A note on the preconditioner Pm=(I+Sm)

  • Authors:

  • Toshiyuki Kohno;Hiroshi Niki

  • Affiliations:

  • Facutly of Informatics, Okayama University of Science, Japan;Facutly of Informatics, Okayama University of Science, Japan

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2009

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Abstract

Kotakemori et al. [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+S"m"a"x), Journal of Computational and Applied Mathematics 145 (2002) 373-378] have reported that the convergence rate of the iterative method with a preconditioner P"m=(I+S"m) was superior to one of the modified Gauss-Seidel method under the condition. These authors derived a theorem comparing the Gauss-Seidel method with the proposed method. However, through application of a counter example, Wen Li [Wen Li, A note on the preconditioned GaussSeidel (GS) method for linear systems, Journal of Computational and Applied Mathematics 182 (2005) 81-91] pointed out that there exists a special matrix that does not satisfy this comparison theorem. In this note, we analyze the reason why such a to counter example may be produced, and propose a preconditioner to overcome this problem.