Advertisement

Numerical Algorithms

, Volume 7, Issue 2, pp 225–251 | Cite as

Non-commutative extrapolation algorithms

  • A. Salam
Article

Abstract

This paper contains two general results. The first is an extension of the theory of general linear extrapolation methods to a non-commutative field (or even a non-commutative unitary ring). The second one, by exploiting these new results, is to solve an old conjecture about Wynn's vector ε-algorithm. Then, by using designants and Clifford algebras, we show how the vectors ∈ k (n) can be written as a ratio of two designants.

This result allow us to find, as a particular case, some well-known results and some others which are new.

Keywords

Designant Clifford algebra extrapolation vector ε-algorithm 

Subject classification

AMS(MOS) 65B05 

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, Edinburgh, 1965).Google Scholar
  2. [2]
    A. Artin,Geometric Algebra (Interscience, New York, 1966).Google Scholar
  3. [3]
    C. Brezinski and M. Redivo Zaglia,Extrapolation Methods. Theory and Practice (North-Holland, Amsterdam, 1991).Google Scholar
  4. [4]
    C. Brezinski, A general extrapolation algorithm, Numer. Math. 35 (1980) 175–187.Google Scholar
  5. [5]
    C. Brezinski, Some results in the theory of the vector ε-algorithm, Lin. Alg. Appl. 8 (1974) 77–86.Google Scholar
  6. [6]
    C. Brezinski, Computation of the eigenelements of a matrix by the ε-algorithm, Lin. Alg. Appl. 11 (1975) 7–20.Google Scholar
  7. [7]
    C. Brezinski, Some determinantal identities in a vector space, with applications, in:Padé Approximation and its Applications, eds. H. Werner and H.J. Bünger, LNM 1071 (Springer, Berlin, 1984) pp. 1–11.Google Scholar
  8. [8]
    C. Brezinski, Application de l'ε-algorithme à la résolution des systèmes non linéaires, C.R. Acad. Sci. Paris 271A (1970) 1174–1177.Google Scholar
  9. [9]
    R. Deheuvels,Formes Quadratiques et Groupes Classiques (Presses Universitaires de France, Paris, 1981).Google Scholar
  10. [10]
    J. Dieudonné, Les déterminants sur un corps non commutatif, Bull. Soc. Math. France 7 (1943) 27–45Google Scholar
  11. [11]
    F.J. Dyson, Quaternion determinants, Helv. Phys. Acta 45 (1972) 289–302.Google Scholar
  12. [12]
    E. Gekeler, On the solution of systems of equations by the epsilon algorithm of Wynn, Math. Comp. 26 (1972) 427–436.Google Scholar
  13. [13]
    R. Godement,Cours d'Algèbre (Hermann, Paris, 1966).Google Scholar
  14. [14]
    P.R. Graves-Morris, Vector-valued rational interpolants I, Numer. Math. 42 (1983) 331–348.Google Scholar
  15. [15]
    P.R. Graves-Morris, Vector-valued rational interpolants II, IMA J. Numer. Anat. 4 (1984) 209–224.Google Scholar
  16. [16]
    P.R. Graves-Morris and C.D. Jenkins, Vector-valued rational interpolants III, Constr. Approx. 2 (1986) 263–289.Google Scholar
  17. [17]
    P.R. Graves-Morris and D.E. Roberts, From matrix to vector Padé approximants, J. Comp. Appl. Math., to appear.Google Scholar
  18. [18]
    T. Håvie, Generalized Neville type extrapolation schemes, BIT 19 (1979) 204–213Google Scholar
  19. [19]
    A. Heyting, Die Theorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplikation, Math. Ann. 98 (1927) 465–490.Google Scholar
  20. [20]
    G.N. Hile and P. Lounesto, Matrix representations of Clifford algebras, Lin. Alg. Appl. 128 (1990) 51–63.Google Scholar
  21. [21]
    M.L. Mehta,Matrix Theory. Selected Topics and Useful Results (Les Editions de Physique, Les Ulis, 1989).Google Scholar
  22. [22]
    J.B. McLeod, A note on the ε-algorithm, Computing 7 (1971) 17–24.Google Scholar
  23. [23]
    O. Ore, Linear equations in non-commutative fields, Ann. Math. 32 (1931) 463–477.Google Scholar
  24. [24]
    R. Penrose, A generalised inverse for matrices, Proc. Cambridge Phil. Soc. 51 (1955) 406–413.Google Scholar
  25. [25]
    I.R. Porteous,Topological Geometry, 2nd ed. (Cambridge University Press, Cambridge, 1981).Google Scholar
  26. [26]
    A. Salam, Extrapolation: extension et nouveaux résultats, Thesis, Université des Sciences et Technologies de Lille (1993).Google Scholar
  27. [27]
    D. Shanks, Nonlinear transformations of divergent and slowly convergent sequences, J.Math. Phys. 34 (1955) 1–42.Google Scholar
  28. [28]
    P. Wynn, On a device for computing thee m(Sn) transformation, MTAC 10 (1956) 91–96.Google Scholar
  29. [29]
    P. Wynn, Vector continued fractions, Lin. Alg. Appl. 1 (1968) 357–395.Google Scholar
  30. [30]
    P. Wynn, Continued fractions whose coefficients obey a non-commutative law of multiplication, Arch. Rational Mech. Anal. 12 (1963) 273–312.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1994

Authors and Affiliations

  • A. Salam
    • 1
  1. 1.Laboratoire d'Analyse Numérique et d'OptimisationUniversité des Sciences et Technologies de Lille, UFR IEEAVilleneuve d'Ascq CedexFrance

Personalised recommendations