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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On accelerating the convergence of infinite double series and integrals

Author: David Levin
Journal: Math. Comp. 35 (1980), 1331-1345
MSC: Primary 65B10; Secondary 65D15
MathSciNet review: 583511
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Abstract: The generalization of Shanks’ e-transformation 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’ e-transformation or its equivalent Wynn’s $\varepsilon$-algorithm. A generalization of the ${[A/S]_R}$ transformation to N-dimensional 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 G-transformation of Gray, Atchison, and McWilliams for infinite 1-D integrals.

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Article copyright: © Copyright 1980 American Mathematical Society