Numerical Algorithms

, Volume 4, Issue 3, pp 361–377 | Cite as

A general projection algorithm for solving systems of linear equations

  • Khalide Jbilou
Article

Abstract

In this paper, we give a general projection algorithm for implementing some known extrapolation methods such as the MPE, the RRE, the MMPE and others. We apply this algorithm to vectors generated linearly and derive new algorithms for solving systems of linear equations. We will show that these algorithms allow us to obtain known projection methods such as the Orthodir or the GCR.

Keywords

Linear systems projection vector sequences 

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References

  1. [1]
    C. Brezinski, Généralisation de la transformation de Shanks, de la table de la Table de Padé et de l'epsilon-algorithme, Calcolo 12 (1975) 317–360.Google Scholar
  2. [2]
    C. Brezinski, Recursive interpolation, extrapolation and projection, J. Comput. Appl. Math. 9 (1983) 369–376.Google Scholar
  3. [3]
    C. Brezinski, Other manifestations of the Schur complement Lin. Alg. Appl. 24 (1988) 231–247.Google Scholar
  4. [4]
    C. Brezinski and H. Sadok, Vector sequence transformation and fixed point methods, in:Numerical Methods in Laminar and Turbulent Flows, eds. C. Taylor et al. (Pineridge, Swansea, 1987) pp. 3–11.Google Scholar
  5. [5]
    S. Cabay and L.W. Jackson, A polynomial extrapolation method for finding limits and antilimits for vector sequences, SIAM J. Numer. Anal. 13 (1976) 734–752.Google Scholar
  6. [6]
    R.P. Eddy, Extrapolation to the limit of a vector sequence, in:Information Linkage Between Applied Mathematics and Industry, ed. P.C.C. Wang, (Academic Press, New York, 1979) pp. 387–396.Google Scholar
  7. [7]
    S.C. Eisenstat, H.C. Elman and M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 20 (1983) 345–357.Google Scholar
  8. [8]
    H.C. Elman, Iterative methods for large sparse nonsymmetric systems of linear equations, Ph.D. thesis, Computer Science Dept., Yale Univ., New Haven, CT (1982).Google Scholar
  9. [9]
    V. Faber and T. Manteufel, Orthogonal Error Methods, SIAM J. Numer Anal. 24 (1987) 170–187.Google Scholar
  10. [10]
    W.D. Ford and A. Sidi, Recursive algorithms for vector extrapolation methods, Appl. Numer. Math. 4 (1988) 477–489.Google Scholar
  11. [11]
    W. Gander, G.H. Golub and D. Gruntz, Solving linear equations by extrapolation, in:Supercomputing, ed. J.S. Kovalic (Nato ASI Series, Springer, 1990).Google Scholar
  12. [12]
    B. Germain-Bonne, Estimation de la limite de suites et formalisation de procédés d'accélération de la convergence, Thèse d'Etat, Université de Lille 1 (1978).Google Scholar
  13. [13]
    Henrici,Elements of Numerical Analysis (Wiley, New York, 1964).Google Scholar
  14. [14]
    K. Jbilou, Méthodes d'extrapolation et de projection. Applications aux suites de vecteurs, Thèse de 3eme cycle, Université de Lille 1 (1988).Google Scholar
  15. [15]
    K. Jbilou and H. Sadok Some results about vector extrapolation methods and related fixed point iterations, J. Comput. Appl. Math. 36 (1991) 385–398.Google Scholar
  16. [16]
    K. Jbilou and H. Sadok, Analysis of the complete and incomplete vector extrapolation methods for solving systems of linear equations, submitted.Google Scholar
  17. [17]
    M. Mesina, Convergence acceleration for the iterative solution ofx=Ax+f, Comput. Meth. Appl. Mech. Eng. 10 (1977) 165–173.Google Scholar
  18. [18]
    B.P. Pugatchev, Acceleration of the convergence of iterative processes and a method for solving systems of nonlinear equations, USSR Comput. Math. Math. Phys. 17 (1978) 199–207.Google Scholar
  19. [19]
    Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comput. 37 (1981) 105–126.Google Scholar
  20. [20]
    H. Sadok, Accélération de la convergence de suites vectorielles et méthodes de point fixe, Thèse, Univ. Lille (1988).Google Scholar
  21. [21]
    A. Sidi, Extrapolation vs. projection methods for linear systems of equations, J. Comput. Appl. Math. 22 (1988) 71–88.Google Scholar
  22. [22]
    A. Sidi, Convergence and stability of minimal polynomial and reduced rank extrapolation algorithms, SIAM J. Numer. Anal. 23 (1986) 197–209.Google Scholar
  23. [23]
    A. Sidi, W.F. Ford and D.A. Smith, Acceleration of convergence of vector sequences, SIAM J. Numer. Anal. 23 (1986) 178–196.Google Scholar
  24. [24]
    D. A. Smith, W.F. Ford and A. Sidi Extrapolation methods for vector sequences, SIAM Rev. 29 (1987) 199–233; Correction, SIAM Rev. 30 (1988) 623–624.Google Scholar
  25. [25]
    D.M. Young and K.C. Jea, Generalized conjugate-gradient acceleration of nonsymmetrisable iterative methods. Lin. Alg. Appl. 34 (1980) 159–194.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Khalide Jbilou
    • 1
  1. 1.Laboratoire d'Analyse Numérique et d'Optimisation, UFRIEEA-M3Université des Sciences et Technologies de LilleVilleneuve d'Ascq CedexFrance

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