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The nonlinear eigenvalue problem*

Published online by Cambridge University Press:  05 May 2017

Stefan Güttel
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK E-mail: stefan.guettel@manchester.ac.uk, francoise.tisseur@manchester.ac.uk
Françoise Tisseur
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK E-mail: stefan.guettel@manchester.ac.uk, francoise.tisseur@manchester.ac.uk

Abstract

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.

Type
Research Article
Copyright
© Cambridge University Press, 2017 

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