Advances in Computational Mathematics

, Volume 2, Issue 4, pp 461–477 | Cite as

A general extrapolation procedure revisited

  • C. Brezinski
  • M. Redivo-Zaglia
Article

Abstract

TheE-algorithm is the most general extrapolation algorithm actually known. The aim of this paper is to provide a new approach to this algorithm. This approach gives a deeper insight into theE-algorithm, and allows one to obtain new properties and to relate it to other algorithms. Some extensions of the procedure are discussed.

Keywords

Convergence acceleration extrapolation 

AMS(MOS) subject classification

65B05 

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References

  1. [1]
    B. Beckermann, A connection between theE-algorithm and the epsilon-algorithm, in:Numerical and Applied Mathematics, ed. C. Brezinski (Baltzer, Basel, 1989) pp. 443–446.Google Scholar
  2. [2]
    C. Brezinski, A general extrapolation algorithm, Numer. Math. 35(1980)175–187.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Brezinski, The Mühlbach-Neville-Aitken algorithm and some extensions, BIT 20(1980) 444–451.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    C. Brezinski, A survey of iterative extrapolation by theE-algorithm, Det Kong. Norske Vid. Selsk. skr. 2(1989)1–26.MATHGoogle Scholar
  5. [5]
    C. Brezinski, Algebraic properties of theE-transformation, in:Numerical Analysis and Mathematical Modelling, Banach Center Publications, Vol. 24 (PWN, Warsaw, 1990) pp. 85–90.Google Scholar
  6. [6]
    C. Brezinski and A.C. Matos, A derivation of extrapolation algorithms based on error estimates, J. Comput. Appl. Math., to appear.Google Scholar
  7. [7]
    C. Brezinski and M. Redivo-Zaglia,Extrapolation Methods. Theory and Practice (North-Holland, Amsterdam, 1991).MATHGoogle Scholar
  8. [8]
    C. Brezinski and A. Salam, Matrix and vector sequence transformations revisited, submitted.Google Scholar
  9. [9]
    C. Brezinski and G. Walz, Sequences of transformations and triangular recursion schemes with applications in numerical analysis, J. Comput. Appl. Math. 34(1991)361–383.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    C. Carstensen, On a general epsilon algorithm, in:Numerical and Applied Mathematics, ed. C. Brezinski (Baltzer, Basel, 1989) pp. 437–441.Google Scholar
  11. [11]
    W.F. Ford and A. Sidi, An algorithm for a generalization of the Richardson extrapolation process, SIAM J. Numer. Anal. 24(1987)1212–1232.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    T. Håvie, Generalized Neville type extrapolation schemes, BIT 19(1979)204–213.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    A.C. Matos and M. Prévost, Acceleration property of theE-algorithm, Numer. Algor. 2(1992) 393–408.MATHCrossRefGoogle Scholar
  14. [14]
    G. Meinardus and G.D. Taylor, Lower estimates for the error of best uniform approximation, J. Approx. Theory 16(1976)150–161.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    G. Mühlbach, A recurrence formula for generalized divided differences and some applications, J. Approx. Theory 9(1973)165–172.MATHCrossRefGoogle Scholar
  16. [16]
    G. Mühlbach, Newton- und Hermite-Interpolation mit Čebyšev-Systemen, Z. Angew. Math. Mech. 54(1974)541–550.MATHMathSciNetGoogle Scholar
  17. [17]
    G. Mühlbach, Neville-Aitken algorithms for interpolation by functions of Čebyšev-systems in the sense of Newton and in a generalized sense of Hermite, in:Theory of Approximation, with Applications, ed. A.G. Law and B.N. Sahney(Academic Press, New York, 1967) pp. 200–212.Google Scholar
  18. [18]
    G. Mühlbach, The general Neville-Aitken algorithm and some applications, Numer. Math. 31(1978)97–110.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    G. Mühlbach, The general recurrence relation for divided differences and the general Newton-interpolation-algorithm with applications to trigonometric interpolation, Numer. Math. 32(1979) 393–408.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Prévost, Acceleration property for theE-algorithm and an application to the summation of series, Adv. Comput. Math. 2(1994)319–341.MATHMathSciNetGoogle Scholar
  21. [21]
    C. Schneider, Vereinfachte Rekursionen zur Richardson-Extrapolation in Spezialfällen, Numer. Math. 24(1975)177–184.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    D. Shanks, Non linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34(1951)1–42.MathSciNetGoogle Scholar
  23. [23]
    A. Sidi, An algorithm for a special case of a generalization of the Richardson extrapolation process, Numer. Math. 38(1982)299–307.MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    A. Sidi, On a generalization of the Richardson extrapolation process, Numer. Math. 57(1990) 365–377.MATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep. 10(1989)189–371.CrossRefGoogle Scholar
  26. [26]
    P. Wynn, On a device for computing theS n) transformation, MTAC 10(1956)91–96.MATHMathSciNetGoogle Scholar
  27. [27]
    P. Wynn, Confluent forms of certain nonlinear algorithms. Arch. Math. 11(1960)223–234.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© J. C. Baltzer AG, Science Publishers 1994

Authors and Affiliations

  • C. Brezinski
    • 1
  • M. Redivo-Zaglia
    • 2
  1. 1.Laboratoire d'Analyse Numérique et d'Optimisation, UFR IEEA-M3Université des Sciences et Technologies de LilleVilleneuve d'Ascq CedexFrance
  2. 2.Dipartimento di Elettronica e InformaticaUniversità degli Studi di PadovaPadovaItaly

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