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Influence of preconditioning and blocking on accuracy in solving Markovian models. (English) Zbl 1170.65022
Summary: We consider the effectiveness of various methods used to solve systems of linear equations (which emerge while modeling computer networks and systems with Markov chains) and the practical influence of the methods applied on accuracy. We consider some hybrids of both direct and iterative methods. Two varieties of the Gauss elimination are considered as an example of direct methods: the LU factorization method and the WZ factorization method. The Gauss-Seidel iterative method is discussed. The paper also shows preconditioning (with the use of incomplete Gauss elimination) and dividing the matrix into blocks where blocks are solved applying direct methods.
The motivation for such hybrids is a very high condition number (which is bad) for coefficient matrices occuring in Markov chains and, thus, slow convergence of traditional iterative methods. Also, the blocking, preconditioning and merging of both are analysed. The paper presents the impact of linked methods on both the time and accuracy of finding vector probability. The results of an experiment are given for two groups of matrices: those derived from some very abstract Markovian models, and those from a general 2D Markov chain.

MSC:
65F10 Iterative numerical methods for linear systems
65F05 Direct numerical methods for linear systems and matrix inversion
65F35 Numerical computation of matrix norms, conditioning, scaling
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65C40 Numerical analysis or methods applied to Markov chains
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