GMRES On (Nearly) Singular Systems

We consider the behavior of the GMRES method for solving a linear system $Ax = b$ when $A$ is singular or nearly so, i.e., ill conditioned. The (near) singularity of $A$ may or may not affect the performance of GMRES, depending on the nature of the system and the initial approximate solution. For si...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 1997-01, Vol.18 (1), p.37-51
Main Authors: Brown, Peter N., Walker, Homer F.
Format: Article
Language:English
Subjects:
Applied mathematics
Laboratories
Linear algebra
Markov analysis
Methods
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Summary:We consider the behavior of the GMRES method for solving a linear system $Ax = b$ when $A$ is singular or nearly so, i.e., ill conditioned. The (near) singularity of $A$ may or may not affect the performance of GMRES, depending on the nature of the system and the initial approximate solution. For singular $A$, we give conditions under which the GMRES iterates converge safely to a least-squares solution or to the pseudoinverse solution. These results also apply to any residual minimizing Krylov subspace method that is mathematically equivalent to GMRES. A practical procedure is outlined for efficiently and reliably detecting singularity or ill conditioning when it becomes a threat to the performance of GMRES.
ISSN:0895-4798
1095-7162
DOI:10.1137/S0895479894262339