The -transformation for infinite double series

and the -transformation for

infinite double integrals

Authors:
Chen Greif and David Levin

Journal:
Math. Comp. **67** (1998), 695-714

MSC (1991):
Primary 65B10; Secondary 40B05, 65D30

DOI:
https://doi.org/10.1090/S0025-5718-98-00955-7

MathSciNet review:
1464144

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate - and -transformations. The -transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the -transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given.

**1.**J. S. R. Chisholm, 1973*Rational approximants defined from double power series*, Math. Comp.,**27**841-848. MR**52:3810****2.**A. Cuyt, 1984*Padé Approximants for operators: Theory and Applications*, Lecture Notes in Mathematics, Vol. 1065, Springer, Berlin. MR**86c:41010****3.**-, 1986*Multivariate Padé approximants revisited*, BIT,**26**71-79. MR**87f:41031****4.**-, 1990*A multivariate convergence theorem of the ``de Montessus de Ballore type'' to multipoles*, J. Comp. Appl. Math.,**32**, 47-57. MR**93h:65012****5.**C. Greif, 1994*Singularity detection and bivariate generalization of the and transformations*, M.Sc. Thesis, Tel Aviv Univ.**6.**D. Levin, 1971*Development of non-linear transformations of series and sequences to increase rate and domain of convergence, and their use for computing results from formal solutions to applied math. problems*, M.Sc. Thesis, Tel Aviv Univ.**7.**-, 1973*Development of non-linear transformations for improving convergence of sequences*, Internat. J. Comput. Math.**B3**371-388. MR**50:11716****8.**-, 1975*Methods for accelerating convergence of infinite series and integrals*, Ph.D. Thesis, Tel Aviv Univ.**9.**-, 1976*General order Padé-type rational approximants defined from double power series*J. Inst. Maths Applics.**18**1-8. MR**55:6066****10.**-, 1980*On accelerating the convergence of infinite double series and integrals*, Maths. Comput.**35**, 1331-1345. MR**82b:65003****11.**D. Levin and A. Sidi, 1981*Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series*, Appl. Math. Comput.**9**, 175-215. MR**83d:65010****12.**-, 1982*Rational approximations from the -transformation*, IMA J. Numer. Anal.**2**153-167. MR**83j:65012****13.**A. Sidi, 1977*Exponential function approximation to Laplace transform inversion and development of non-linear methods for accelerating the convergence of infinite integrals and series*, Ph.D. Thesis, Tel Aviv Univ.**14.**-, 1979*Convergence properties of some non-linear sequence transformations*, Maths. Comput.**33**, 315-326. MR**81h:65003****15.**-1979*Some properties of a generalization of the Richardson extrapolation process*, J. Inst. Maths Applics**24**, 327-346. MR**81a:65011****16.**-1980*Analysis of convergence of the -transformation for power series*, Maths. Comput.**35**, 833-850. MR**83d:41039****17.**-, 1995*Convergence analysis for a generalized Richardson extrapolation process with an application to the -transformation on convergent and divergent logarithmic sequences*, Maths. Comput.**64**, no. 212, 1627-1657. MR**96a:65009****18.**P. Wynn, 1956*On a device for computing the transformation*, Math. Tables Aids Comput.**10**91-96. MR**18:801e**

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Additional Information

**Chen Greif**

Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Address at time of publication:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T-1Z2

Email:
greif@math.ubc.ca

**David Levin**

Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Email:
levin@math.tau.ac.il

DOI:
https://doi.org/10.1090/S0025-5718-98-00955-7

Received by editor(s):
November 21, 1995

Received by editor(s) in revised form:
July 19, 1996, and January 8, 1997

Article copyright:
© Copyright 1998
American Mathematical Society