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:

$$\begin{gathered} T_0(x)=1 \\ {T}_1(x)=x \\ {T}_{k+1}(x)=2x{T}_k(x)-T_{k-1}(x) \end{gathered}$$

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:

$$\begin{gathered} H_0(x)=1 \\ {H}_1(x)=2x \\ {H}_{k+1}(x)=2x{H}_k(X)-2k{H}_{k-1}(x) \end{gathered}$$

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:

$$\begin{gathered} L_0(x)=1 \\ {L}_1(x)=1-x \\ (k+1)L_{k+1}(x)=(2k+1-x)L_k(x)-k{L}_{k-1}(x) \end{gathered}$$

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:

$$\begin{gathered} P_0(x)=1 \\ {P}_1(x)=x \\ (k+1)P_{k+1}(x)=(2k+1)x{P}_k(x)-k{P}_{k-1}(x) \end{gathered}$$

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:

$$\begin{gathered} P_0^{vw}(x)=1 \\ {P}_{-1}^{vw}(x)=0 \\ 2k(k+v+w)(2k+v+w-2)P_k^{vw}(x)= \\ (2k+v+w-1)[(2k+v+w)(2k+v+w-2)x+v^2-w^2]P_{k-1}^{vw}(x) \\ -2(k+v-1)(k+w-1)(2k+v+w)P_{k-2}^{vw}(x) \end{gathered}$$

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 $\mathrm{\Delta }=$ shift and let $P_s(x)=P(x+\mathrm{\Delta })$. 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 $\sum _{i=0}^{n-1}{b}_i{x}^i=\sum _{i=0}^{n-1}{a}_i(x+\mathrm{\Delta }{)}^i$ for all $x$.

Parameters:

coefficients - Coefficients of the original polynomial.

shift - Shift value.

Returns:

the coefficients $b_i$ of the shifted polynomial.