Skip to main content
SpringerLink
Log in

Adaptive improved block SOR method with orderings

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

Recently, a generalized SOR method with multiple relaxation parameters were considered for solving a linear system of equations and it was shown that if a pair of parameter values is computed from the pivots of the Gaussian elimination applied to the system, then the spectral radius of the iterative matrix is reduced to zero. A proper choice of orderings and starting vectors for the iteration were also proposed.

In this paper, we apply the above method to two-dimensional cases, and propose the “adaptive improved block SOR method with orderings” for block tridiagonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special multiple relaxation parameters are selected with this method for ann ×n block tridiagonal matrix whose block matrices are alln ×n matrices, then this iterative method converges at mostn 2 iterations. Hence this is a direct method. In particular, if we select proper orderings and apply the admissible error bounds, then convergence occurs at fewer iterations (for example,O(n) iterations) thann 2 iterations. Results of several numerical examples show this efficiency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.E. Bank and D.J. Rose, Marching algorithms for elliptic boundary value problems. I: The constant coefficient case. SIAM J. Numer. Anal.,14 (1977), 792–829.

    Article  MATH  MathSciNet  Google Scholar 

  2. R.E. Bank, Marching algorithms for elliptic boundary value problems. II: The variable coefficient case. SIAM J. Numer. Anal.,14 (1977), 950–970.

    Article  MATH  MathSciNet  Google Scholar 

  3. J.J. Buoni and R.S. Varga, Theorems of Stein-Rosenberg type. Numerical Mathematics (eds. R. Ansorge, K. Glashoff, and B. Werner), Birkhauser, Basel, 1979, 65–75.

    Google Scholar 

  4. L.W. Ehrlich, An Ad-Hoc SOR method. J. Comput. Phys.,44 (1981), 31–45.

    Article  MATH  Google Scholar 

  5. L.W. Ehrlich, A local relaxation scheme (Ad-Hoc SOR) applied to nine point and block difference equations. Iterative Methods for Large Linear Systems (eds. D. Kincaid and L. Hayes), Academic Press, 1990, 81–90.

  6. H.C. Elman and M.P. Chernesky, Ordering effects on relaxation methods applied to the discrete convection-diffusion. Recent Advances in Iterative Methods (eds. G. Golub, A. Greenbaum and M. Luskin), 1994, 45–57.

  7. B. Friedman, The iterative solution of elliptic difference equations. Report NYO-7698, Institute of Mathematical Science, New York University, 1957.

  8. H. Han, V.P. Il’in, W. Yuan and R.B. Kellogg, Analysis of flow directed iterations. J. Comput. Math.,10 (1992), 57–76.

    MATH  MathSciNet  Google Scholar 

  9. J. Heller, Ordering properties of linear successive iteration schemes applied to multidiagonal type linear systems. J. Soc. Indust. Appl. Math.,5 (1957), 238–243.

    Article  MATH  MathSciNet  Google Scholar 

  10. K. Ishihara, Y. Muroya and T. Yamamoto, On nonlinear SOR-like methods, II — convergence of the SOR-Newton method for mildly nonlinear equations. Japan J. Indust. Appl. Math.,14 (1997), 99–110.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Ishiwata and Y. Muroya, Improved SOR-like method with orderings for non-symmetric linear equations derived from singular perturbation problems. Numerical Analysis of Ordinary Differential Equations and its Applications, World Scientific Publishers, Singapore, 1995, 59–73.

    Google Scholar 

  12. E. Ishiwata and Y. Muroya, Main convergence theorems for the improved SOR method with orderings. Intern. J. Comput. Math.,66, No.1–2 (1998), 123–147.

    Article  MATH  MathSciNet  Google Scholar 

  13. E. Ishiwata and Y. Muroya, Improved SOR method with orderings and direct methods. Japan J. Indust. Appl. Math.,16 (1999), 175–193.

    Article  MATH  MathSciNet  Google Scholar 

  14. E. Ishiwata and Y. Muroya, Precise error estimates of the improved SOR method with orderings for tridiagonal matrices. Tech. Report 95-43, Waseda University, 1995.

  15. K.R. James, Convergence of matrix iterations subject to diagonal dominance. SIAM J. Numer. Anal.,10 (1973), 478–484.

    Article  MATH  MathSciNet  Google Scholar 

  16. K.R. James and A. Riha, Convergence criteria for successive overrelaxation. SIAM J. Numer. Anal.,12 (1975), 137–143.

    Article  MATH  MathSciNet  Google Scholar 

  17. C.C.J. Kuo, B.C. Levy and B.R. Musicus, A local relaxation method for solving elliptic PDEs on mesh-connected arrays. SIAM J. Sci. Stat. Comput.,8 (1987), 550–573.

    Article  MATH  MathSciNet  Google Scholar 

  18. R.E. Lynch, J.R. Rice and D.H. Thomas, Direct solution of partial difference equations by tensor product methods. Numer. Math.,6 (1964), 185–199.

    Article  MATH  MathSciNet  Google Scholar 

  19. R.E. Lynch, J.R. Rice and D.H. Thomas, Tensor product analysis of partial difference equations. Bull. Amer. Math. Soc.,70 (1964), 378–384.

    Article  MATH  MathSciNet  Google Scholar 

  20. R.S. Varga, Matrix Iterative Analysis. Prentice-Hall Inc. Englewood Cliffs, New Jersey, 1962.

    Google Scholar 

  21. R.S. Varga, On recurring theorems on diagonal dominance. Linear Algebra and Its Applications,13 (1976), 1–9.

    Article  MATH  Google Scholar 

  22. T. Yamamoto, On Lanczos’ algorithm for tri-diagonalization. J. Sci. Hiroshima Univ. Ser. A-I,32 (1968), 259–284.

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Emiko Ishiwata

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

  2. Kotaro Isogai

    Present address: AP Solution Department, System Division 1, Fujitsu Ltd., Fujitsu Oimachi Bldg., 1-20-10, Oi, Shinagawa-ku, 140-8508, Tokyo, Japan

Authors and Affiliations

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

    Emiko Ishiwata & Yoshiaki Muroya

  2. Graduate School of Information Systems, The University of Electro-Communications, 1-5-1, Chofugaoka, Chofu-shi, 182, Tokyo, Japan

    Kotaro Isogai

Authors
  1. Emiko Ishiwata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshiaki Muroya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kotaro Isogai
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was partially supported by Grant-in-Aid for Science Research (c) program of the Japanese Ministry of Education, Science, Sports and Culture.

About this article

Cite this article

Ishiwata, E., Muroya, Y. & Isogai, K. Adaptive improved block SOR method with orderings. Japan J. Indust. Appl. Math. 16, 443–466 (1999). https://doi.org/10.1007/BF03167368

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167368

Key words

Search

Navigation