Nonlinear eigenvalue problems

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

Two-parametric nonlinear eigenvalue problems

Eigenvalue problems of the form x′′ = −λf(x) + μg(x−), (i), x(0) = 0, x(1) = 0, (ii) are considered, where x and x− are the positive and negative parts of x respectively. We are looking for (λ, μ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuč́ık problem for piece-wise linear equations. In our considerations functions f and g may be nonlinear f...

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...

Nonlinear Eigenvalue Problems with Specified Eigenvalues

This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T , we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. While the usual resolvent norm addresses this question for a single eigenvalue of multiplicity one, the general sett...