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.IX Title "Math::GSL::Sum 3pm"
.TH Math::GSL::Sum 3pm 2024-03-07 "perl v5.38.2" "User Contributed Perl Documentation"
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.SH NAME
Math::GSL::Sum \- Sum series with the Levin u\-transform
.SH SYNOPSIS
.IX Header "SYNOPSIS"
.Vb 1
\&    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);
.Ve
.SH DESCRIPTION
.IX Header "DESCRIPTION"
These functions accelerate the convergence of a series using the Levin u\-transform.
.IP \(bu 4
gsl_sum_levin_u_alloc($n)
.Sp
This function allocates a workspace for a Levin u\-transform of \f(CW$n\fR terms.
.IP \(bu 4
gsl_sum_levin_u_free($w)
.Sp
\&\- This function frees the memory associated with the workspace \f(CW$w\fR.
.IP \(bu 4
gsl_sum_levin_u_accel($array, \f(CW$array_size\fR, \f(CW$w\fR)
.Sp
This function takes the terms of a series in the array reference \f(CW$array\fR of size
\&\f(CW$array_size\fR and computes the extrapolated limit of the series using a Levin
u\-transform. Additional working space must be provided in \f(CW$w\fR. 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 \f(CW$w\fR\->{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.
.IP \(bu 4
gsl_sum_levin_u_minmax
.IP \(bu 4
gsl_sum_levin_u_step
.IP \(bu 4
gsl_sum_levin_utrunc_alloc($n)
.Sp
This function allocates a workspace for a Levin u\-transform of \f(CW$n\fR terms,
without error estimation.
.IP \(bu 4
gsl_sum_levin_utrunc_free($w)
.Sp
This function frees the memory associated with the workspace \f(CW$w\fR.
.IP \(bu 4
gsl_sum_levin_utrunc_accel($array, \f(CW$array_size\fR, \f(CW$w\fR)
.Sp
This function takes the terms of a series in the array reference \f(CW$array\fR of size
\&\f(CW$array_size\fR and computes the extrapolated limit of the series using a Levin
u\-transform. Additional working space must be provided in \f(CW$w\fR. 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 \f(CW$w\fR\->{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.
.IP \(bu 4
gsl_sum_levin_utrunc_minmax
.IP \(bu 4
gsl_sum_levin_utrunc_step
.SH AUTHORS
.IX Header "AUTHORS"
Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>
.SH "COPYRIGHT AND LICENSE"
.IX Header "COPYRIGHT AND LICENSE"
Copyright (C) 2008\-2023 Jonathan "Duke" Leto and Thierry Moisan
.PP
This program is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.