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.

\[\label{eq:levin} \function{levin_sum}\code{(a)} = (A, \epsilon) \qquad\text{where} \qquad A \approx \sum_{n=0}^{\infty} a_{n} \pm \epsilon,\]

\(\code{a} = [a_{0}, a_{1}, \ldots, a_{n}]\), 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 for each term (as defined in the GNU Scientific Library source code).

Function list

levin_suma, truncate=False, info_dict=None Return (\(A, \epsilon\)) where \(A\) is the approximated sum of the series ([eq:levin]) and \(\epsilon\) is its absolute error estimate.

The calculation of the error in the extrapolated value is an O\((N^2)\) process, which is expensive in time and memory. A full table of intermediate values and derivatives through to O\((N)\) must be computed and stored, but this does give a reliable error estimate.

A faster but less reliable method which estimates the error from the convergence of the extrapolated value is employed if is . This attempts to estimate the error from the “truncation error” in the extrapolation, the difference between the final two approximations. Using this method avoids the need to compute an intermediate table of derivatives because the error is estimated from the behavior of the extrapolated value itself. Consequently this algorithm is an O\((N)\) process and only requires O\((N)\) terms of storage. If the series converges sufficiently fast then this procedure can be acceptable. It is appropriate to use this method when there is a need to compute many extrapolations of series with similar convergence properties at high-speed. For example, when numerically integrating a function defined by a parameterized series where the parameter varies only slightly. A reliable error estimate should be computed first using the full algorithm described above in order to verify the consistency of the results.

If a dictionary is passed as , then two entries will be added: will be the number of terms used [1]_ and will be the sum of these terms without acceleration.

Further Reading

For details on the underlying implementation of these functions please consult the GNU Scientific Library reference manual. The algorithms used by these functions are described Fessler et al. (1983). The theory of the \(u\)-transform was presented Levin in 1973, and a review paper from 2000 by Homeier is available online.