Class PolynomialsUtils


  • public class PolynomialsUtils
    extends Object
    A collection of static methods that operate on or return polynomials.
    • 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.