Abstract
This paper reveals the inner links between two known frameworks of multisplitting relaxation methods as completely as possible. By meticulously investigating the specific structures of these two frameworks, the asymptotic convergence rates as well as the monotone convergence rates of them are compared theoretically. This then definitely answers the question that which converges faster between these two frameworks of parallel matrix multisplitting relaxation methods from the standpoint of pure mathematics. At last, an example further confirms the correctness of the theoretical results.
Similar content being viewed by others
References
Bai, Z. Z., Variants of the synchronous and asynchronous matrix multisplitting unsymmetric AOR methods, J. Fudan Univ. (Natural Sci.) 2 (1995), 139–148.
Bai, Z. Z. andWang, D. R., Generalized matrix multisplitting relaxation methods and their convergence, Numer. Math. J. Chinese Univ. (English Ser.) 2 (1993), 87–100.
Evans, D. J., Wang, D. R. andBai, Z. Z., Matrix multi-splitting multiparameter relaxation methods. Intern. J. Computer Math. 43 (1992), 173–188.
Frommer, A. andMayer, G., Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl. 119 (1989), 141–152.
Neumann, M. andPlemmons, R. J., Convergence of parallel multisplitting methods for M-matrices, Linear Algebra Appl. 88/89 (1987), 559–573.
O'Leary, D. P. andWhite, R. E., Multi-splittings of matrices and parallel solution of linear systems, SIAM J. Alg. Disc. Meth. 6 (1985), 630–640.
Varga, R. S. Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N. J., 1962.
Lanzkron, P. J., Rose, D. J. andSzyld, D. B., Convergence of nested classical iterative methods for linear systems, Numer. Math. 58 (1991), 685–702.
Bai, Z. Z., On the convergence of the generalized matrix multisplitting relaxed methods, Commun. Numer. Methods Eng. 11 (1995), 363–371.
Bai, Z. Z., Comparisons of the convergence and divergence rates of the parallel matrix multisplitting iteration methods, Chinese J. Eng. Math. 11 (1994), 99–102.
Evans, D. J. andWang, D. R., An asynchronous parallel algorithm for solving a class of nonlinear simultaneous equations, Parallel Comput. 17 (1991), 161–180.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bai, ZZ. On the comparisons of the multisplitting unsymmetric aor methods for M-matrices. Calcolo 32, 207–220 (1995). https://doi.org/10.1007/BF02575836
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02575836