Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities
Provides theoretical tools for analyzing sensitivity and stability in a specific class of nonlinear eigenvalue problems, which is incremental for numerical linear algebra researchers.
The paper derives explicit and computable expressions for eigenvalue condition numbers and backward errors in symmetric nonlinear eigenvalue problems with eigenvector nonlinearities, showing that these problems require additional care compared to linear or eigenvalue-nonlinear cases.
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.