16.12 Examples

The integrator QAGS will handle a large class of definite integrals. For example, consider the following integral, which has an algebraic-logarithmic singularity at the origin,

     \int_0^1 x^{-1/2} log(x) dx = -4

The program below computes this integral to a relative accuracy bound of 1e-7.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_integration.h>
     
     double f (double x, void * params) {
       double alpha = *(double *) params;
       double f = log(alpha*x) / sqrt(x);
       return f;
     }
     
     int
     main (void)
     {
       gsl_integration_workspace * w 
         = gsl_integration_workspace_alloc (1000);
       
       double result, error;
       double expected = -4.0;
       double alpha = 1.0;
     
       gsl_function F;
       F.function = &f;
       F.params = &alpha;
     
       gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                             w, &result, &error); 
     
       printf ("result          = % .18f\n", result);
       printf ("exact result    = % .18f\n", expected);
       printf ("estimated error = % .18f\n", error);
       printf ("actual error    = % .18f\n", result - expected);
       printf ("intervals =  %d\n", w->size);
     
       gsl_integration_workspace_free (w);
     
       return 0;
     }

The results below show that the desired accuracy is achieved after 8 subdivisions.

     $ ./a.out
     result          = -3.999999999999973799
     exact result    = -4.000000000000000000
     estimated error =  0.000000000000246025
     actual error    =  0.000000000000026201
     intervals =  8

In fact, the extrapolation procedure used by QAGS produces an accuracy of almost twice as many digits. The error estimate returned by the extrapolation procedure is larger than the actual error, giving a margin of safety of one order of magnitude.