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PreprintCC BY 4.0arXiv

Compensated Splitting For Generalized Lyapunov Equations

  • Hongjia Chen
  • Ren-Cang Li
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Abstract

The generalized Lyapunov equation has a natural splitting that leads to the standard fixed-point iteration (sFPI) for its numerical solution. sFPI is convergent, for an arbitrarily given initial guess, if and only if the spectral radius of the associated linear operator is smaller than $1$. This means that sFPI may diverge if the spectral radius is $1$ or bigger. In this paper, we propose a compensated splitting scheme that aims to reduce the spectral radius so that the resulting compensated fixed-point iteration (cFPI) has a better convergence property, namely, cFPI may still converge even if sFPI does not or cFPI converges faster than sFPI does when the latter is also convergent. Numerical results are presented to demonstrate the superiority of cFPI to sFPI.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026compensated,
  title = {{Compensated Splitting For Generalized Lyapunov Equations}},
  author = {Hongjia Chen and Ren-Cang Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.22245},
  eprint = {2606.22245},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.22245},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Compensated Splitting For Generalized Lyapunov Equations
AU  - Hongjia Chen
AU  - Ren-Cang Li
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2606.22245
UR  - https://arxiv.org/abs/2606.22245
AB  - The generalized Lyapunov equation has a natural splitting that leads to the standard fixed-point iteration (sFPI) for its numerical solution. sFPI is convergent, for an arbitrarily given initial guess, if and only if the spectral radius of the associated linear operator is smaller than $1$. This means that sFPI may diverge if the spectral radius is $1$ or bigger. In this paper, we propose a compensated splitting scheme that aims to reduce the spectral radius so that the resulting compensated fixed-point iteration (cFPI) has a better convergence property, namely, cFPI may still converge even if sFPI does not or cFPI converges faster than sFPI does when the latter is also convergent. Numerical results are presented to demonstrate the superiority of cFPI to sFPI.
ER  -