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# Manual Reference Pages  -  MATH::GSL::SUM (3)

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### NAME

Math::GSL::Sum - Sum series with the Levin u-transform

### SYNOPSIS

```

use Math::GSL::Sum qw/:all/;

my \$w = gsl_sum_levin_u_alloc(5);
\$values = [8,2,3,4,6];
my (\$status, \$sum_accel, \$abserr) = gsl_sum_levin_u_accel(\$values, 5, \$w);
gsl_sum_levin_u_free(\$w);

my \$w2 = gsl_sum_levin_utrunc_alloc(5);
my (\$status2, \$sum_accel2, \$abserr_trunc) = gsl_sum_levin_utrunc_accel(\$values, 5, \$w2);
gsl_sum_levin_utrunc_free(\$w);

```

### DESCRIPTION

These functions accelerate the convergence of a series using the Levin u-transform.
o gsl_sum_levin_u_alloc(\$n)

This function allocates a workspace for a Levin u-transform of \$n terms.

o gsl_sum_levin_u_free(\$w)

- This function frees the memory associated with the workspace \$w.

o gsl_sum_levin_u_accel(\$array, \$array_size, \$w)

This function takes the terms of a series in the array reference \$array of size \$array_size and computes the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in \$w. The function returns multiple values in this order : 0 if the operation succeeded, 1 otherwise, the extrapolated sum and an estimate of the absolute error. The actual term-by-term sum is returned in \$w->{sum_plain}. The algorithm calculates the truncation error (the difference between two successive extrapolations) and round-off error (propagated from the individual terms) to choose an optimal number of terms for the extrapolation. All the terms of the series passed in through array should be non-zero.

o gsl_sum_levin_u_minmax
o gsl_sum_levin_u_step
o gsl_sum_levin_utrunc_alloc(\$n)

This function allocates a workspace for a Levin u-transform of \$n terms, without error estimation.

o gsl_sum_levin_utrunc_free(\$w)

This function frees the memory associated with the workspace \$w.

o gsl_sum_levin_utrunc_accel(\$array, \$array_size, \$w)

This function takes the terms of a series in the array reference \$array of size \$array_size and computes the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in \$w. The function returns multiple values in this order : 0 if the operation succeeded, 1 otherwise, the extrapolated sum and an estimate of the error. The actual term-by-term sum is returned in \$w->{sum_plain}. The algorithm terminates when the difference between two successive extrapolations reaches a minimum or is sufficiently small. To improve the reliability of the algorithm the extrapolated values are replaced by moving averages when calculating the truncation error, smoothing out any fluctuations.

o gsl_sum_levin_utrunc_minmax
o gsl_sum_levin_utrunc_step

### AUTHORS

Jonathan Duke Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>