Package org.hipparchus.analysis.polynomials

Class PolynomialsUtils

java.lang.Object
org.hipparchus.analysis.polynomials.PolynomialsUtils
public class PolynomialsUtils extends Object
A collection of static methods that operate on or return polynomials.

  • Method Details

    • 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.