Quadratic matrix equations of the kind A_1 X^2 + A_0 X + 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.
Solving Quadratic Matrix Equations Arising in Random Walks in the Quarter Plane
Bini, Dario A.
;Meini, Beatrice;
2020-01-01
Abstract
Quadratic matrix equations of the kind A_1 X^2 + A_0 X + 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
