Package org.apache.commons.math4.legacy.analysis.polynomials

Class PolynomialsUtils

java.lang.Object

org.apache.commons.math4.legacy.analysis.polynomials.PolynomialsUtils

public final class PolynomialsUtils
extends Object

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

Since:

2.0

Method Summary

Modifier and Type	Method	Description
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

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.