Numerical Algorithms

, Volume 22, Issue 2, pp 193–212 | Cite as

Some results about GMRES in the singular case

  • L. Smoch
Article

Abstract

In this paper, we study the Generalized Minimal Residual (GMRES) method for solving singular linear systems, particularly when the necessary and sufficient condition to obtain a Krylov solution is not satisfied. Thanks to some new results which may be applied in exact arithmetic or in finite precision, we analyze the convergence of GMRES and restarted GMRES. These formulas can also be used in the case when the systems are nonsingular. In particular, it allows us to understand what is often referred to as stagnation of the residual norm of GMRES.

GMRES restarted GMRES Krylov subspace singular system QR‐factorization Ritz values 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P.N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Statist. Comput. 12 (1991) 58–78.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    P.N. Brown and H.F. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl. 18 (1997) 37–51.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    S.L. Campbell, I.C.F. Ipsen, C.T. Kelley and C.D. Meyer, GMRES and the minimal polynomial, BIT 36(4) (1996) 664–675.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    R.W. Freund and M. Hochbruck, On the use of two QMR algorithms to solve singular systems and applications in Markov chains modeling, Numer. Linear Algebra Appl. 1 (1994) 403–420.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    I.C.F. Ipsen and C.D. Meyer, The idea behind Krylov methods, Amer. Math. Monthly, to appear.Google Scholar
  6. [6]
    Y. Saad, Variations on Arnoldi's method for solving eigenelements of large unsymmetric matrices, Linear Algebra Appl. 34 (1980) 269–295.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Y. Saad, Krylov subspace method for solving unsymmetric linear systems, Math. Comp. 37 (1981) 105–126.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Y. Saad and M.H. Schultz, GMRES: A generalized residual method for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856–869.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    H. Sadok, Analysis of the convergence of the minimal and the orthogonal residual methods, Technical Report LMPA, Université du Littoral, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (1998).Google Scholar
  10. [10]
    H.A. Van der Vorst and C. Vuik, The superlinear convergence behaviour of GMRES, J. Comput. Appl. Math. 48 (1993) 327–341.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • L. Smoch
    • 1
  1. 1.Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniversité du Littoral, zone universitaire de la Mi‐voix, bâtiment H. PoincarréCalais CedexFrance

Personalised recommendations