- publishing free software manuals
 GNU Scientific Library Reference Manual - Third Edition (v1.12) by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. RossiPaperback (6"x9"), 592 pages, 60 figuresISBN 0954612078RRP £24.95 ($39.95) ## 29.3 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.

 ISBN 0954612078 GNU Scientific Library Reference Manual - Third Edition (v1.12) See the print edition