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Improved SOR method with orderings and direct methods

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Abstract

A generalized SOR method with multiple relaxation parameters is considered for solving a linear system of equations. Optimal choices of the parameters are examined under the assumption that the coefficient matrix is tridiagonal and regular. It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which is computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.

When the system is well-conditioned, it is solved stably by Gaussian elimination; there is little advantage in using an iterative method. However, when the system is ill-conditioned, the direct method is not necessarily stable and it is often required to improve the numerical solution by a certain iterative algorithm. Some numerical examples are presented which show that the proposed method is more efficient than the standard iterative refinement method.

It is also discussed how to apply our technique to a class of systems which includes Hessenberg systems.

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Author notes

  1. Emiko Ishiwata

    Present address: Department of Information Sciences, Toho University, Miyama 2-2-1, Funabashi-city, 274-8510, Chiba, Japan

Authors and Affiliations

  1. Department of Mathematics, Waseda University, Ohkubo 3-4-1 Shinjuku-ku, 169, Tokyo, Japan

    Emiko Ishiwata & Yoshiaki Muroya

Authors
  1. Emiko Ishiwata
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  2. Yoshiaki Muroya
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Additional information

This work was partially supported by a Science Research Grant-in-Aid from the Japanese Ministry of Education, Science, Sports and Culture.

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Ishiwata, E., Muroya, Y. Improved SOR method with orderings and direct methods. Japan J. Indust. Appl. Math. 16, 175–193 (1999). https://doi.org/10.1007/BF03167325

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  • DOI: https://doi.org/10.1007/BF03167325

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