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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The generalization of Shanks' *e*-transformation to double series is discussed and a class of nonlinear transformations, the 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 -algorithm. A generalization of the 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 -algorithm of Wynn and the *G*-transformation of Gray, Atchison, and McWilliams for infinite 1-D integrals.

**[1]**N. K. BOSE & S. BASU, "2-D matrix Padé approximants: existence, non-uniqueness and recursive computation,"*IEEE Trans. Automat. Control.*(To appear.)**[2]**J. S. R. Chisholm,*Rational approximants defined from double power series*, Math. Comp.**27**(1973), 841–848. MR**0382928**, 10.1090/S0025-5718-1973-0382928-6**[3]**P. R. Graves-Morris, R. Hughes Jones, and G. J. Makinson,*The calculation of some rational approximants in two variables*, J. Inst. Math. Appl.**13**(1974), 311–320. MR**0375739****[4]**H. L. Gray, T. A. Atchison, and G. V. McWilliams,*Higher order 𝐺-transformations*, SIAM J. Numer. Anal.**8**(1971), 365–381. MR**0288933****[5]**R. Hughes Jones and G. J. Makinson,*The generation of Chisholm rational polynomial approximants to power series in two variables*, J. Inst. Math. Appl.**13**(1974), 299–310. MR**0375738****[6]**D. Levin,*General order Padé-type rational approximants defined from double power series*, J. Inst. Math. Appl.**18**(1976), no. 1, 1–8. MR**0433087****[7]**David Levin and Avram Sidi,*Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series*, Appl. Math. Comput.**9**(1981), no. 3, 175–215. MR**650681**, 10.1016/0096-3003(81)90028-X**[8]**Daniel Shanks,*Non-linear transformations of divergent and slowly convergent sequences*, J. Math. and Phys.**34**(1955), 1–42. MR**0068901****[9]**P. Wynn,*On a device for computing the 𝑒_{𝑚}(𝑆_{𝑛}) tranformation*, Math. Tables Aids Comput.**10**(1956), 91–96. MR**0084056**, 10.1090/S0025-5718-1956-0084056-6**[10]**P. Wynn,*Upon a second confluent form of the 𝜖-algorithm*, Proc. Glasgow Math. Assoc.**5**(1962), 160–165 (1962). MR**0139253**

Retrieve articles in *Mathematics of Computation*
with MSC:
65B10,
65D15

Retrieve articles in all journals with MSC: 65B10, 65D15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0583511-3

Article copyright:
© Copyright 1980
American Mathematical Society