The Weighted Averages Algorithm Revisited

Abstract

The classic weighted averages (WA) algorithm for the evaluation of Sommerfeld-like integrals is reviewed and reappraised. As a result, a new version of the WA algorithm, called generalized WA, is introduced. The new version can be considered as a generalization of the well established Hölder and Cèsaro means, used to sum divergent series. Generalized WA exhibits a more compact formulation, devoid of iterative and recursive steps, and a wider range of applications. It is more robust, as it provides a unique formulation, valid for monotonic and oscillating functions. The implementation of the new version is easier and more economical in terms of basic operations. Preliminary numerical examples show that generalized WA also outperforms in terms of accuracy the classic WA algorithm, which is currently recognized as the most competitive algorithm to evaluate Sommerfeld integral tails.

Near-Field Time-Domain Shielding Effectiveness of Thin Conductive Screens

“…To accelerate the calculation of the possibly oscillatory integrals (3) and (4), several methods can be applied [14].…”

Efficient Algorithms for Computing Sommerfeld Integral Tails

“…In the original version of the WA method [19], the remainders are expanded into infinite series using integration by parts, which is truncated for numerical purposes (18) where (19) The weights obtained in this way, although different, are asymptotically equivalent to those in (15). In the new WA method [22], the same remainder expansion is used. But now, instead of acting always on two consecutive partial integrals , the new WA method acts simultaneously on members of the sequence in (7).…”

“…But now, instead of acting always on two consecutive partial integrals , the new WA method acts simultaneously on members of the sequence in (7). Indeed, the system of equations obtained by writing (18) for different values of is solved using Cramer's rule, yielding after some algebraic manipulations to the following final expression for the best possible estimation that can be found, out of the , according the new WA method [22] (20)…”

“…For a long time it was considered inefficient for practical purposes, since it calls for computation of the determinants, which is usually a cumbersome and time-consuming task. Then in [22] it was noticed that those determinants, in the case of SIs, are indeed Vandermonde's type determinants, for which an analytical solution exists.…”

“…The method has been proven to be one of the most efficient extrapolation methods for the convergence acceleration of the sequences obtained when the integration-then-summation technique is applied to SIs. In this paper we use the classic Mosig-Michalski version but also a very interesting and recently introduced new version of WA [22].…”