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Computational Technology Reviews
ISSN 2044-8430
Computational Technology Reviews
Volume 2, 2010
Large Sparse Eigenvalue Solvers for the Determination of Hopf Bifurcations of Nonlinear Dynamical Systems
K. Meerbergen

Department of Computer Science, Katholieke Universiteit Leuven, Belgium

Full Bibliographic Reference for this paper
K. Meerbergen, "Large Sparse Eigenvalue Solvers for the Determination of Hopf Bifurcations of Nonlinear Dynamical Systems", Computational Technology Reviews, vol. 2, pp. 223-246, 2010. doi:10.4203/ctr.2.9
Keywords: algebraic eigenvalue problems, Arnoldi method, implicit restarts, dynamical systems, inverse iteration, shift-and-invert transform, Lyapunov equation.

Summary
We review methods for computing the right-most eigenvalues of a matrix pair. Applications are the stability analysis of dynamical systems. During the last two decades, iterative solvers for computing a few eigenvalues of the nonsymmetric algebraic eigenvalue problem have been developed. Best known are the Krylov methods, in particular the Arnoldi method. We review these methods with the application of the determination of right-most eigenvalues in mind. We show that these methods are reliable if they are used carefully.

In addition to a review of eigenvalue solvers, we will discuss some new ideas that have been proposed the last five years. They are based on parameterized eigenvalue problems. We will mainly discuss the most recent idea that directly computes the values of the parameter whose Jacobian matrix has a pair of purely imaginary eigenvalues, rather than monitoring eigenvalues as a function of the parameter.

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