Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-10T20:07:50.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Extrapolation of sequences using a generalized epsilon-algorithm

Published online by Cambridge University Press:  17 February 2009

Michael N. Barber  and
C. J. Hamer
Michael N. Barber
Affiliation:
Department of Applied Mathematics, University of New South Wales, P. O. Box 1, Kensington, N.S.W. 2033
C. J. Hamer
Affiliation:
Department of Theoretical Physics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 2600
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The problem of estimating the limit f∞ of a sequence fn converging as fn − f∞ = O(n−λ) as n → ∞, where λ > 0, is discussed. Using the generalization of the ε-algorithm proposed recently by Vanden Broeck and Schwartz, an acceleration scheme is developed. The method is illustrated on several test sequences and compared to other acceleration procedures.

Type
Research Article
Information
The ANZIAM Journal , Volume 23 , Issue 3 , January 1982 , pp. 229 - 240
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Brezinski, C., “Accélération de suites à convergence logarithmique”, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A727–A730.Google Scholar
[2]Brezinski, C., Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol. 584 (Springer-Verlag, Berlin, 1977).CrossRefGoogle Scholar
[3]Cordellier, F., “Caracterisation des suites que la première étape du θ-algorithme transformé en suites constants”, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), A389–A392.Google Scholar
[4]Drummond, J. E., “Summing a common type of slowly convergent series of positive terms”, J. Austral. Math. Soc. Ser. B 19 (1976), 416–421.CrossRefGoogle Scholar
[5]Hamer, C. J. and Barber, M. N., “Finite-lattice extrapolations for Z3 and Z5 models”, J. Phys. A 14 (1981), 2009–2025.Google Scholar
[6]Levin, D., “Development of non-linear transformations for improving the convergence of sequences”, Internat. J. Comput. Math. Ser. B 3 (1973), 371–388.CrossRefGoogle Scholar
[7]Lubkin, S., “A method of summing infinite series”, J. Res. Nat. Bur. Standards 48 (1952), 228–254.CrossRefGoogle Scholar
[8]Nienhuis, B., Riedel, E. K. and Schick, M., “Magnetic exponents of the two-dimensional q-state Potts model”, J. Phys. A 13 (1980), L189–L192.Google Scholar
[9]Ralston, A., A first course in numerical analysis (McGraw-Hill, New York, 1965).Google Scholar
[10]Shanks, D., “Non-linear transformations of divergent and slowly convergent sequences”, J. Math. and Phys. 34 (1955), 1–42.CrossRefGoogle Scholar
[11]Smith, D. A. and Ford, W. F., “Acceleration of linear and logarithmic convergence”, SIAM J. Numer. Anal. 16 (1979), 223–240.CrossRefGoogle Scholar
[12]Broeck, J.-M. Vanden and Schwartz, L. W., “A one-parameter family of sequence transformations”, SIAM J. Math. Anal. 10 (1979), 658–666.CrossRefGoogle Scholar
[13]Wynn, P., “Upon systems of recursions which obtain among the quotients of the Padé table”, Numer. Math. 8 (1966), 264–269.CrossRefGoogle Scholar