - Point at which the function value is to be computed.

Returns:

the function value.

Throws:

DimensionMismatchException - if x and y have different lengths.

NonMonotonicSequenceException - if x is not sorted in strictly increasing order.

NumberIsTooSmallException - if the size of x is less than 2.

degree

public int degree()

Returns the degree of the polynomial.

Returns:

the degree of the polynomial

getInterpolatingPoints

public double[] getInterpolatingPoints()

Returns a copy of the interpolating points array.

Changes made to the returned copy will not affect the polynomial.

Returns:

a fresh copy of the interpolating points array

getInterpolatingValues

public double[] getInterpolatingValues()

Returns a copy of the interpolating values array.

Changes made to the returned copy will not affect the polynomial.

Returns:

a fresh copy of the interpolating values array

getCoefficients

public double[] getCoefficients()

Returns a copy of the coefficients array.

Changes made to the returned copy will not affect the polynomial.

Note that coefficients computation can be ill-conditioned. Use with caution and only when it is necessary.

Returns:

a fresh copy of the coefficients array

evaluate

public static double evaluate(double[] x,
              double[] y,
              double z)
                       throws DimensionMismatchException,
                              NumberIsTooSmallException,
                              NonMonotonicSequenceException

Evaluate the Lagrange polynomial using Neville's Algorithm. It takes O(n^2) time.

Parameters:

x - Interpolating points array.

y - Interpolating values array.

z - Point at which the function value is to be computed.

Returns:

the function value.

Throws:

DimensionMismatchException - if x and y have different lengths.

NonMonotonicSequenceException - if x is not sorted in strictly increasing order.

NumberIsTooSmallException - if the size of x is less than 2.

evaluateInternal

private static double evaluateInternal(double[] x,
                      double[] y,
                      double z)

Evaluate the Lagrange polynomial using Neville's Algorithm. It takes O(n^2) time.

Parameters:

x - Interpolating points array.

y - Interpolating values array.

z - Point at which the function value is to be computed.

Returns:

the function value.

Throws:

DimensionMismatchException - if x and y have different lengths.

NonMonotonicSequenceException - if x is not sorted in strictly increasing order.

NumberIsTooSmallException - if the size of x is less than 2.

computeCoefficients

protected void computeCoefficients()

Calculate the coefficients of Lagrange polynomial from the interpolation data. It takes O(n^2) time. Note that this computation can be ill-conditioned: Use with caution and only when it is necessary.

verifyInterpolationArray

public static boolean verifyInterpolationArray(double[] x,
                               double[] y,
                               boolean abort)
                                        throws DimensionMismatchException,
                                               NumberIsTooSmallException,
                                               NonMonotonicSequenceException

Check that the interpolation arrays are valid. The arrays features checked by this method are that both arrays have the same length and this length is at least 2.

Parameters:

x - Interpolating points array.

y - Interpolating values array.

abort - Whether to throw an exception if x is not sorted.

Returns:

false if the x is not sorted in increasing order, true otherwise.

Throws:

DimensionMismatchException - if the array lengths are different.

NumberIsTooSmallException - if the number of points is less than 2.

NonMonotonicSequenceException - if x is not sorted in strictly increasing order and abort is true.

See Also:

evaluate(double[], double[], double), computeCoefficients()