Conference abstracts

Session B7 - Numerical Linear Algebra

July 14, 17:30 ~ 18:00

On the solution of quadratic matrix equations with infinite-dimensional structured coefficients

Beatrice Meini

University of Pisa, Italy   -   beatrice.meini@unipi.it

Matrix equations of the kind $A_{-1}+A_0 X+A_1 X^2=X$, where both the matrix coefficients and the unknown are semi-infinite matrices belonging to a Banach algebra, are considered. These equations, where coefficients are nonnegative quasi-Toeplitz matrices, are encountered in certain Quasi-Birth-Death (QBD) stochastic processes, as the tandem Jackson queue or the reflecting random walk in the quarter plane. We provide a numerical framework for approximating the minimal nonnegative solution of these equations which relies on semi-infinite quasi-Toeplitz matrix arithmetic. This arithmetic does not perform any finite size truncation, instead it approximates the infinite matrices through the sum of a banded Toeplitz matrix and a compact correction. In particular, we show that the algorithm of Cyclic Reduction can be effectively applied and can approximate the infinite dimensional solutions with quadratic convergence at a cost which is comparable to that of the finite case. This way, we may compute a finite approximation of the sought solution, as well as of the invariant probability measure of the associated QBD process, within a given accuracy. Numerical experiments, performed on a collection of benchmarks, confirm the theoretical analysis.

Joint work with D.A. Bini (University of Pisa), S. Massei (Scuola Normale Superiore, Pisa) and L. Robol (CNR, Pisa).

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