Acceleration of Slowly Convergent Series via the Generalized Weighted-Averages Method

Abstract

Abstract-A generalized version of the weighted-averages method is presented for the acceleration of convergence of sequences and series over a wide range of test problems, including linearly and logarithmically convergent series as well as monotone and alternating series.This method was originally developed in a partitionextrapolation procedure for accelerating the convergence of semiinfinite range integrals with Bessel function kernels (Sommerfeld-type integrals), which arise in computational electromagnetics problems involving scattering/radiation in planar stratified media. In this paper, the generalized weighted-averages method is obtained by incorporating the optimal remainder estimates already available in the literature. Numerical results certify its comparable and in many cases superior performance against not only the traditional weighted-averages method but also against the most proven extrapolation methods often used to speed up the computation of slowly convergent series.

A linear transformation to accelerate the convergence of the negative binomial series

“…Eq. 5) and the fact that neither complex algorithms nor any estimate of the residuals have been necessary, as instead it is the case when using other types of transformations [2][3][4].…”

“…Although nonlinear transformations are in general expected to be more effective than linear transformations, the latter have the advantage of being easier to apply and can, in some cases, be very effective too [3][4][5]. Accordingly, in this work a linear sequence transformation is introduced to improve the convergence properties of the binomial series when r < 0 and x and y have the same sign.…”

Weighted Averages and Double Exponential Algorithms

“…The decisive step here was to consider the implied integrals as defined in the Abel's sense and to apply to them the well known Cèsaro and Hölder means [8], [9]. In addition to Sommerfeld tails [10], generalized WA algorithms can cope now with divergent series [11] and with much more complex integrals, like those involving products of Bessel functions [12]. Thus, the transmutation of WA from a very specific tool for a particular electromagnetic problem to a generic numerical algorithm is being achieved.…”