Numerical Algorithms

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

Some results about GMRES in the singular case

  • L. Smoch


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 


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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

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