GMRES convergence bounds for eigenvalue problems

Melina A. Freitag, Patrick Kürschner, Jennifer Pestana

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
73 Downloads (Pure)


The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g. tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.
Original languageEnglish
Pages (from-to)203-222
Number of pages20
JournalComputational Methods in Applied Mathematics
Issue number2
Early online date7 Jun 2017
Publication statusPublished - 30 Apr 2018


  • convergence analysis
  • inexact inverse iteration
  • inexact subspace iteration
  • Krylov subspace methods
  • block Krylov methods
  • preconditioning


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