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

Journal article

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.

Authors: Titley-Peloquin, D; Pestana, J; Wathen, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Oxford University Press
• Journal: IMA Journal of Numerical Analysis
• Volume: 34
• Issue: 2
• Pages: 462-479
• Publication date: 2013-07-25
• DOI: 10.1093/imanum/drt025
• EISSN: 1464-3642
• ISSN: 0272-4979
• Keywords: Krylov subspace methods; linear systems; convergence analysis; GMRES; SBTMR; iterative methods