An integral method for solving nonlinear eigenvalue problems

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 with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured nonlinear eigenvalue problems arise and some numerical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

For the nonlinear eigenvalue problem T(λ)x=0 we propose an iterative projection method for computing a few eigenvalues close to a given parameter. The current search space is expanded by a generalization of the shift-and-invert Arnoldi method. The resulting projected eigenproblems of small dimension are solved by inverse iteration. The method is applied to a rational eigenvalue problem governing damped vibrations of a structure.

https://tore.tuhh.de/bitstream/11420/147/1/rep56.pdf

An Arnoldi method for nonlinear eigenvalue problems

Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

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The nonlinear eigenvalue problem

http://eprints.ma.man.ac.uk/630/01/covered/MIMS_ep2006_391.pdf

A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems

The theory of Rayleigh functionals for non-linear eigenvalue problems T(λ) u = 0 is extended to cases where the functional is defined only on a proper subset. The theory applies to problems which do not satisfy an overdamping condition and yields a minimax characterization of eigenvalues. Applications to damped free vibrations of an elastic body are discussed.

B. Werner

Papers

A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems