Advances in Computational Mathematics

, Volume 2, Issue 3, pp 319–341 | Cite as

Acceleration property for the E-algorithm and an application to the summation of series

  • Marc Prévost
Article

Abstract

The E-algorithm is the most general sequence transformation actually known, since it contains as particular cases almost all the sequence transformations discovered so far: Richardson polynomial extrapolation, Shanks’ transformation, summation processes, Germain-Bonne transformation, Levin’s generalized transformations, the processp and rational extrapolation. In [10] some results concerning the columns of the E-algorithm were proved. In this paper, by adding conditions about determinants, we prove that the diagonal of this algorithm also accelerates the convergence of the initial sequence.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A.C. Aitken,Determinants and Matrices (Oliver and Boyd, 1951).Google Scholar
  2. [2]
    C. Brezinski, A general extrapolation algorithm, Numer. Math. 35 (1980) 175–187.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Brezinski, The asymptotic behaviour of sequences and new series transformations based on the Cauchy product, Rocky Mt. J. Math. 21 (1991).Google Scholar
  4. [4]
    T.J. Bromwich,An Introduction to the Theory of Infinite Series (MacMillan, London, 1949).Google Scholar
  5. [5]
    R.A. Brualdi and H. Schneider, Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Laplace, Binet, Muir and Cayley, Lin. Alg. Appl. 52–53 (1983) 769–791.Google Scholar
  6. [6]
    P.J. Davis,Spirals: From Theodorus of Cyrene to Meta-Chaos, The Hedric Lectures (Math. Assoc. America, 1990).Google Scholar
  7. [7]
    T. Havie, Generalized Neville type extrapolation schemes, BIT 19 (1979) 204–213.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    S. Karlin,Total Positivity, Vol. 1 (Stanford University Press, Stanford, 1962).MATHGoogle Scholar
  9. [9]
    P.J. Laurent, Un théorème de convergence pour le procédé d’extrapolation de Richardson, C. R. Acad. Sci. Paris 256 (1963) 1435–1437.MATHMathSciNetGoogle Scholar
  10. [10]
    A. Matos and M. Prévost, Acceleration property for the columns of the E-algorithm, Numer. Algor. 2 (1992) 393–408.MATHCrossRefGoogle Scholar
  11. [11]
    A. Sidi, On a generalization of the Richardson extrapolation process, Numer. Math. 57 (1990) 357–365.MathSciNetCrossRefGoogle Scholar
  12. [12]
    D.V. Widder,The Laplace Transform (Princeton University Press, 1946).Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1994

Authors and Affiliations

  • Marc Prévost
    • 1
    • 2
  1. 1.Laboratoire d’Analyse Numérique et d’OptimisationUniversité des Sciences et Technologies de LilleVilleneuve d’Ascq CédexFrance
  2. 2.Bat. H. PoincarréUniversité du Littoral, zone de la mi-voixCalaisFrance

Personalised recommendations