Factorized Krylov subspace methods for solving large Sylvester equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917540209360896 |
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| author | Satake, Yuki Fukaya, Takeshi Sogabe, Tomohiro Zhang, Shao-Liang |
| author_facet | Satake, Yuki Fukaya, Takeshi Sogabe, Tomohiro Zhang, Shao-Liang |
| contents | Krylov subspace methods, such as the Conjugate Gradient (CG) and BiCGSTAB methods, are widely used in scientific computing for solving linear systems. In this study, we propose a new framework for solving large Sylvester equations in a low-rank format by reconstructing matrix-oriented Krylov subspace methods. The framework realizes efficient algorithms that are mathematically equivalent to the matrix-oriented Krylov subspace methods by exploiting the mathematical properties of the Sylvester operator and the low-rank structure of the right-hand side. Specifically, by leveraging these properties, approximate solutions can be expressed in a low-rank factorized form, enabling efficient computation and reduced memory requirements. The effectiveness of our algorithms is demonstrated through numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Factorized Krylov subspace methods for solving large Sylvester equations Satake, Yuki Fukaya, Takeshi Sogabe, Tomohiro Zhang, Shao-Liang Numerical Analysis Krylov subspace methods, such as the Conjugate Gradient (CG) and BiCGSTAB methods, are widely used in scientific computing for solving linear systems. In this study, we propose a new framework for solving large Sylvester equations in a low-rank format by reconstructing matrix-oriented Krylov subspace methods. The framework realizes efficient algorithms that are mathematically equivalent to the matrix-oriented Krylov subspace methods by exploiting the mathematical properties of the Sylvester operator and the low-rank structure of the right-hand side. Specifically, by leveraging these properties, approximate solutions can be expressed in a low-rank factorized form, enabling efficient computation and reduced memory requirements. The effectiveness of our algorithms is demonstrated through numerical experiments. |
| title | Factorized Krylov subspace methods for solving large Sylvester equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2605.28274 |