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arXiv · 2511.02598

A Block-Shifted Cyclic Reduction Algorithm for Solving a Class of Quadratic Matrix Equations

Xu Li·Beatrice Meini
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Abstract

The cyclic reduction (CR) algorithm is an efficient method for solving quadratic matrix equations that arise in quasi-birth-death (QBD) stochastic processes. However, its convergence is not guaranteed when the associated matrix polynomial has more than one eigenvalue on the unit circle. To address this limitation, we introduce a novel iteration method, referred to as the Block-Shifted CR algorithm, that improves the CR algorithm by utilizing singular value decomposition (SVD) and block shift-and-deflate techniques. This new approach extends the applicability of existing solvers to a broader class of quadratic matrix equations. Numerical experiments demonstrate the effectiveness and robustness of the proposed method.

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Xu Li, Beatrice Meini. 2025-11-04. A Block-Shifted Cyclic Reduction Algorithm for Solving a Class of Quadratic Matrix Equations. https://doi.org/10.1016/j.laa.2026.04.010

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