The Newton-Shamanskii method for solving a quadratic matrix equation arising in quasi-birth-death problems

2014/06/04 by Pei-Chang Guo, Guo, Peichang
Business, Management and Accounting · Computer Science · Mathematics · #65F30 65H10 #Advanced Queuing Theory Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1406.1075

openalex publication_date 2014/06/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01

Abstract

In order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. We apply the Newton-Shamanskii method for solving the equation. We show that the sequence of matrices generated by the Newton-Shamanskii method is monotonically increasing and converges to the minimal nonnegative solution of the equation. Numerical experiments show the effectiveness of our method.

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