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Mathematical Problems in Engineering
Volume 2017, Article ID 4018239, 8 pages
Research Article

A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains

School of Science, China University of Geosciences, Beijing 100083, China

Correspondence should be addressed to Pei-Chang Guo; nc.ude.bguc@gnahciep

Received 16 June 2017; Revised 18 September 2017; Accepted 28 September 2017; Published 19 October 2017

Academic Editor: Nunzio Salerno

Copyright © 2017 Pei-Chang Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 or can be found by Newton-like methods. We prove monotone convergence results for the Newton-Shamanskii iteration for this class of 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. A Schur decomposition method is used to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.