Localization Theorems for Nonlinear Eigenvalue Problems

Localization Theorems for Nonlinear Eigenvalue Problems

Abstract. Let T : Ω → C be a matrix-valued function that is analytic on some simplyconnected domain Ω ⊂ C. A point λ ∈ 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 non...

Nonlinear Eigenvalue Problems

Heinrich Voss Hamburg University of Technology 115.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-2 115.2 Analytic matrix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-3 115.3 Variational Characterization of Eigenvalues . . . . . . . . 115-7 115.4 General Rayleigh Functionals . . . . . . . . . . . . . . . . . . . ...

Solving nonlinear eigenvalue problems

Chebyshev interpolation for nonlinear eigenvalue problems

This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix valued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE-eigenvalue problems and typically exclude the use of establis...

Nonlinear Eigenvalue Problems: A Challenge for Modern Eigenvalue Methods

We discuss the state of the art in numerical solution methods for large scale polynomial or rational eigenvalue problems. We present the currently available solution methods such as the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and wi...