Solving quadratic matrix equations arising in random walks in the quarter plane

07/23/2019

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by   Dario A. Bini, et al.

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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.