A Block-Shifted Cyclic Reduction Algorithm for Solving a Class of Quadratic Matrix Equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918454068510720 |
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| author | Li, Xu Meini, Beatrice |
| author_facet | Li, Xu Meini, Beatrice |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Block-Shifted Cyclic Reduction Algorithm for Solving a Class of Quadratic Matrix Equations Li, Xu Meini, Beatrice Numerical Analysis 15A24, 65F45, 65B99 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. |
| title | A Block-Shifted Cyclic Reduction Algorithm for Solving a Class of Quadratic Matrix Equations |
| topic | Numerical Analysis 15A24, 65F45, 65B99 |
| url | https://arxiv.org/abs/2511.02598 |