Some observations on weighted GMRES

Stefan Guettel, Jennifer Pestana

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We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present a new alternative implementation of the weighted Arnoldi algorithm which under known circumstances will be favourable in terms of computational complexity. These implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used.
Original languageEnglish
Pages (from-to)733-752
Number of pages21
JournalNumerical Algorithms
Issue number4
Early online date10 Jan 2014
Publication statusPublished - Dec 2014


  • weighted GMRES
  • linear systems
  • Krylov subspace method
  • harmonic Ritz values
  • algorithms
  • algebra


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