On the choice of preconditioner for minimum residual methods for non-Hermitian matrices

Andrew J. Wathen, Jennifer Pestana

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We consider the solution of left preconditioned linear systems P−1Cx=P−1c, where P,C∈Cn×n are non-Hermitian, c∈Cn, and C, P, and P−1C are diagonalisable with spectra symmetric about the real line. We prove that, when P and C are self-adjoint with respect to the same Hermitian sesquilinear form, the convergence of a minimum residual method in a particular nonstandard inner product applied to the preconditioned linear system is bounded by a term that depends only on the spectrum of P−1C. The inner product is related to the spectral decomposition of P. When P is self-adjoint with respect to a nearby Hermitian sesquilinear form to C, the convergence of a minimum residual method in this nonstandard inner product applied to the preconditioned linear system is bounded by a term involving the eigenvalues of P−1C and a constant factor. The size of this factor is related to the nearness of the Hermitian sesquilinear forms. Numerical experiments indicate that for certain matrices eigenvalue-dependent convergence is observed both for the nonstandard method and for standard GMRES.
Original languageEnglish
Pages (from-to)57-68
Number of pages12
JournalJournal of Computational and Applied Mathematics
Volume249
DOIs
Publication statusPublished - 2013

Keywords

  • nonstandard inner products
  • non-Hermitian matrices
  • GMRES
  • preconditioning

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