Spectral behaviour of GMRES applied to singular systems

Author: Smoch, Laurent

Source: Advances in Computational Mathematics, Volume 27, Number 2, August 2007 , pp. 151-166(16)

Publisher: Springer

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Abstract:

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.

Keywords: GMRES; Krylov subspace; singular system; Hessenberg matrix; geometric multiplicities; breakdowns; 15A03; 15A18; 65F10

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10444-007-9029-4

Affiliations: 1: Email: smoch@lmpa.univ-littoral.fr

Publication date: 2007-08-01