On accelerating the convergence of infinite double series and integrals
Author:
David Levin
Journal:
Math. Comp. 35 (1980), 13311345
MSC:
Primary 65B10; Secondary 65D15
DOI:
https://doi.org/10.1090/S00255718198005835113
MathSciNet review:
583511
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Abstract  References  Similar Articles  Additional Information
Abstract: The generalization of Shanks’ etransformation to double series is discussed and a class of nonlinear transformations, the ${[A/S]_R}$ transformations, for accelerating the convergence of infinite double series is presented. It is constructed so as to sum exactly infinite double series whose terms satisfy certain finite linear double difference equations; in that sense it is a generalization of Shanks’ etransformation or its equivalent Wynn’s $\varepsilon$algorithm. A generalization of the ${[A/S]_R}$ transformation to Ndimensional series is also presented and their application to power series is discussed and exemplified. Some transformations for accelerating the convergence of infinite double integrals are also obtained, generalizing the confluent $\varepsilon$algorithm of Wynn and the Gtransformation of Gray, Atchison, and McWilliams for infinite 1D integrals.

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Article copyright:
© Copyright 1980
American Mathematical Society