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How to make Simpler GMRES and GCR more stable

by P. Jiranek and M. Rozloznik and M. H. Gutknecht

(Report number 2008-10)

Abstract
In this paper we analyze the numerical behavior of several minimum residual methods, which are mathematically equivalent to the GMRES method. Two main approaches are compared: the one that computes the approximate solution (similar to GMRES) in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and the one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of the resulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and, in a sense they explain the experimentally observed fact that the GCR (ORTHOMIN) method delivers very accurate approximate solutions when it converges fast enough without stagnation.

Keywords: Large-scale nonsymmetric linear systems, Krylov subspace methods, minimum residual methods, numerical stability, rounding errors.

@Techreport{JRG08_375,
  author = {P. Jiranek and M. Rozloznik and M. H. Gutknecht},
  title = {How to make Simpler GMRES and GCR more stable},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-10.pdf },
  year = {2008}
}

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