Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

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}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).