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Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems

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Abstract

Invariant pairs have been proposed as a numerically robust means to represent and compute several eigenvalues along with the corresponding (generalized) eigenvectors for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In this work, we consider nonlinear eigenvalue problems that depend on an additional parameter and our interest is to track several eigenvalues as this parameter varies. Based on the concept of invariant pairs, a theoretically sound and reliable numerical continuation procedure is developed. Particular attention is paid to the situation when the procedure approaches a singularity, that is, when eigenvalues included in the invariant pair collide with other eigenvalues. For the real generic case, it is proven that such a singularity only occurs when two eigenvalues collide on the real axis. It is shown how this situation can be handled numerically by an appropriate expansion of the invariant pair. The viability of our continuation procedure is illustrated by a numerical example.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Wolf-Jürgen Beyn

  2. Seminar for Applied Mathematics, D-MATH, ETH Zurich, Raemistrasse 101, 8092, Zurich, Switzerland

    Cedric Effenberger

  3. MATHICSE, EPF Lausanne, Station 8, 1015, Lausanne, Switzerland

    Daniel Kressner

Authors
  1. Wolf-Jürgen Beyn
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  2. Cedric Effenberger
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  3. Daniel Kressner
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Corresponding author

Correspondence to Daniel Kressner.

Additional information

W.-J. Beyn and D. Kressner were supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. C. Effenberger was supported by the SNF research module Robust numerical methods for solving nonlinear eigenvalue problems within the SNF ProDoc Efficient Numerical Methods for Partial Differential Equations.

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Beyn, WJ., Effenberger, C. & Kressner, D. Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems. Numer. Math. 119, 489–516 (2011). https://doi.org/10.1007/s00211-011-0392-1

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  • DOI: https://doi.org/10.1007/s00211-011-0392-1

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