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Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems

Beyn, Wolf-Jürgen ; Effenberger, Cedric ; Kressner, Daniel

In: Numerische Mathematik, 2011, vol. 119, no. 3, p. 489-516

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    Title
    • Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems
    Author
    • Beyn, Wolf-Jürgen. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany
    • Effenberger, Cedric. Seminar for Applied Mathematics, D-MATH, ETH Zurich, Raemistrasse 101, 8092, Zurich, Switzerland
    • Kressner, Daniel. MATHICSE, EPF Lausanne, Station 8, 1015, Lausanne, Switzerland
    Document Type
    • Postprint
    Classification
    • Computer science
    OAI-PMH Identifier
    • oai:doc.rero.ch:318466
    Summary
    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