## Abstract

Matrix equations of the kind A_{1}X^{2}+A_{0}X+A_{−1}=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 quasi-Toeplitz matrices, are encountered in certain quasi-birth–death processes as the tandem Jackson queue or in any other processes that can be modeled as a reflecting random walk in the quarter plane. We provide a numerical framework for approximating the minimal nonnegative solution of these equations that relies on semi-infinite quasi-Toeplitz matrix arithmetic. 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 that is comparable to that of the finite case. This way, we may compute a finite approximation of the sought solution and of the invariant probability measure of the associated quasi-birth–death process, within a given accuracy. Numerical experiments, performed on a collection of benchmarks, confirm the theoretical analysis.

Original language | English |
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Article number | e2128 |

Journal | Numerical Linear Algebra with Applications |

Volume | 25 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:Copyright © 2017 John Wiley & Sons, Ltd.

Copyright:

Copyright 2019 Elsevier B.V., All rights reserved.

## Keywords

- cyclic reduction
- quadratic matrix equations
- quasi-birth-and-death processes
- Toeplitz matrices