10. pygsl.sum
Series acceleration

This chapter describes the use of the series acceleration tools based on the Levin $ u$ -transform. This method takes a small number of terms from the start of a series and uses a systematic approximation to compute an extrapolated value and an estimate of its error. The $ u$ -transform works for both convergent and divergent series, including asymptotic series.

$\displaystyle \function{levin_sum}\code{(a)} = (A, \epsilon)$   where$\displaystyle \qquad A \approx \sum_{n=0}^{\infty} a_{n} \pm \epsilon,$ (10.1)

$ \code{a} = [a_{0}, a_{1}, \ldots, a_{n}]$ , and $ \epsilon$ is an estimate of the absolute error.

Note: This function is intended for summing analytic series where each term is known to high accuracy, and the rounding errors are assumed to originate from finite precision. They are taken to be relative errors of order GSL_DBL_EPSILON for each term (as defined in the GNU Scientific Library source code).