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

TL;DR

This paper introduces an accelerated Newton-Shamanskii iteration method for efficiently computing minimal nonnegative solutions of matrix equations in Markov chain analysis, demonstrating improved convergence through numerical experiments.

Contribution

The paper applies and accelerates the Newton-Shamanskii iteration for matrix equations in Markov chains, offering a faster alternative to existing methods.

Findings

Converges monotonically to the minimal nonnegative solution.

Accelerated iteration improves computational efficiency.

Numerical examples confirm effectiveness.

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.