## Example of accelerating a series

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.

    bash$ ./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.