source: branches/HeuristicLab.ALGLIB-2.5.0/ALGLIB-2.5.0/legendre.cs @ 5192

/*************************************************************************
>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses

>>> END OF LICENSE >>>
*************************************************************************/


namespace alglib {
  public class legendre {
    /*************************************************************************
    Calculation of the value of the Legendre polynomial Pn.

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Legendre polynomial Pn at x
    *************************************************************************/
    public static double legendrecalculate(int n,
        double x) {
      double result = 0;
      double a = 0;
      double b = 0;
      int i = 0;

      result = 1;
      a = 1;
      b = x;
      if (n == 0) {
        result = a;
        return result;
      }
      if (n == 1) {
        result = b;
        return result;
      }
      for (i = 2; i <= n; i++) {
        result = ((2 * i - 1) * x * b - (i - 1) * a) / i;
        a = b;
        b = result;
      }
      return result;
    }


    /*************************************************************************
    Summation of Legendre polynomials using Clenshaw’s recurrence formula.

    This routine calculates
        c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x)

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Legendre polynomial at x
    *************************************************************************/
    public static double legendresum(ref double[] c,
        int n,
        double x) {
      double result = 0;
      double b1 = 0;
      double b2 = 0;
      int i = 0;

      b1 = 0;
      b2 = 0;
      for (i = n; i >= 0; i--) {
        result = (2 * i + 1) * x * b1 / (i + 1) - (i + 1) * b2 / (i + 2) + c[i];
        b2 = b1;
        b1 = result;
      }
      return result;
    }


    /*************************************************************************
    Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N

    Input parameters:
        N   -   polynomial degree, n>=0

    Output parameters:
        C   -   coefficients
    *************************************************************************/
    public static void legendrecoefficients(int n,
        ref double[] c) {
      int i = 0;

      c = new double[n + 1];
      for (i = 0; i <= n; i++) {
        c[i] = 0;
      }
      c[n] = 1;
      for (i = 1; i <= n; i++) {
        c[n] = c[n] * (n + i) / 2 / i;
      }
      for (i = 0; i <= n / 2 - 1; i++) {
        c[n - 2 * (i + 1)] = -(c[n - 2 * i] * (n - 2 * i) * (n - 2 * i - 1) / 2 / (i + 1) / (2 * (n - i) - 1));
      }
    }
  }
}