Log In Sign Up

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

by   Dario A. Bini, et al.

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.


page 1

page 2

page 3

page 4


A computational framework for two-dimensional random walks with restarts

The treatment of two-dimensional random walks in the quarter plane leads...

Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations

We consider a class of linear matrix equations involving semi-infinite m...

A family of fast fixed point iterations for M/G/1-type Markov chains

We consider the problem of computing the minimal nonnegative solution G ...

Computing eigenvalues of semi-infinite quasi-Toeplitz matrices

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form A=T(a...

Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

We study solutions to the stochastic fixed point equation Xd=AX+B when t...