On accelerating the convergence of infinite double series and integrals
Author:
David Levin
Journal:
Math. Comp. 35 (1980), 13311345
MSC:
Primary 65B10; Secondary 65D15
MathSciNet review:
583511
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Abstract: The generalization of Shanks' etransformation 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' etransformation or its equivalent Wynn's algorithm. A generalization of the 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 algorithm of Wynn and the Gtransformation of Gray, Atchison, and McWilliams for infinite 1D integrals.
 [1]
N. K. BOSE & S. BASU, "2D matrix Padé approximants: existence, nonuniqueness 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
(52 #3810), http://dx.doi.org/10.1090/S00255718197303829286
 [3]
P.
R. GravesMorris, 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
(51 #11929)
 [4]
H.
L. Gray, T.
A. Atchison, and G.
V. McWilliams, Higher order 𝐺transformations, SIAM J.
Numer. Anal. 8 (1971), 365–381. MR 0288933
(44 #6128)
 [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
(51 #11928)
 [6]
D.
Levin, General order Padétype rational approximants defined
from double power series, J. Inst. Math. Appl. 18
(1976), no. 1, 1–8. MR 0433087
(55 #6066)
 [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
(83d:65010), http://dx.doi.org/10.1016/00963003(81)90028X
 [8]
Daniel
Shanks, Nonlinear transformations of divergent and slowly
convergent sequences, J. Math. and Phys. 34 (1955),
1–42. MR
0068901 (16,961e)
 [9]
P.
Wynn, On a device for computing the
𝑒_{𝑚}(𝑆_{𝑛}) tranformation, Math. Tables Aids Comput. 10 (1956), 91–96. MR 0084056
(18,801e), http://dx.doi.org/10.1090/S00255718195600840566
 [10]
P.
Wynn, Upon a second confluent form of the 𝜖algorithm,
Proc. Glasgow Math. Assoc. 5 (1962), 160–165 (1962).
MR
0139253 (25 #2689)
 [1]
 N. K. BOSE & S. BASU, "2D matrix Padé approximants: existence, nonuniqueness and recursive computation," IEEE Trans. Automat. Control. (To appear.)
 [2]
 J. S. R. CHISHOLM, "Rational approximants defined from double power series," Math. Comp., v. 27, 1973, pp. 841848. MR 0382928 (52:3810)
 [3]
 P. R. GRAVESMORRIS, R. HUGHESJONES & G. J. MAKINSON, "The calculation of some rational approximants in two variables," J. Inst. Math. Appl., v. 13, 1974, pp. 311320. MR 0375739 (51:11929)
 [4]
 H. L. GRAY, T. A. ATCHISON & G. V. McWILLIAMS, "Higher order Gtransformations," SIAM J. Numer. Anal., v. 8, 1971, pp. 365381. MR 0288933 (44:6128)
 [5]
 R. HUGHESJONES & G. J. MAKINSON, "The generation of Chisholm rational polynomial approximants to power series in two variables," J. Inst. Math. Appl., v. 13, 1974, pp. 299310. MR 0375738 (51:11928)
 [6]
 D. LEVIN, "General order Padétype rational approximants defined from double power series," J. Inst. Math. Appl., v. 18, 1976, pp. 18. MR 0433087 (55:6066)
 [7]
 D. LEVIN & A. SIDI, "Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series," Appl. Math. and Comp. (To appear.) MR 650681 (83d:65010)
 [8]
 D. SHANKS, "Nonlinear transformations of divergent and slowly convergent sequences," J. Math. Phys., v. 34, 1955, pp. 142. MR 0068901 (16:961e)
 [9]
 P. WYNN, "On a device for computing the transformation," MTAC, v. 10, 1956, pp. 9196. MR 0084056 (18:801e)
 [10]
 P. WYNN, "Upon a second confluent form of the algorithm," Proc. Glasgow Math. Soc., v. 5, 1962, pp. 160165. MR 0139253 (25:2689)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005835113
PII:
S 00255718(1980)05835113
Article copyright:
© Copyright 1980
American Mathematical Society
