Highly Accurate Latouche-Ramaswami Logarithmic Reduction Algorithm for Quasi-Birth-and-Death Process
Year: 2022
Author: Guiding Gu, Wang Li, Ren-Cang Li
Journal of Mathematical Study, Vol. 55 (2022), Iss. 2 : pp. 180–194
Abstract
This paper is concerned with the quadratic matrix equation $A_0+A_1X+A_2X^2$ $=X$ where $I-A_0-A_1-A_2$ is a regular $M$-matrix, i.e., there exists an entrywise positive vector u such that $(I-A_0-A_1-A_2)$u $\ge 0$ entrywise. It broadly includes those originally arising from the quasi-birth-and-death (QBD) process as a special case where $I-A_0-A_1-A_2$ is an irreducible singular $M$-matrix and $(A_0+A_1+A_2)$1=1 with 1 being the vector of all ones. A highly accurate implementation of Latouche-Ramaswami logarithmic reduction algorithm [Journal of Applied Probability, 30(3):650-674, 1993] is proposed to compute the unique minimal nonnegative solution of the matrix equation with high entrywise relative accuracy as it deserves. Numerical examples are presented to demonstrate the effectiveness of the proposed implementation.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v55n2.22.05
Journal of Mathematical Study, Vol. 55 (2022), Iss. 2 : pp. 180–194
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Quadratic matrix equation M-matrix quasi-birth-and-death process minimal nonnegative solution entrywise relative accuracy.