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Examples

This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in x and y, and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter.

     int
     main (void)
     {
       size_t iter = 0;
       int status;
     
       const gsl_multimin_fdfminimizer_type *T;
       gsl_multimin_fdfminimizer *s;
     
       /* Position of the minimum (1,2). */
       double par[2] = { 1.0, 2.0 };
     
       gsl_vector *x;
       gsl_multimin_function_fdf my_func;
     
       my_func.f = &my_f;
       my_func.df = &my_df;
       my_func.fdf = &my_fdf;
       my_func.n = 2;
       my_func.params = ∥
     
       /* Starting point, x = (5,7) */
       x = gsl_vector_alloc (2);
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);
     
       T = gsl_multimin_fdfminimizer_conjugate_fr;
       s = gsl_multimin_fdfminimizer_alloc (T, 2);
     
       gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);
     
       do
         {
           iter++;
           status = gsl_multimin_fdfminimizer_iterate (s);
     
           if (status)
             break;
     
           status = gsl_multimin_test_gradient (s->gradient, 1e-3);
     
           if (status == GSL_SUCCESS)
             printf ("Minimum found at:\n");
     
           printf ("%5d %.5f %.5f %10.5f\n", iter,
                   gsl_vector_get (s->x, 0),
                   gsl_vector_get (s->x, 1),
                   s->f);
     
         }
       while (status == GSL_CONTINUE && iter < 100);
     
       gsl_multimin_fdfminimizer_free (s);
       gsl_vector_free (x);
     
       return 0;
     }
     

The initial step-size is chosen as 0.01, a conservative estimate in this case, and the line minimization parameter is set at 0.0001. The program terminates when the norm of the gradient has been reduced below 0.001. The output of the program is shown below,

              x       y         f
         1 4.99629 6.99072  687.84780
         2 4.98886 6.97215  683.55456
         3 4.97400 6.93501  675.01278
         4 4.94429 6.86073  658.10798
         5 4.88487 6.71217  625.01340
         6 4.76602 6.41506  561.68440
         7 4.52833 5.82083  446.46694
         8 4.05295 4.63238  261.79422
         9 3.10219 2.25548   75.49762
        10 2.85185 1.62963   67.03704
        11 2.19088 1.76182   45.31640
        12 0.86892 2.02622   30.18555
     Minimum found at:
        13 1.00000 2.00000   30.00000
     

Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting the successive points.

The conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic. Additional iterations would be needed for a more complicated function.

Here is another example using the Nelder-Mead Simplex algorithm to minimize the same example object function, as above.

     int
     main(void)
     {
       size_t np = 2;
       double par[2] = {1.0, 2.0};
     
       const gsl_multimin_fminimizer_type *T =
         gsl_multimin_fminimizer_nmsimplex;
       gsl_multimin_fminimizer *s = NULL;
       gsl_vector *ss, *x;
       gsl_multimin_function minex_func;
     
       size_t iter = 0, i;
       int status;
       double size;
     
       /* Initial vertex size vector */
       ss = gsl_vector_alloc (np);
     
       /* Set all step sizes to 1 */
       gsl_vector_set_all (ss, 1.0);
     
       /* Starting point */
       x = gsl_vector_alloc (np);
     
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);
     
       /* Initialize method and iterate */
       minex_func.f = &my_f;
       minex_func.n = np;
       minex_func.params = (void *)&par;
     
       s = gsl_multimin_fminimizer_alloc (T, np);
       gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
     
       do
         {
           iter++;
           status = gsl_multimin_fminimizer_iterate(s);
     
           if (status)
             break;
     
           size = gsl_multimin_fminimizer_size (s);
           status = gsl_multimin_test_size (size, 1e-2);
     
           if (status == GSL_SUCCESS)
             {
               printf ("converged to minimum at\n");
             }
     
           printf ("%5d ", iter);
           for (i = 0; i < np; i++)
             {
               printf ("%10.3e ", gsl_vector_get (s->x, i));
             }
           printf ("f() = %7.3f size = %.3f\n", s->fval, size);
         }
       while (status == GSL_CONTINUE && iter < 100);
     
       gsl_vector_free(x);
       gsl_vector_free(ss);
       gsl_multimin_fminimizer_free (s);
     
       return status;
     }
     

The minimum search stops when the Simplex size drops to 0.01. The output is shown below.

         1  6.500e+00  5.000e+00 f() = 512.500 size = 1.082
         2  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
         3  5.250e+00  4.000e+00 f() = 290.625 size = 1.372
         4  5.500e+00  1.000e+00 f() = 252.500 size = 1.372
         5  2.625e+00  3.500e+00 f() = 101.406 size = 1.823
         6  3.469e+00  1.375e+00 f() = 98.760  size = 1.526
         7  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
         8  1.820e+00  3.156e+00 f() = 63.467  size = 1.105
         9  1.016e+00  2.812e+00 f() = 43.206  size = 1.105
        10  2.041e+00  2.008e+00 f() = 40.838  size = 0.645
        11  1.236e+00  1.664e+00 f() = 32.816  size = 0.645
        12  1.236e+00  1.664e+00 f() = 32.816  size = 0.447
        13  5.225e-01  1.980e+00 f() = 32.288  size = 0.447
        14  1.103e+00  2.073e+00 f() = 30.214  size = 0.345
        15  1.103e+00  2.073e+00 f() = 30.214  size = 0.264
        16  1.103e+00  2.073e+00 f() = 30.214  size = 0.160
        17  9.864e-01  1.934e+00 f() = 30.090  size = 0.132
        18  9.190e-01  1.987e+00 f() = 30.069  size = 0.092
        19  1.028e+00  2.017e+00 f() = 30.013  size = 0.056
        20  1.028e+00  2.017e+00 f() = 30.013  size = 0.046
        21  1.028e+00  2.017e+00 f() = 30.013  size = 0.033
        22  9.874e-01  1.985e+00 f() = 30.006  size = 0.028
        23  9.846e-01  1.995e+00 f() = 30.003  size = 0.023
        24  1.007e+00  2.003e+00 f() = 30.001  size = 0.012
     converged to minimum at
        25  1.007e+00  2.003e+00 f() = 30.001  size = 0.010
     

The simplex size first increases, while the simplex moves towards the minimum. After a while the size begins to decrease as the simplex contracts around the minimum.