Abstract
We consider the convergence theory of the Successive Overrelaxation (SOR) iterative method for the solution of linear systems Ax = b, when the matrix A has block a p × p partitioned p-cyclic form. Our purpose is to extend much of the p-cyclic SOR theory for nonsingular A to consistent singular systems and to apply the results to the solution of large scale systems arising, e.g., in queueing network problems in Markov analysis. Markov chains and queueing models lead to structured singular linear systems and are playing an increasing role in the understanding of complex phenomena arising in computer, communication and transportation systems.
For certain important classes of singular problems, we develop a convergence theory for p-cyclic SOR, and show how to repartition for optimal convergence. Recent results by Kontovasilis, Plemmons and Stewart on the new concept of convergence of SOR in an extended sense are further analyzed and applied to the solution of periodic Markov chains.
Research supported in part by NSF grant CCR-86-19817 and by the US Air Force under grant no. AFOSR-88-10243.
Research supported by the US Air Force under grant no. AFOSR-91-0163.
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References
A. Berman and R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979.
F. Bonhoure, Y. Dallery, and W. Stewart. On the efficient use of periodicy properties for the efficient numerical solution of certain Markov chains. MASI Tech. Rept. 91–40, Université de Paris, France, 1991.
M. Eiermann, W. Niethammer, and A. Ruttan. Optimal successive overrelaxation iterative methods for p-cyclic matrices. Numer. Math., 57:593–606, 1990.
S. Galanis and A. Hadjidimos. How to repartition a block p-cyclic consistently ordered matrix for optimal SOR convergence. SIAM J. Matrix Anal. Appl., 13:102–120, 1992.
S. Galanis, A. Hadjidimos, and D. Noutsos. On the equivalence of the k-step iterative Euler methods and successive overrelaxation (SOR) methods for k-cyclic matrices. Math. Comput. Simulation, 30:213–230, 1988.
G. H. Golub and J. de Pillis. Toward an effective two-parameter SOR method. in Iterative Methods for Large Lin. Syst., D. Kincaid and L. Hayes, Eds., Academic Press, New York, 107–118, 1988.
A. Hadjidimos. On the optimization of the classical iterative schemes for the solution of complex singular linear systems. SIAM J. Alg. Disc. Meth., 6(4):555–566, 1985.
K. Kontovasilis, R. J. Plemmons and W. J. Stewart. Block cyclic SOR for Markov chains with p-cyclic infintesimal generator. Linear Algebra Appl., 154–156:145–223, 1991.
T. L. Markham, M. Neumann, and R. J. Plemmons. Convergence of a direct-iterative method for large-scale least squares problems Linear Algebra Appl., 69:155–167, 1985.
D. J. Pierce, A. Hadjidimos, and R. J. Plemmons. Optimality relationships for p-cyclic SOR. Numer. Math., 56:635–643, 1990.
R. S. Varga. p-cyclic matrices: A generalization of the Young-Frankel successive overrelaxation scheme. Pacific J. Math., 9:617–628, 1959.
R. S. Varga. Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs, NJ, 1962.
D. M. Young. Iterative methods for solving partial differential equations of elliptic type. Trans. Amer. Math. Soc., 76:92–111, 1954.
D. M. Young. Iterative Solution of Large Linear Systems. Academic Press, New York, 1971.
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© 1993 Springer-Verlag New York, Inc.
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Hadjidimos, A., Plemmons, R.J. (1993). Analysis of P-Cyclic Iterations for Markov Chains. In: Meyer, C.D., Plemmons, R.J. (eds) Linear Algebra, Markov Chains, and Queueing Models. The IMA Volumes in Mathematics and its Applications, vol 48. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8351-2_8
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DOI: https://doi.org/10.1007/978-1-4613-8351-2_8
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