Localization theorems for nonlinear eigenvalue problems
Provides theoretical tools for analyzing eigenvalue locations in nonlinear problems, benefiting researchers in numerical linear algebra and applied mathematics.
The paper develops new localization theorems for nonlinear eigenvalue problems, extending classical results like Gershgorin's theorem and the Bauer-Fike theorem. These results are applied to three example problems, demonstrating their utility in bounding eigenvalues.
Let $T : Ω\rightarrow \bbC^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $Ω\subset \bbC$. A point $λ\in Ω$ is an eigenvalue if the matrix $T(λ)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.