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.

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

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

Method Summary

Modifier and Type	Method and Description
private static PolynomialFunction	Returns: coefficients array buildPolynomial(final int degree of the polynomial degree, final List<BigFraction> list where the computed coefficients are stored coefficients, final PolynomialsUtils.RecurrenceCoefficientsGenerator recurrence coefficients generator generator) Deprecated as a consequence of `PolynomialFunction` being deprecated. Get the coefficients array for a given degree.
private static void	computeUpToDegree(final int maximal degree degree, final int current maximal degree maxDegree, final PolynomialsUtils.RecurrenceCoefficientsGenerator recurrence coefficients generator generator, final List<BigFraction> list where the computed coefficients should be appended coefficients) Compute polynomial coefficients up to a given degree.
public static PolynomialFunction	Returns: Chebyshev polynomial of specified degree createChebyshevPolynomial(final int degree of the polynomial degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. Create a Chebyshev polynomial of the first kind.
public static PolynomialFunction	Returns: Hermite polynomial of specified degree createHermitePolynomial(final int degree of the polynomial degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. Create a Hermite polynomial.
public static PolynomialFunction	Returns: Jacobi polynomial of specified degree createJacobiPolynomial(final int degree of the polynomial degree, final int first exponent v, final int second exponent w) Deprecated as a consequence of `PolynomialFunction` being deprecated. Create a Jacobi polynomial.
public static PolynomialFunction	Returns: Laguerre polynomial of specified degree createLaguerrePolynomial(final int degree of the polynomial degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. Create a Laguerre polynomial.
public static PolynomialFunction	Returns: Legendre polynomial of specified degree createLegendrePolynomial(final int degree of the polynomial degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. Create a Legendre polynomial.
public static double[]	Returns: the coefficients \(b_i\) of the shifted polynomial. shift(final double[] Coefficients of the original polynomial. coefficients, final double Shift value. 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`.

Inherited from java.lang.Object:: clone equals finalize getClass hashCode notify notifyAll toString wait wait wait

Field Detail

CHEBYSHEV_COEFFICIENTS	back to summary
private static final List<BigFraction> CHEBYSHEV_COEFFICIENTS Coefficients for Chebyshev polynomials.

HERMITE_COEFFICIENTS	back to summary
private static final List<BigFraction> HERMITE_COEFFICIENTS Coefficients for Hermite polynomials.

JACOBI_COEFFICIENTS	back to summary
private static final Map<PolynomialsUtils.JacobiKey, List<BigFraction>> JACOBI_COEFFICIENTS Coefficients for Jacobi polynomials.

LAGUERRE_COEFFICIENTS	back to summary
private static final List<BigFraction> LAGUERRE_COEFFICIENTS Coefficients for Laguerre polynomials.

LEGENDRE_COEFFICIENTS	back to summary
private static final List<BigFraction> LEGENDRE_COEFFICIENTS Coefficients for Legendre polynomials.

Constructor Detail

PolynomialsUtils	back to summary
private PolynomialsUtils() Private constructor, to prevent instantiation.

Method Detail

buildPolynomial back to summary

buildPolynomial	back to summary
private static PolynomialFunction buildPolynomial(final int degree, final List<BigFraction> coefficients, final PolynomialsUtils.RecurrenceCoefficientsGenerator generator) Deprecated as a consequence of `PolynomialFunction` being deprecated. See corresponding docs for further information. Get the coefficients array for a given degree. Parameters degree:int degree of the polynomial coefficients:List<BigFraction> list where the computed coefficients are stored generator:PolynomialsUtils.RecurrenceCoefficientsGenerator recurrence coefficients generator Returns:PolynomialFunction coefficients array

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

Deprecated

as a consequence of PolynomialFunction being deprecated. See corresponding docs for further information.

Get the coefficients array for a given degree.

Parameters

degree:int: degree of the polynomial
coefficients:List<BigFraction>: list where the computed coefficients are stored
generator:PolynomialsUtils.RecurrenceCoefficientsGenerator: recurrence coefficients generator

Returns:PolynomialFunction

coefficients array

computeUpToDegree	back to summary
private static void computeUpToDegree(final int degree, final int maxDegree, final PolynomialsUtils.RecurrenceCoefficientsGenerator generator, final List<BigFraction> coefficients) Compute polynomial coefficients up to a given degree. Parameters degree:int maximal degree maxDegree:int current maximal degree generator:PolynomialsUtils.RecurrenceCoefficientsGenerator recurrence coefficients generator coefficients:List<BigFraction> list where the computed coefficients should be appended

computeUpToDegree

back to summary

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

Compute polynomial coefficients up to a given degree.

Parameters

degree:int: maximal degree
maxDegree:int: current maximal degree
generator:PolynomialsUtils.RecurrenceCoefficientsGenerator: recurrence coefficients generator
coefficients:List<BigFraction>: list where the computed coefficients should be appended

createChebyshevPolynomial back to summary

createChebyshevPolynomial	back to summary
public static PolynomialFunction createChebyshevPolynomial(final int degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. See corresponding docs for further information. 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:int degree of the polynomial Returns:PolynomialFunction Chebyshev polynomial of specified degree

public static PolynomialFunction createChebyshevPolynomial(final int degree)

Deprecated

as a consequence of PolynomialFunction being deprecated. See corresponding docs for further information.

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:int: degree of the polynomial

Returns:PolynomialFunction

Chebyshev polynomial of specified degree

createHermitePolynomial back to summary

createHermitePolynomial	back to summary
public static PolynomialFunction createHermitePolynomial(final int degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. See corresponding docs for further information. 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:int degree of the polynomial Returns:PolynomialFunction Hermite polynomial of specified degree

public static PolynomialFunction createHermitePolynomial(final int degree)

Deprecated

as a consequence of PolynomialFunction being deprecated. See corresponding docs for further information.

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:int: degree of the polynomial

Returns:PolynomialFunction

Hermite polynomial of specified degree

createJacobiPolynomial back to summary

createJacobiPolynomial	back to summary
public static PolynomialFunction createJacobiPolynomial(final int degree, final int v, final int w) Deprecated as a consequence of `PolynomialFunction` being deprecated. See corresponding docs for further information. 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:int degree of the polynomial v:int first exponent w:int second exponent Returns:PolynomialFunction Jacobi polynomial of specified degree

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

Deprecated

as a consequence of PolynomialFunction being deprecated. See corresponding docs for further information.

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:int: degree of the polynomial
v:int: first exponent
w:int: second exponent

Returns:PolynomialFunction

Jacobi polynomial of specified degree

createLaguerrePolynomial back to summary

createLaguerrePolynomial	back to summary
public static PolynomialFunction createLaguerrePolynomial(final int degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. See corresponding docs for further information. 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:int degree of the polynomial Returns:PolynomialFunction Laguerre polynomial of specified degree

public static PolynomialFunction createLaguerrePolynomial(final int degree)

Deprecated

as a consequence of PolynomialFunction being deprecated. See corresponding docs for further information.

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:int: degree of the polynomial

Returns:PolynomialFunction

Laguerre polynomial of specified degree

createLegendrePolynomial back to summary

createLegendrePolynomial	back to summary
public static PolynomialFunction createLegendrePolynomial(final int degree) Deprecated as a consequence of `PolynomialFunction` being deprecated. See corresponding docs for further information. 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:int degree of the polynomial Returns:PolynomialFunction Legendre polynomial of specified degree

public static PolynomialFunction createLegendrePolynomial(final int degree)

Deprecated

as a consequence of PolynomialFunction being deprecated. See corresponding docs for further information.

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:int: degree of the polynomial

Returns:PolynomialFunction

Legendre polynomial of specified degree

shift back to summary

shift	back to summary
public static double[] shift(final double[] coefficients, final 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:double[] Coefficients of the original polynomial. shift:double Shift value. Returns:double[] the coefficients \(b_i\) of the shifted polynomial.

public static double[] shift(final double[] coefficients, final 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:double[]: Coefficients of the original polynomial.
shift:double: Shift value.

Returns:double[]

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

org.apache.commons.math3.analysis.polynomials back to summary

private Class PolynomialsUtils.JacobiKey

Class Inheritance

extends Object

java.lang.Object
org.apache.commons.math3.analysis.polynomials.PolynomialsUtils.JacobiKey

Inner class for Jacobi polynomials keys.

Field Summary

Modifier and Type	Field and Description
private final int	v First exponent.
private final int	w Second exponent.

Constructor Summary

Access	Constructor and Description
pack-priv	JacobiKey(final int first exponent v, final int second exponent w) Simple constructor.

Method Summary

Modifier and Type	Method and Description
public boolean	Returns: true if the instance and the other key refer to the same polynomial equals(final Object other key key) Overrides java.lang.Object.equals. Check if the instance represent the same key as another instance.
public int	Returns: hash code hashCode() Overrides java.lang.Object.hashCode. Get hash code.

Inherited from java.lang.Object:: clone finalize getClass notify notifyAll toString wait wait wait

Field Detail

v	back to summary
private final int v First exponent.

w	back to summary
private final int w Second exponent.

Constructor Detail

JacobiKey	back to summary
pack-priv JacobiKey(final int v, final int w) Simple constructor. Parameters v:int first exponent w:int second exponent

JacobiKey

back to summary

pack-priv JacobiKey(final int v, final int w)

Simple constructor.

Parameters

v:int: first exponent
w:int: second exponent

Method Detail

equals	back to summary
public boolean equals(final Object key) Overrides java.lang.Object.equals. Check if the instance represent the same key as another instance. Parameters key:Object other key Returns:boolean true if the instance and the other key refer to the same polynomial Annotations @Override

equals

back to summary

public boolean equals(final Object key)

Overrides java.lang.Object.equals.

Check if the instance represent the same key as another instance.

Parameters

key:Object: other key

Returns:boolean

true if the instance and the other key refer to the same polynomial

Annotations

@Override

hashCode	back to summary
public int hashCode() Overrides java.lang.Object.hashCode. Get hash code. Returns:int hash code Annotations @Override

hashCode

back to summary

public int hashCode()

Overrides java.lang.Object.hashCode.

Get hash code.

Returns:int

hash code

Annotations

@Override

public Class PolynomialsUtils

Nested and Inner Type Summary

Field Summary

Constructor Summary

Method Summary

Field Detail

Constructor Detail

Method Detail

private Class PolynomialsUtils.JacobiKey

Field Summary

Constructor Summary

Method Summary

Field Detail

Constructor Detail

Method Detail

private Interface PolynomialsUtils.RecurrenceCoefficientsGenerator

Method Summary

Method Detail