fr.cnes.sirius.patrius.math.analysis.polynomials

Class PolynomialsUtils

java.lang.Object

fr.cnes.sirius.patrius.math.analysis.polynomials.PolynomialsUtils

public final class PolynomialsUtils
extends Object

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

Since:

2.0

Version:

$Id: PolynomialsUtils.java 18108 2017-10-04 06:45:27Z bignon $

Method Summary

Modifier and Type	Method and 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 Ps(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:

  T0(X)   = 1
  T1(X)   = X
  Tk+1(X) = 2X Tk(X) - Tk-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:

  H0(X)   = 1
  H1(X)   = 2X
  Hk+1(X) = 2X Hk(X) - 2k Hk-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:

        L0(X)   = 1
        L1(X)   = 1 - X
  (k+1) Lk+1(X) = (2k + 1 - X) Lk(X) - k Lk-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:

        P0(X)   = 1
        P1(X)   = X
  (k+1) Pk+1(X) = (2k+1) X Pk(X) - k Pk-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:

        P0vw(X)   = 1
        P-1vw(X)  = 0
  2k(k + v + w)(2k + v + w - 2) Pkvw(X) =
  (2k + v + w - 1)[(2k + v + w)(2k + v + w - 2) X + v2 - w2] Pk-1vw(X)
  - 2(k + v - 1)(k + w - 1)(2k + v + w) Pk-2vw(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 Ps(x) whose values at point x will be the same as the those from the original polynomial P(x) when computed at x + shift. Thus, if P(x) = Σi ai xi, then

  

   

    
Ps(x)
    
= Σi bi xi
   
   

    

    
= Σi ai (x + shift)i
   
  
 

Parameters:

coefficients - Coefficients of the original polynomial.

shift - Shift value.

Returns:

the coefficients bi of the shifted polynomial.