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

Xu Li, Beatrice Meini

TL;DR

This work addresses solving quadratic matrix equations arising in quasi-birth-death processes, where classical cyclic reduction can fail when more than one eigenvalue lies on the unit circle. The authors propose the Block-Shifted CR (BS-CR) algorithm, which integrates singular value decomposition with block shift-and-deflate to separate invariant subspaces corresponding to eigenvalues inside and on the unit circle, reducing the core difficulty to a small $ ext{$ ext{ell} imes ext{ell}$}$ quadratic matrix equation solved via the QZ algorithm. Their approach yields a principled deflation of non-unit-circle eigenvalues and a reduced system that preserves the critical unit-circle spectrum, enabling reliable convergence and improved accuracy. Numerical experiments on null-recurrent QBD-related QMEs demonstrate that BS-CR achieves higher accuracy with fewer iterations compared to CR, S-CR, and fixed-point iterations, highlighting its practical value for large-scale problems and challenging spectral configurations. The method thus broadens the class of QMEs that can be solved efficiently and robustly in applications such as stochastic models and control, where unit-circle eigenvalues complicate classical solvers.

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.

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

TL;DR

This work addresses solving quadratic matrix equations arising in quasi-birth-death processes, where classical cyclic reduction can fail when more than one eigenvalue lies on the unit circle. The authors propose the Block-Shifted CR (BS-CR) algorithm, which integrates singular value decomposition with block shift-and-deflate to separate invariant subspaces corresponding to eigenvalues inside and on the unit circle, reducing the core difficulty to a small ext{ell} imes ext{ell} quadratic matrix equation solved via the QZ algorithm. Their approach yields a principled deflation of non-unit-circle eigenvalues and a reduced system that preserves the critical unit-circle spectrum, enabling reliable convergence and improved accuracy. Numerical experiments on null-recurrent QBD-related QMEs demonstrate that BS-CR achieves higher accuracy with fewer iterations compared to CR, S-CR, and fixed-point iterations, highlighting its practical value for large-scale problems and challenging spectral configurations. The method thus broadens the class of QMEs that can be solved efficiently and robustly in applications such as stochastic models and control, where unit-circle eigenvalues complicate classical solvers.

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.

Paper Structure

This paper contains 13 sections, 9 theorems, 87 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and assumptions
  3. Cyclic Reduction (CR)
  4. Assumptions
  5. Block shift-and-deflate technique
  6. Deflation of the eigenvalues not lying on the unit circle
  7. Convergence properties of CR in the case of more than one eigenvalue on the unit circle
  8. Combination of CR, block shift-and-deflate, and QZ algorithm
  9. Numerical results
  10. Example 1
  11. Example 2
  12. Example 3
  13. Conclusions

Key Result

Theorem 1

Assume that $|\lambda_m|<1<|\lambda_{m+1}|$ and the QME eq:qme and $A_0 X^2 + A_1 X+ A_2=0$ have solution $G$ and $S$, respectively, with $\rho(G)<1$ and $\rho(S)<1$. If CR can be carried out without a breakdown, then where $G_k=-\left(\widehat{A}_{1}^{(k)}\right)^{-1}A_{0}$ and $R_k=-A_2 \left(\widehat{A}_{1}^{(k)}\right)^{-1}$.

Figures (2)

  • Figure 1: Residual errors for the different versions of CR applied to Example 2
  • Figure 2: CPU time and number of iterations for the different versions of CR applied to Example 2

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 4
  • proof
  • Proposition 3
  • proof
  • ...and 3 more