GMRES convergence bounds that depend on the right-hand-side vector

David Titley-Peloquin, Jennifer Pestana, Andrew J. Wathen

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


We consider the convergence of the algorithm GMRES of Saad and Schultz for solving linear equations Bx=b, where B ∈ ℂn×n is nonsingular and diagonalizable, and b ∈ ℂn. Our analysis explicitly includes the initial residual vector r0. We show that the GMRES residual norm satisfies a weighted polynomial least-squares problem on the spectrum of B, and that GMRES convergence reduces to an ideal GMRES problem on a rank-1 modification of the diagonal matrix of eigenvalues of B. Numerical experiments show that the new bounds can accurately describe GMRES convergence.
Original languageEnglish
Pages (from-to)462-479
Number of pages8
JournalIMA Journal of Numerical Analysis
Issue number2
Early online date25 Jul 2013
Publication statusPublished - 1 Apr 2014


  • convergence analysis
  • iterative methods
  • linear systems
  • Krylov subspace methods

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