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arXiv · 1508.06341

A fast Newton-Shamanskii iteration for M/G/1-type and GI/M/1-type Markov chains

Pei-Chang Guo
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Abstract

For the nonlinear matrix equations arising in the analysis of M/G/1-type and GI/M/1-type Markov chains, the minimal nonnegative solution $G$ or $R$ can be found by Newton-like methods. Recently a fast Newton's iteration is proposed in \cite{Houdt2}. We apply the Newton-Shamanskii iteration to the equations. Starting with zero initial guess or some other suitable initial guess, the Newton-Shamanskii iteration provides a monotonically increasing sequence of nonnegative matrices converging to the minimal nonnegative solution. We use the technique in \cite{houdt2} to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.

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Pei-Chang Guo. 2015-08-26. A fast Newton-Shamanskii iteration for M/G/1-type and GI/M/1-type Markov chains. https://arxiv.org/abs/1508.06341

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