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Numerical Solution of Large-Scale Low-Rank Coupled Generalized Sylvester Equations

  • Umair Zulfiqar
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Abstract

This paper develops projection methods for computing low-rank solutions to large-scale coupled generalized Sylvester equations. The dominant subspaces of these solutions are shown to be spanned by specific resolvent directions, motivating a Petrov--Galerkin projection framework. Conditions for the unique solvability of the projected coupled generalized Sylvester equations are also discussed. Based on these dominant subspaces, two algorithms are proposed: an extended Krylov-based projection algorithm and a rational Krylov-based projection algorithm. The former uses fixed shifts, whereas the latter uses adaptive shifts. Numerical experiments on large-scale benchmark problems demonstrate that the proposed methods achieve high accuracy and fast convergence.

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Cite

BibTeXRIS
BibTeX
@misc{zulfiqar2026numerical,
  title = {{Numerical Solution of Large-Scale Low-Rank Coupled Generalized Sylvester Equations}},
  author = {Umair Zulfiqar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30804},
  eprint = {2610.30804},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30804},
}
RIS
TY  - UNPB
TI  - Numerical Solution of Large-Scale Low-Rank Coupled Generalized Sylvester Equations
AU  - Umair Zulfiqar
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30804
UR  - https://arxiv.org/abs/2610.30804
AB  - This paper develops projection methods for computing low-rank solutions to large-scale coupled generalized Sylvester equations. The dominant subspaces of these solutions are shown to be spanned by specific resolvent directions, motivating a Petrov--Galerkin projection framework. Conditions for the unique solvability of the projected coupled generalized Sylvester equations are also discussed. Based on these dominant subspaces, two algorithms are proposed: an extended Krylov-based projection algorithm and a rational Krylov-based projection algorithm. The former uses fixed shifts, whereas the latter uses adaptive shifts. Numerical experiments on large-scale benchmark problems demonstrate that the proposed methods achieve high accuracy and fast convergence.
ER  -