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

TL;DR

This paper introduces a Block-Shifted Cyclic Reduction algorithm that enhances the solution of quadratic matrix equations, especially when traditional methods struggle with eigenvalues on the unit circle.

Contribution

The paper presents a novel Block-Shifted CR algorithm using SVD and shift-and-deflate techniques to improve convergence and applicability for quadratic matrix equations.

Findings

The new method extends the class of solvable quadratic matrix equations.

Numerical experiments show improved robustness and effectiveness.

The approach addresses convergence issues related to eigenvalues on the unit circle.

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.