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35.8 Examples

The following program uses the Brent algorithm to find the minimum of the function f(x) = \cos(x) + 1, which occurs at x = \pi. The starting interval is (0,6), with an initial guess for the minimum of 2.

#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_min.h>

double fn1 (double x, void * params)
{
  (void)(params); /* avoid unused parameter warning */
  return cos(x) + 1.0;
}

int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_min_fminimizer_type *T;
  gsl_min_fminimizer *s;
  double m = 2.0, m_expected = M_PI;
  double a = 0.0, b = 6.0;
  gsl_function F;

  F.function = &fn1;
  F.params = 0;

  T = gsl_min_fminimizer_brent;
  s = gsl_min_fminimizer_alloc (T);
  gsl_min_fminimizer_set (s, &F, m, a, b);

  printf ("using %s method\n",
          gsl_min_fminimizer_name (s));

  printf ("%5s [%9s, %9s] %9s %10s %9s\n",
          "iter", "lower", "upper", "min",
          "err", "err(est)");

  printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
          iter, a, b,
          m, m - m_expected, b - a);

  do
    {
      iter++;
      status = gsl_min_fminimizer_iterate (s);

      m = gsl_min_fminimizer_x_minimum (s);
      a = gsl_min_fminimizer_x_lower (s);
      b = gsl_min_fminimizer_x_upper (s);

      status 
        = gsl_min_test_interval (a, b, 0.001, 0.0);

      if (status == GSL_SUCCESS)
        printf ("Converged:\n");

      printf ("%5d [%.7f, %.7f] "
              "%.7f %+.7f %.7f\n",
              iter, a, b,
              m, m - m_expected, b - a);
    }
  while (status == GSL_CONTINUE && iter < max_iter);

  gsl_min_fminimizer_free (s);

  return status;
}

Here are the results of the minimization procedure.

$ ./a.out 
using brent method
 iter [    lower,     upper]       min        err  err(est)
    0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
    1 [2.0000000, 6.0000000] 3.5278640 +0.3862713 4.0000000
    2 [2.0000000, 3.5278640] 3.1748217 +0.0332290 1.5278640
    3 [2.0000000, 3.1748217] 3.1264576 -0.0151351 1.1748217
    4 [3.1264576, 3.1748217] 3.1414743 -0.0001183 0.0483641
    5 [3.1414743, 3.1748217] 3.1415930 +0.0000004 0.0333474
Converged:
    6 [3.1414743, 3.1415930] 3.1415927 +0.0000000 0.0001187