Affiliations: [a] School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail: [email protected] | [b] Computer Science Department, Technion - Israel Institute of Technology, Haifa 32000, Israel. E-mail: [email protected]
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.
