Advances in Computational Mathematics

, Volume 27, Issue 2, pp 151–166 | Cite as

Spectral behaviour of GMRES applied to singular systems



The purpose of this paper is to develop a spectral analysis of the Hessenberg matrix obtained by the GMRES algorithm used for solving a linear system with a singular matrix. We prove that the singularity of the Hessenberg matrix depends on the nature of A and some other criteria such as the zero eigenvalue multiplicity and the projection of the initial residual on particular subspaces. We also show some new results about the distinct kinds of breakdown which may occur in the algorithm when the system is singular.


GMRES Krylov subspace singular system Hessenberg matrix geometric multiplicities breakdowns 

Mathematics Subject Classifications (2000)

15A03 15A18 65F10 


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité du Littoral, zone universitaire de la Mi-voix, bâtiment H. PoincarréCalais CedexFrance

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