Volume 55, Issue 2
Highly Accurate Latouche-Ramaswami Logarithmic Reduction Algorithm for Quasi-Birth-and-Death Process

J. Math. Study, 55 (2022), pp. 180-194.

Published online: 2022-04

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• 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.

• Keywords

Quadratic matrix equation, M-matrix, quasi-birth-and-death process, minimal nonnegative solution, entrywise relative accuracy.

15A24, 65F30, 65H10

guiding@mail.shufe.edu.cn (Guiding Gu)

li.wang@uta.edu (Wang Li)

rcli@uta.edu (Ren-Cang Li)

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• RIS
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@Article{JMS-55-180, author = {Guiding and Gu and guiding@mail.shufe.edu.cn and 6718 and School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China and Guiding Gu and Wang and Li and li.wang@uta.edu and 23080 and Department of Mathematics, University of Texas at Arlington, P.O. Box 19408, Arlington, TX 76019, USA and Wang Li and Ren-Cang and Li and rcli@uta.edu and 13355 and Department of Mathematics, University of Texas at Arlington, P.O. Box 19408, Arlington, TX 76019, USA and Ren-Cang Li}, title = {Highly Accurate Latouche-Ramaswami Logarithmic Reduction Algorithm for Quasi-Birth-and-Death Process}, journal = {Journal of Mathematical Study}, year = {2022}, volume = {55}, number = {2}, pages = {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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v55n2.22.05}, url = {http://global-sci.org/intro/article_detail/jms/20494.html} }
TY - JOUR T1 - Highly Accurate Latouche-Ramaswami Logarithmic Reduction Algorithm for Quasi-Birth-and-Death Process AU - Gu , Guiding AU - Li , Wang AU - Li , Ren-Cang JO - Journal of Mathematical Study VL - 2 SP - 180 EP - 194 PY - 2022 DA - 2022/04 SN - 55 DO - http://doi.org/10.4208/jms.v55n2.22.05 UR - https://global-sci.org/intro/article_detail/jms/20494.html KW - Quadratic matrix equation, M-matrix, quasi-birth-and-death process, minimal nonnegative solution, entrywise relative accuracy. AB -

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.

Guiding Gu, Wang Li & Ren-Cang Li. (2022). Highly Accurate Latouche-Ramaswami Logarithmic Reduction Algorithm for Quasi-Birth-and-Death Process. Journal of Mathematical Study. 55 (2). 180-194. doi:10.4208/jms.v55n2.22.05
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