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Spectral behaviour of GMRES applied to singular systems

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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.

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  1. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, zone universitaire de la Mi-voix, bâtiment H. Poincarré, 50 rue F. Buisson, BP 699, F-62228, Calais Cedex, France

    Laurent Smoch

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  1. Laurent Smoch
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Correspondence to Laurent Smoch.

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Communicated by L. Reichel.

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Smoch, L. Spectral behaviour of GMRES applied to singular systems. Adv Comput Math 27, 151–166 (2007). https://doi.org/10.1007/s10444-007-9029-4

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  • DOI: https://doi.org/10.1007/s10444-007-9029-4

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