### 31.3 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 %zu terms\n",
w->sum_plain, w->terms_used);

printf ("exact value      = % .16f\n", zeta_2);
printf ("accelerated sum  = % .16f using %zu 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  =  1.6449340669228176 using 13 terms
estimated error  =  0.0000000000888360
actual error     =  0.0000000000745912


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.