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<title>GNU Scientific Library &ndash; Reference Manual: Example of accelerating a series</title>

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<a name="Example-of-accelerating-a-series"></a>
<div class="header">
<p>
Next: <a href="Series-Acceleration-References.html#Series-Acceleration-References" accesskey="n" rel="next">Series Acceleration References</a>, Previous: <a href="Acceleration-functions-without-error-estimation.html#Acceleration-functions-without-error-estimation" accesskey="p" rel="previous">Acceleration functions without error estimation</a>, Up: <a href="Series-Acceleration.html#Series-Acceleration" accesskey="u" rel="up">Series Acceleration</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Examples-22"></a>
<h3 class="section">31.3 Examples</h3>

<p>The following code calculates an estimate of <em>\zeta(2) = \pi^2 / 6</em>
using the series,
</p>
<div class="example">
<pre class="example">\zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
</pre></div>

<p>After <var>N</var> terms the error in the sum is <em>O(1/N)</em>, making direct
summation of the series converge slowly.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_sum.h&gt;

#define N 20

int
main (void)
{
  double t[N];
  double sum_accel, err;
  double sum = 0;
  int n;
  
  gsl_sum_levin_u_workspace * w 
    = gsl_sum_levin_u_alloc (N);

  const double zeta_2 = M_PI * M_PI / 6.0;
  
  /* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */

  for (n = 0; n &lt; N; n++)
    {
      double np1 = n + 1.0;
      t[n] = 1.0 / (np1 * np1);
      sum += t[n];
    }
  
  gsl_sum_levin_u_accel (t, N, w, &amp;sum_accel, &amp;err);

  printf (&quot;term-by-term sum = % .16f using %d terms\n&quot;, 
          sum, N);

  printf (&quot;term-by-term sum = % .16f using %zu terms\n&quot;, 
          w-&gt;sum_plain, w-&gt;terms_used);

  printf (&quot;exact value      = % .16f\n&quot;, zeta_2);
  printf (&quot;accelerated sum  = % .16f using %zu terms\n&quot;, 
          sum_accel, w-&gt;terms_used);

  printf (&quot;estimated error  = % .16f\n&quot;, err);
  printf (&quot;actual error     = % .16f\n&quot;, 
          sum_accel - zeta_2);

  gsl_sum_levin_u_free (w);
  return 0;
}
</pre></div>

<p>The output below shows that the Levin <em>u</em>-transform is able to obtain an 
estimate of the sum to 1 part in 
<em>10^10</em> using the first eleven terms of the series.  The
error estimate returned by the function is also accurate, giving
the correct number of significant digits. 
</p>
<div class="example">
<pre class="example">$ ./a.out 
</pre><pre class="verbatim">term-by-term sum =  1.5961632439130233 using 20 terms
term-by-term sum =  1.5759958390005426 using 13 terms
exact value      =  1.6449340668482264
accelerated sum  =  1.6449340669228176 using 13 terms
estimated error  =  0.0000000000888360
actual error     =  0.0000000000745912
</pre></div>

<p>Note that a direct summation of this series would require 
<em>10^10</em> terms to achieve the same precision as the accelerated 
sum does in 13 terms.
</p>
<hr>
<div class="header">
<p>
Next: <a href="Series-Acceleration-References.html#Series-Acceleration-References" accesskey="n" rel="next">Series Acceleration References</a>, Previous: <a href="Acceleration-functions-without-error-estimation.html#Acceleration-functions-without-error-estimation" accesskey="p" rel="previous">Acceleration functions without error estimation</a>, Up: <a href="Series-Acceleration.html#Series-Acceleration" accesskey="u" rel="up">Series Acceleration</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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