GMRES On (Nearly) Singular Systems

  • Authors:

  • Peter N. Brown;Homer F. Walker

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Matrix Analysis and Applications
  • Year:
  • 1997

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Abstract

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.