A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains. (19th October 2017)

• Record Type: Journal Article
• Authors: Guo, Pei-Chang
• Other Names: Salerno Nunzio Academic Editor.
• Is Part Of: Mathematical problems in engineering. Volume 2017(2017)
• Publication Date: 2017-10-19
• Subjects: Engineering mathematics -- Periodicals; 510.2462
• DOI: 10.1155/2017/4018239
• Languages: English
• ISSNs: 1024-123X

Abstract

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. 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.