Numerical Algorithms

, Volume 68, Issue 1, pp 143–165 | Cite as

The role eigenvalues play in forming GMRES residual norms with non-normal matrices

Original Paper

Abstract

In this paper we give explicit expressions for the norms of the residual vectors generated by the GMRES algorithm applied to a non-normal matrix. They involve the right-hand side of the linear system, the eigenvalues, the eigenvectors and, in the non-diagonalizable case, the principal vectors. They give a complete description of how eigenvalues contribute in forming residual norms and offer insight in what quantities can prevent GMRES from being governed by eigenvalues.

Keywords

GMRES convergence Non-normal matrix Eigenvalues Residual norms 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.ParisFrance
  2. 2.Institute of Computer Science, Academy of Sciences of the Czech RepublicPragueCzech Republic
  3. 3.Faculty of Pharmacy in Hradec KrálovéCharles University in PragueHradec KrálovéCzech Republic

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