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PreprintarXiv licencearXiv

Factorized Krylov subspace methods for solving large Sylvester equations

  • Yuki Satake
  • Takeshi Fukaya
  • Tomohiro Sogabe
  • Shao-Liang Zhang
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{satake2026factorized,
  title = {{Factorized Krylov subspace methods for solving large Sylvester equations}},
  author = {Yuki Satake and Takeshi Fukaya and Tomohiro Sogabe and Shao-Liang Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.28274},
  eprint = {2605.28274},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.28274},
}
RIS
TY  - UNPB
TI  - Factorized Krylov subspace methods for solving large Sylvester equations
AU  - Yuki Satake
AU  - Takeshi Fukaya
AU  - Tomohiro Sogabe
AU  - Shao-Liang Zhang
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2605.28274
UR  - https://arxiv.org/abs/2605.28274
AB  - 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.
ER  -