Summary.
In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above.
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Received August 19, 1997 / Revised version received August 20, 1998 / Published online January 27, 2000
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Song, Y. Semiconvergence of nonnegative splittings for singular matrices. Numer. Math. 85, 109–127 (2000). https://doi.org/10.1007/s002110050479
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DOI: https://doi.org/10.1007/s002110050479