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On Convergence and Implementation of Minimal Residual KrylovSubspace Methods for Unsymmetric Linear Systems

by J. Liesen and M. Rozloznik and Z. Strakos

(Report number 2000-11)

Abstract
Consider linear algebraic systems A x = b with a general unsymmetric nonsingular matrix A. We study Krylov subspace methods for solving such systems that minimize the norm of the residual at each step. Such methods are often formulated in terms of a sequence of least squares problems of increasing dimension. Therefore we begin with an overdetermined least squares problem Bu \approx c and present several basic identities and bounds for the least squares residual r = c - By. Then we apply these results to minimal residual Krylov subspace methods, and formulate several theoretical consequences about their convergence. We consider possible implementations, in particular various forms of the GMRES method [26], and discuss their numerical properties. Finally, we illustrate our findings by numerical examples and draw conclusions.

Keywords: linear systems, least squares problems, Krylov subspace methods, minimal residual methods, GMRES, convergence, rounding errors

@Techreport{LRS00_270,
  author = {J. Liesen and M. Rozloznik and Z. Strakos},
  title = {On Convergence and Implementation of Minimal Residual KrylovSubspace Methods for Unsymmetric Linear Systems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2000-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2000/2000-11.pdf },
  year = {2000}
}

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