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<h3 class="section">16.12 Examples</h3>
<p>The integrator <code>QAGS</code> will handle a large class of definite
integrals. For example, consider the following integral, which has a
algebraic-logarithmic singularity at the origin,
<pre class="example"> \int_0^1 x^{-1/2} log(x) dx = -4
</pre>
<p class="noindent">The program below computes this integral to a relative accuracy bound of
<code>1e-7</code>.
<pre class="example"><pre class="verbatim"> #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;
}
</pre></pre>
<p class="noindent">The results below show that the desired accuracy is achieved after 8
subdivisions.
<pre class="example"> $ ./a.out
<pre class="verbatim"> result = -3.999999999999973799
exact result = -4.000000000000000000
estimated error = 0.000000000000246025
actual error = 0.000000000000026201
intervals = 8
</pre></pre>
<p class="noindent">In fact, the extrapolation procedure used by <code>QAGS</code> 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.
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