Example of accelerating a series - GNU Scientific Library -- Reference Manual
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### 30.3 Examples

The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using the series,

         \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...

After N terms the error in the sum is O(1/N), making direct summation of the series converge slowly.

         #include <stdio.h>
         #include <gsl/gsl_math.h>
         #include <gsl/gsl_sum.h>

         #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 < N; n++)
             {
               double np1 = n + 1.0;
               t[n] = 1.0 / (np1 * np1);
               sum += t[n];
             }

           gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);

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

           printf ("term-by-term sum = % .16f using %d terms\n",
                   w->sum_plain, w->terms_used);

           printf ("exact value      = % .16f\n", zeta_2);
           printf ("accelerated sum  = % .16f using %d terms\n",
                   sum_accel, w->terms_used);

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

           gsl_sum_levin_u_free (w);
           return 0;
         }

The output below shows that the Levin u-transform is able to obtain an estimate of the sum to 1 part in 10^10 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.

         $ ./a.out
         term-by-term sum =  1.5961632439130233 using 20 terms
         term-by-term sum =  1.5759958390005426 using 13 terms
         exact value      =  1.6449340668482264
         accelerated sum  =  1.6449340668166479 using 13 terms
         estimated error  =  0.0000000000508580
         actual error     = -0.0000000000315785

Note that a direct summation of this series would require 10^10 terms to achieve the same precision as the accelerated sum does in 13 terms.

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