GNU Scientific Library – Reference Manual: Minimization Examples

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### 34.8 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)
    {
      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

        0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
        1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
        2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
        3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
        4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
        5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
        6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
        7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
        8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
        9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
       10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
    Converged:
       11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175

Next: Minimization References and Further Reading, Previous: Minimization Algorithms, Up: One dimensional Minimization [Index]