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PreprintCC BY-NC-ND 4.0arXiv

Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities

  • Vilhelm Peterson Lithell
  • Victor Janssens
  • Elias Jarlebring
  • Karl Meerbergen
  • Wim Michiels
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Abstract

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.

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Cite

BibTeXRIS
BibTeX
@misc{lithell2026conditioning,
  title = {{Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities}},
  author = {Vilhelm Peterson Lithell and Victor Janssens and Elias Jarlebring and Karl Meerbergen and Wim Michiels},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.20933},
  eprint = {2605.20933},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.20933},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities
AU  - Vilhelm Peterson Lithell
AU  - Victor Janssens
AU  - Elias Jarlebring
AU  - Karl Meerbergen
AU  - Wim Michiels
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2605.20933
UR  - https://arxiv.org/abs/2605.20933
AB  - We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.
ER  -