org.apache.commons.math3.analysis.polynomials

Class PolynomialsUtils

java.lang.Object

org.apache.commons.math3.analysis.polynomials.PolynomialsUtils

public class PolynomialsUtils
extends java.lang.Object

A collection of static methods that operate on or return polynomials.

Since:

2.0

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
private static class	PolynomialsUtils.JacobiKey Inner class for Jacobi polynomials keys.
private static interface	PolynomialsUtils.RecurrenceCoefficientsGenerator Interface for recurrence coefficients generation.

Field Summary

Fields

Modifier and Type	Field and Description
private static java.util.List<BigFraction>	CHEBYSHEV_COEFFICIENTS Coefficients for Chebyshev polynomials.
private static java.util.List<BigFraction>	HERMITE_COEFFICIENTS Coefficients for Hermite polynomials.
private static java.util.Map<PolynomialsUtils.JacobiKey,java.util.List<BigFraction>>	JACOBI_COEFFICIENTS Coefficients for Jacobi polynomials.
private static java.util.List<BigFraction>	LAGUERRE_COEFFICIENTS Coefficients for Laguerre polynomials.
private static java.util.List<BigFraction>	LEGENDRE_COEFFICIENTS Coefficients for Legendre polynomials.

Constructor Summary

Constructors

Modifier	Constructor and Description
private	PolynomialsUtils() Private constructor, to prevent instantiation.

Method Summary

Methods

Modifier and Type	Method and Description
private static PolynomialFunction	buildPolynomial(int degree, java.util.List<BigFraction> coefficients, PolynomialsUtils.RecurrenceCoefficientsGenerator generator) Get the coefficients array for a given degree.
private static void	computeUpToDegree(int degree, int maxDegree, PolynomialsUtils.RecurrenceCoefficientsGenerator generator, java.util.List<BigFraction> coefficients) Compute polynomial coefficients up to a given degree.
static PolynomialFunction	createChebyshevPolynomial(int degree) Create a Chebyshev polynomial of the first kind.
static PolynomialFunction	createHermitePolynomial(int degree) Create a Hermite polynomial.
static PolynomialFunction	createJacobiPolynomial(int degree, int v, int w) Create a Jacobi polynomial.
static PolynomialFunction	createLaguerrePolynomial(int degree) Create a Laguerre polynomial.
static PolynomialFunction	createLegendrePolynomial(int degree) Create a Legendre polynomial.
static double[]	shift(double[] coefficients, double shift) Compute the coefficients of the polynomial \(P_s(x)\) whose values at point x will be the same as the those from the original polynomial \(P(x)\) when computed at x + shift.

Methods inherited from class java.lang.Object

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

Field Detail

CHEBYSHEV_COEFFICIENTS

private static final java.util.List<BigFraction> CHEBYSHEV_COEFFICIENTS

Coefficients for Chebyshev polynomials.

HERMITE_COEFFICIENTS

private static final java.util.List<BigFraction> HERMITE_COEFFICIENTS

Coefficients for Hermite polynomials.

LAGUERRE_COEFFICIENTS

private static final java.util.List<BigFraction> LAGUERRE_COEFFICIENTS

Coefficients for Laguerre polynomials.

LEGENDRE_COEFFICIENTS

private static final java.util.List<BigFraction> LEGENDRE_COEFFICIENTS

Coefficients for Legendre polynomials.

JACOBI_COEFFICIENTS

private static final java.util.Map<PolynomialsUtils.JacobiKey,java.util.List<BigFraction>> JACOBI_COEFFICIENTS

Coefficients for Jacobi polynomials.

Constructor Detail

PolynomialsUtils

private PolynomialsUtils()

Private constructor, to prevent instantiation.

Method Detail

createChebyshevPolynomial

public static PolynomialFunction createChebyshevPolynomial(int degree)

Create a Chebyshev polynomial of the first kind.

Chebyshev polynomials of the first kind are orthogonal polynomials. They can be defined by the following recurrence relations:

\( T_0(x) = 1 \\ T_1(x) = x \\ T_{k+1}(x) = 2x T_k(x) - T_{k-1}(x) \)

Parameters:

degree - degree of the polynomial

Returns:

Chebyshev polynomial of specified degree

createHermitePolynomial

public static PolynomialFunction createHermitePolynomial(int degree)

Create a Hermite polynomial.

Hermite polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

\( H_0(x) = 1 \\ H_1(x) = 2x \\ H_{k+1}(x) = 2x H_k(X) - 2k H_{k-1}(x) \)

Parameters:

degree - degree of the polynomial

Returns:

Hermite polynomial of specified degree

createLaguerrePolynomial

public static PolynomialFunction createLaguerrePolynomial(int degree)

Create a Laguerre polynomial.

Laguerre polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

\( L_0(x) = 1 \\ L_1(x) = 1 - x \\ (k+1) L_{k+1}(x) = (2k + 1 - x) L_k(x) - k L_{k-1}(x) \)

Parameters:

degree - degree of the polynomial

Returns:

Laguerre polynomial of specified degree

createLegendrePolynomial

public static PolynomialFunction createLegendrePolynomial(int degree)

Create a Legendre polynomial.

Legendre polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

\( P_0(x) = 1 \\ P_1(x) = x \\ (k+1) P_{k+1}(x) = (2k+1) x P_k(x) - k P_{k-1}(x) \)

Parameters:

degree - degree of the polynomial

Returns:

Legendre polynomial of specified degree

createJacobiPolynomial

public static PolynomialFunction createJacobiPolynomial(int degree,
                                        int v,
                                        int w)

Create a Jacobi polynomial.

Jacobi polynomials are orthogonal polynomials. They can be defined by the following recurrence relations:

\( P_0^{vw}(x) = 1 \\ P_{-1}^{vw}(x) = 0 \\ 2k(k + v + w)(2k + v + w - 2) P_k^{vw}(x) = \\ (2k + v + w - 1)[(2k + v + w)(2k + v + w - 2) x + v^2 - w^2] P_{k-1}^{vw}(x) \\ - 2(k + v - 1)(k + w - 1)(2k + v + w) P_{k-2}^{vw}(x) \)

Parameters:

degree - degree of the polynomial

v - first exponent

w - second exponent

Returns:

Jacobi polynomial of specified degree

shift

public static double[] shift(double[] coefficients,
             double shift)

Compute the coefficients of the polynomial \(P_s(x)\) whose values at point x will be the same as the those from the original polynomial \(P(x)\) when computed at x + shift.

More precisely, let \(\Delta = \) shift and let \(P_s(x) = P(x + \Delta)\). The returned array consists of the coefficients of \(P_s\). So if \(a_0, ..., a_{n-1}\) are the coefficients of \(P\), then the returned array \(b_0, ..., b_{n-1}\) satisfies the identity \(\sum_{i=0}^{n-1} b_i x^i = \sum_{i=0}^{n-1} a_i (x + \Delta)^i\) for all \(x\).

Parameters:

coefficients - Coefficients of the original polynomial.

shift - Shift value.

Returns:

the coefficients \(b_i\) of the shifted polynomial.

buildPolynomial

private static PolynomialFunction buildPolynomial(int degree,
                                 java.util.List<BigFraction> coefficients,
                                 PolynomialsUtils.RecurrenceCoefficientsGenerator generator)

Get the coefficients array for a given degree.

Parameters:

degree - degree of the polynomial

coefficients - list where the computed coefficients are stored

generator - recurrence coefficients generator

Returns:

coefficients array

computeUpToDegree

private static void computeUpToDegree(int degree,
                     int maxDegree,
                     PolynomialsUtils.RecurrenceCoefficientsGenerator generator,
                     java.util.List<BigFraction> coefficients)

Compute polynomial coefficients up to a given degree.

Parameters:

degree - maximal degree

maxDegree - current maximal degree

generator - recurrence coefficients generator

coefficients - list where the computed coefficients should be appended