On the use of incomplete semiiterative methods for singular systems and applications in Markov chain modeling

Authors:
Yimin Wei;Hebing Wu
Affiliations:
Department of Mathematics, Fudan University, Shanghai 200433, People's Republic of China;Institute of Mathematics, Fudan University, Shanghai 200433, People's Republic of China
Venue:
Applied Mathematics and Computation
Year:
2002

Citing 4
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Matrix analysis

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Index splitting for the Drazin inverse and the singular linear system

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Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index

Journal of Computational and Applied Mathematics

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Abstract

There are several methods for solving linear fixed point problem x = Tx + c, where T ∈ CN,N is a square matrix and I - T is possibly singular. Such problems arise if one splits the coefficient matrix of a singular system Ax = b of algebraic equations according to A= M - N (M nonsingular) which leads to x = M-1Nx + M-1b =: Tx +c. The basic iteration x0 ∈ CN, xm = Txm-1 + c (m ≥ 1) requires the modulus of every eigenvalue of the iteration matrix T except 1 is less than 1 and q = index(I-T), the index of I - T is equal to 1 for convergence. In this paper, we try to use the incomplete semiiterative methods (ISIM) to solve x = Tx + c when c ∈ R(I - T)q. Usually the special semiiterative methods are convergent even when the spectral radius of the iteration matrix is greater than 1 and q ≥ 1. Then the use of the ISIM in the Markov chain modeling is considered. Finally, numerical examples are reported.