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<a name="Numerical-integration-examples"></a>
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Next: <a href="Numerical-integration-References-and-Further-Reading.html#Numerical-integration-References-and-Further-Reading" accesskey="n" rel="next">Numerical integration References and Further Reading</a>, Previous: <a href="Numerical-integration-error-codes.html#Numerical-integration-error-codes" accesskey="p" rel="previous">Numerical integration error codes</a>, Up: <a href="Numerical-Integration.html#Numerical-Integration" accesskey="u" rel="up">Numerical Integration</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Examples-10"></a>
<h3 class="section">17.14 Examples</h3>

<p>The integrator <code>QAGS</code> will handle a large class of definite
integrals.  For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,
</p>
<div class="example">
<pre class="example">\int_0^1 x^{-1/2} log(x) dx = -4
</pre></div>

<p>The program below computes this integral to a relative accuracy bound of
<code>1e-7</code>.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;math.h&gt;
#include &lt;gsl/gsl_integration.h&gt;

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 = &amp;f;
  F.params = &amp;alpha;

  gsl_integration_qags (&amp;F, 0, 1, 0, 1e-7, 1000,
                        w, &amp;result, &amp;error); 

  printf (&quot;result          = % .18f\n&quot;, result);
  printf (&quot;exact result    = % .18f\n&quot;, expected);
  printf (&quot;estimated error = % .18f\n&quot;, error);
  printf (&quot;actual error    = % .18f\n&quot;, result - expected);
  printf (&quot;intervals       = %zu\n&quot;, w-&gt;size);

  gsl_integration_workspace_free (w);

  return 0;
}
</pre></div>

<p>The results below show that the desired accuracy is achieved after 8
subdivisions. 
</p>
<div class="example">
<pre class="example">$ ./a.out 
</pre><pre class="verbatim">result          = -4.000000000000085265
exact result    = -4.000000000000000000
estimated error =  0.000000000000135447
actual error    = -0.000000000000085265
intervals       = 8
</pre></div>

<p>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.
</p>




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