Comments on the GMRES Convergence for Preconditioned Systems

  • Nabil Gmati
  • Bernard Philippe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)


The purpose of this paper is to comment a frequent observation by the engineers studying acoustic scattering. It is related to the convergence of the GMRES method when solving systems Ax = b with A = I − B. The paper includes a theorem which expresses the convergence rate when some eigenvalues of B have modulus larger than one; that rate depends on the rate measured when solving the system obtained by spectral projection onto the invariant subspace corresponding to the other eigenvalues. The conclusion of the theorem is illustrated on the Helmholtz equation.


Unit Disk Invariant Subspace Error Matrix Spectral Projection Jordan Block 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Nabil Gmati
    • 1
  • Bernard Philippe
    • 2
  1. 1.ENITLAMSIN.Tunis BelvédèreTunisie
  2. 2.INRIA/IRISARENNES CedexFrance

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