Solving quadratic matrix equations arising in random walks in the quarter plane
Abstract
Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution $G$. In many queueing models, described by random walks in the quarter plane, the coefficients $A_1,A_0,A_{-1}$ are infinite tridiagonal matrices with an almost Toeplitz structure. Here, we analyze some fixed point iterations, including Newton's iteration, for the computation of $G$ and introduce effective algorithms and acceleration strategies which fully exploit the Toeplitz structure of the matrix coefficients and of the current approximation. Moreover, we provide a structured perturbation analysis for the solution $G$. The results of some numerical experiments which demonstrate the effectiveness of our approach are reported.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2019
- DOI:
- 10.48550/arXiv.1907.09796
- arXiv:
- arXiv:1907.09796
- Bibcode:
- 2019arXiv190709796B
- Keywords:
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- Mathematics - Numerical Analysis
- E-Print:
- SIAM Journal on Matrix Analysis and Applications, 2020, 41(2), pp. 691-714