1: static char help[] = "Calculates moments for Gaussian functions.\n\n";

  3: #include <petscviewer.h>
  4: #include <petscdt.h>
  5: #include <petscvec.h>

  7: #include <gsl/gsl_sf_hermite.h>
  8: #include <gsl/gsl_randist.h>

 10: int main(int argc, char **argv)
 11: {
 12:   int        s, n         = 15;
 13:   PetscInt   tick, moment = 0, momentummax = 7;
 14:   PetscReal *zeros, *weights, scale, h, sigma = 1 / sqrt(2), g = 0, mu = 0;

 16:   PetscFunctionBeginUser;
 17:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));

 19:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-n", &n, NULL));
 20:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-moment_max", &momentummax, NULL));
 21:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-sigma", &sigma, NULL));
 22:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &mu, NULL));

 24:   /* calculate zeros and roots of Hermite Gauss quadrature */
 25:   PetscCall(PetscMalloc1(n, &zeros));
 26:   zeros[0] = 0;
 27:   tick     = n % 2;
 28:   for (s = 0; s < n / 2; s++) {
 29:     zeros[2 * s + tick]     = -gsl_sf_hermite_zero(n, s + 1);
 30:     zeros[2 * s + 1 + tick] = gsl_sf_hermite_zero(n, s + 1);
 31:   }

 33:   PetscCall(PetscDTFactorial(n, &scale));
 34:   scale = exp2(n - 1) * scale * PetscSqrtReal(PETSC_PI) / (n * n);
 35:   PetscCall(PetscMalloc1(n + 1, &weights));
 36:   for (s = 0; s < n; s++) {
 37:     h          = gsl_sf_hermite(n - 1, (double)zeros[s]);
 38:     weights[s] = scale / (h * h);
 39:   }
 40:   /* zeros and weights verified up to n = 5 with http://mathworld.wolfram.com/Hermite-GaussQuadrature.html */

 42:   for (moment = 0; moment < momentummax; moment++) {
 43:     /* http://www.wouterdenhaan.com/numerical/integrationslides.pdf */
 44:     /* https://en.wikipedia.org/wiki/Gauss-Hermite_quadrature */
 45:     /*
 46:        int_{-infinity}^{infinity} \frac{1}{sigma sqrt(2pi)} exp(- \frac{(x - mu)^2}{2 sigma^2) h(x) dx

 48:        then approx equals 1/pi sum_i w_i h( sqrt(2) sigma x_i + mu)
 49:     */
 50:     g = 0;
 51:     for (s = 0; s < n; s++) g += weights[s] * PetscPowRealInt(sqrt(2) * sigma * zeros[s] + mu, moment);
 52:     g /= sqrt(PETSC_PI);
 53:     /* results confirmed with https://en.wikipedia.org/wiki/Normal_distribution#Moments sigma^p * (p-1)!!*/
 54:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Moment %" PetscInt_FMT " %g \n", moment, (double)g));
 55:   }
 56:   PetscCall(PetscFree(zeros));
 57:   PetscCall(PetscFree(weights));
 58:   PetscCall(PetscFinalize());
 59:   return 0;
 60: }

 62: /*TEST

 64:   build:
 65:     requires: gsl double

 67:   test:

 69: TEST*/