Extrapolation Methods for Infinite Multiple Series and Integrals

Abstract

Methods for the numerical evaluation of multi-dimensional infinite-range integrals and infinite series are reviewed. Particular emphasis is put on those methods that are based on the generalized Richardson extrapolation (GREP) of Sidi and the Levin-Sidi D- and d-transformations for one-dimensional integrals and series respectively. After summarizing in detail the essentials of the D- and d-transformations, the paper first discusses an approach due to Sidi that makes sequential use of the D- and d-transformations in multidimensional problems. Next a multi-dimensional version of GREP is introduced and recent generalizations of the D- and d-transformations that are due to Greif and Levin are discussed. All these are based on a careful analysis of the asymptotic expansions of suitably defined remainders.