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Stable and effective restarting of the SS--RR and SS--RRR methods for computing partial eigenpairs of Hermitian matrices

  • Zhongxiao Jia
  • Tianhang Liu
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Abstract

We consider stable and effective restarts of the SS--RR and SS--RRR methods for computing the eigenpairs of a large Hermitian matrix A∈ℂn× n with the nev eigenvalues in a given interval. Taking the CJ--SS--RR and CJ--SS--RRR methods for instance, we prove that the commonly used restart scheme and its orthonormalized variant, named the natural restart schemes, are not numerically stable, thereby lowering attainable accuracy of approximate eigenpairs and missing some of the nev desired eigenpairs. To cure these deficiencies, we propose a new restart scheme, called the combination restart, that takes the n× ℓ starting matrix to be the product of the matrix, which consists of the selected nev approximate eigenvectors, and an nev× ℓ Gaussian random matrix. We prove the remarkable mathematical and numerical superiority of the combination restart over the natural restarts, then present a hybrid restart scheme that fully exploits the advantages of the natural restart and combination restart, and finally develop CJ--SS--RR and CJ--SS--RRR algorithms with the hybrid restart. Numerical experiments confirm that the new algorithms are numerically backward stable to the working precision and more efficient than the CJ--SS--RRR algorithm with the natural restart. The combination restart and hybrid restart schemes can be straightforwardly extended to the SS--RR and SS--RRR methods for general standard and generalized eigenvalue problems.

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Cite

BibTeXRIS
BibTeX
@misc{jia2026stable,
  title = {{Stable and effective restarting of the SS--RR and SS--RRR methods for computing partial eigenpairs of Hermitian matrices}},
  author = {Zhongxiao Jia and Tianhang Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14669},
  eprint = {2610.14669},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14669},
}
RIS
TY  - UNPB
TI  - Stable and effective restarting of the SS--RR and SS--RRR methods for computing partial eigenpairs of Hermitian matrices
AU  - Zhongxiao Jia
AU  - Tianhang Liu
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14669
UR  - https://arxiv.org/abs/2610.14669
AB  - We consider stable and effective restarts of the SS--RR and SS--RRR methods for computing the eigenpairs of a large Hermitian matrix A∈ℂ^n× n with the n_ev eigenvalues in a given interval. Taking the CJ--SS--RR and CJ--SS--RRR methods for instance, we prove that the commonly used restart scheme and its orthonormalized variant, named the natural restart schemes, are not numerically stable, thereby lowering attainable accuracy of approximate eigenpairs and missing some of the n_ev desired eigenpairs. To cure these deficiencies, we propose a new restart scheme, called the combination restart, that takes the n× ℓ starting matrix to be the product of the matrix, which consists of the selected n_ev approximate eigenvectors, and an n_ev× ℓ Gaussian random matrix. We prove the remarkable mathematical and numerical superiority of the combination restart over the natural restarts, then present a hybrid restart scheme that fully exploits the advantages of the natural restart and combination restart, and finally develop CJ--SS--RR and CJ--SS--RRR algorithms with the hybrid restart. Numerical experiments confirm that the new algorithms are numerically backward stable to the working precision and more efficient than the CJ--SS--RRR algorithm with the natural restart. The combination restart and hybrid restart schemes can be straightforwardly extended to the SS--RR and SS--RRR methods for general standard and generalized eigenvalue problems.
ER  -