A convergence and stability study of the iterated Lubkin transformation and the -algorithm

Author:
Avram Sidi

Journal:
Math. Comp. **72** (2003), 419-433

MSC (2000):
Primary 65B05, 65B10, 40A05, 40A25, 41A60

Published electronically:
May 1, 2002

MathSciNet review:
1933829

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we analyze the convergence and stability of the iterated Lubkin transformation and the -algorithm as these are being applied to sequences whose members behave like as , where and are complex scalars and is a nonnegative integer. We study the three different cases in which (i) , , and (logarithmic sequences), (ii) and (linear sequences), and (iii) (factorial sequences). We show that both methods accelerate the convergence of all three types of sequences. We show also that both methods are stable on linear and factorial sequences, and they are unstable on logarithmic sequences. On the basis of this analysis we propose ways of improving accuracy and stability in problematic cases. Finally, we provide a comparison of these results with analogous results corresponding to the Levin -transformation.

**1.**Siddhartha Bhowmick, Ranjan Bhattacharya, and Dhiranjan Roy,*Iterations of convergence accelerating nonlinear transforms*, Comput. Phys. Comm.**54**(1989), no. 1, 31–46. MR**996929**, 10.1016/0010-4655(89)90030-1**2.**Claude Brezinski,*Accélération de suites à convergence logarithmique*, C. R. Acad. Sci. Paris Sér. A-B**273**(1971), A727–A730 (French). MR**0305544****3.**J. E. Drummond,*Summing a common type of slowly convergent series of positive terms*, J. Austral. Math. Soc. Ser. B**19**(1975/76), no. 4, 416–421. MR**0478540****4.**Walter B. Ford,*Studies on divergent series and summability & The asymptotic developments of functions defined by Maclaurin series*, Chelsea Publishing Co., New York, 1960. MR**0115035****5.**David Levin,*Development of non-linear transformations of improving convergence of sequences*, Internat. J. Comput. Math.**3**(1973), 371–388. MR**0359261****6.**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**7.**Samuel Lubkin,*A method of summing infinite series*, J. Research Nat. Bur. Standards**48**(1952), 228–254. MR**0051576****8.**Paul Sablonnière,*Asymptotic behaviour of iterated modified Δ² and 𝜃₂ transforms on some slowly convergent sequences*, Numer. Algorithms**3**(1992), no. 1-4, 401–409. Extrapolation and rational approximation (Puerto de la Cruz, 1992). MR**1199386**, 10.1007/BF02141947**9.**Daniel Shanks,*Non-linear transformations of divergent and slowly convergent sequences*, J. Math. and Phys.**34**(1955), 1–42. MR**0068901****10.**Avram Sidi,*Convergence properties of some nonlinear sequence transformations*, Math. Comp.**33**(1979), no. 145, 315–326. MR**514827**, 10.1090/S0025-5718-1979-0514827-6**11.**Avram Sidi,*Analysis of convergence of the 𝑇-transformation for power series*, Math. Comp.**35**(1980), no. 151, 833–850. MR**572860**, 10.1090/S0025-5718-1980-0572860-0**12.**Avram Sidi,*Acceleration of convergence of (generalized) Fourier series by the 𝑑-transformation*, Ann. Numer. Math.**2**(1995), no. 1-4, 381–406. Special functions (Torino, 1993). MR**1343544****13.**Avram Sidi,*Convergence analysis for a generalized Richardson extrapolation process with an application to the 𝑑⁽¹⁾-transformation on convergent and divergent logarithmic sequences*, Math. Comp.**64**(1995), no. 212, 1627–1657. MR**1312099**, 10.1090/S0025-5718-1995-1312099-5**14.**Avram Sidi,*The Richardson extrapolation process with a harmonic sequence of collocation points*, SIAM J. Numer. Anal.**37**(2000), no. 5, 1729–1746 (electronic). MR**1759914**, 10.1137/S0036142998340137**15.**David A. Smith and William F. Ford,*Acceleration of linear and logarithmic convergence*, SIAM J. Numer. Anal.**16**(1979), no. 2, 223–240. MR**526486**, 10.1137/0716017**16.**V. A. Pesoshin, V. M. Tarasov, and O. I. Dapin,*Generation of random numbers with a given law of distribution*, Voprosy Kibernet. (Moscow)**82**(1981), 157–162 (Russian). Probabilistic computational tools and methods. MR**650464****17.**Andrew H. Van Tuyl,*Acceleration of convergence of a family of logarithmically convergent sequences*, Math. Comp.**63**(1994), no. 207, 229–246. MR**1234428**, 10.1090/S0025-5718-1994-1234428-2**18.**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

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65B05,
65B10,
40A05,
40A25,
41A60

Retrieve articles in all journals with MSC (2000): 65B05, 65B10, 40A05, 40A25, 41A60

Additional Information

**Avram Sidi**

Affiliation:
Computer Science Department, Technion - Israel Institute of Technology, Haifa 32000, Israel

Email:
asidi@cs.technion.ac.il

DOI:
https://doi.org/10.1090/S0025-5718-02-01433-3

Received by editor(s):
March 21, 2001

Published electronically:
May 1, 2002

Article copyright:
© Copyright 2002
American Mathematical Society