• Han, Yu-Du ;
  • Yun, Jae-Heon
  • Received : 2011.05.19
  • Published : 2012.07.01


In this paper, we first study convergence of a special type of multisplitting methods with preweighting, and then we provide some comparison results of those multisplitting methods. Next, we propose both parallel implementation of an SOR-like multisplitting method with preweighting and an application of the SOR-like multisplitting method with preweighting to a parallel preconditioner of Krylov subspace method. Lastly, we provide parallel performance results of both the SOR-like multisplitting method with preweighting and Krylov subspace method with the parallel preconditioner to evaluate parallel efficiency of the proposed methods.


multisplitting method;preweighting;preconditioner;Krylov subspace method;parallel performance


  1. A. Berman and R. J. Plemmons, Nonnegative matrices in the Mathematical sciences, Academic Press, New York, 1979.
  2. L. Elsner, Comparisons of weak regular splitting and multisplitting methods, Numer. Math. 56 (1989), no. 2-3, 283-289.
  3. A. Frommer and G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl. 119 (1989), 141-152.
  4. A. Frommer and G. Mayer, On the theory and practice of multisplitting methods in parallel computation, Computing 49 (1992), no. 1, 63-74
  5. A. Frommer and D. B. Szyld, H-splittings and two-stage iterative methods, Numer. Math. 63 (1992), no. 3, 345-356.
  6. M. Neumann and R. J. Plemmons, Convergence of parallel multisplitting iterative methods for M-matrices, Linear Algebra Appl. 88/89 (1987), 559-574.
  7. D. P. O'Leary and R. E. White, Multisplittings of matrices and parallel solution of linear systems, SIAM J. Algebraic Discrete Methods 6 (1985), no. 4, 630-640.
  8. OpenMP Architecture Review Board, OpenMP Fortran Application Program Interface Version 2.0, 2000.
  9. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, 1996.
  10. D. B. Szyld and M. T. Jones, Two-stage and multisplitting methods for the parallel solution of linear systems, SIAM J. Matrix Anal. Appl. 13 (1992), no. 2, 671-679.
  11. R. S. Varga, Matrix Iterative Analysis, Springer, Berlin, 2000.
  12. C. L. Wang and J. H. Zhao, Further results on regular splittings and multisplittings, Int. J. Comput. Math. 82 (2005), no. 4, 421-431.
  13. R. E. White, Multisplitting with different weighting schemes, SIAM J. Matrix Anal. Appl. 10 (1989), no. 4, 481-493.
  14. R. E. White, Multisplitting of a symmetric positive definite matrix, SIAM J. Matrix Anal. Appl. 11 (1990), no. 1, 69-82.
  15. J. H. Yun, Performance of ILU factorization preconditioners based on multisplittings, Numer. Math. 95 (2003), no. 4, 761-779.


Supported by : Korea Research Foundation(KRF)