Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems

Beyn W-J, Thümmler V (2009)
SIAM Journal on Matrix Analysis and Applications 31(3): 1361-1381.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Beyn, Wolf-JürgenUniBi; Thümmler, Vera
Abstract / Bemerkung
We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and parameterized nonlinear wave equations. The aim of the paper is to present and analyze a continuation method for invariant subspaces that belong to a group of eigenvalues, the number of which is much smaller than the dimension of the system. The continuation method is of predictor-corrector type, similar to the approach for the linear eigenvalue problem in [Beyn, Kless, and Thummler, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001], but we avoid linearizing the problem, which will double the dimension and change the sparsity pattern. The matrix equations that occur in the predictor and the corrector step are solved by a bordered version of the Bartels-Stewart algorithm. Furthermore, we set up an update procedure that handles the transition from real to complex conjugate eigenvalues, which occurs when eigenvalues from inside the continued cluster collide with eigenvalues from outside. The method is demonstrated on several numerical examples: a homotopy between random matrices, a fluid conveying pipe problem, and a traveling wave of a damped wave equation.
Stichworte
quadratic eigenvalue problem; numerical continuation; sparse matrices
Erscheinungsjahr
2009
Zeitschriftentitel
SIAM Journal on Matrix Analysis and Applications
Band
31
Ausgabe
3
Seite(n)
1361-1381
ISSN
0895-4798
Page URI
https://pub.uni-bielefeld.de/record/1796580

Zitieren

Beyn W-J, Thümmler V. Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications. 2009;31(3):1361-1381.
Beyn, W. - J., & Thümmler, V. (2009). Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 31(3), 1361-1381. https://doi.org/10.1137/080723107
Beyn, W. - J., and Thümmler, V. (2009). Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications 31, 1361-1381.
Beyn, W.-J., & Thümmler, V., 2009. Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 31(3), p 1361-1381.
W.-J. Beyn and V. Thümmler, “Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems”, SIAM Journal on Matrix Analysis and Applications, vol. 31, 2009, pp. 1361-1381.
Beyn, W.-J., Thümmler, V.: Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications. 31, 1361-1381 (2009).
Beyn, Wolf-Jürgen, and Thümmler, Vera. “Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems”. SIAM Journal on Matrix Analysis and Applications 31.3 (2009): 1361-1381.

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