SIAM Journal on Applied Mathematics
Krylov subspace methods for solving large Lyapunov equations
SIAM Journal on Numerical Analysis
Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics
SIAM Journal on Matrix Analysis and Applications
Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems
Applied Numerical Mathematics
Matrix computations (3rd ed.)
SIAM Journal on Numerical Analysis
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Elements of applied bifurcation theory (2nd ed.)
Elements of applied bifurcation theory (2nd ed.)
Using Generalized Cayley Transformations within an Inexact Rational Krylov Sequence Method
SIAM Journal on Matrix Analysis and Applications
Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils
SIAM Journal on Scientific Computing
Numerical methods for bifurcations of dynamical equilibria
Numerical methods for bifurcations of dynamical equilibria
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
A Jacobi--Davidson Type Method for a Right Definite Two-Parameter Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Defining Functions for Multiple Hopf Bifurcations
SIAM Journal on Numerical Analysis
A Jacobi--Davidson Type Method for the Two-Parameter Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Fast Methods for Estimating the Distance to Uncontrollability
SIAM Journal on Matrix Analysis and Applications
A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations
SIAM Journal on Scientific Computing
Use of near-breakdowns in the block Arnoldi method for solving large Sylvester equations
Applied Numerical Mathematics
Stability and Stabilization of Time-Delay Systems (Advances in Design & Control) (Advances in Design and Control)
Convergence Analysis of Projection Methods for the Numerical Solution of Large Lyapunov Equations
SIAM Journal on Numerical Analysis
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The detection of a Hopf bifurcation in a large-scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues of a sequence of large sparse eigenvalue problems. Guckenheimer, Gueron, and Harris-Warrick [SIAM J. Numer. Anal., 34 (1997), pp. 1-21] proposed a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilizes a certain sum of Kronecker products and involves the solution of matrices of squared dimension, which is impractical for large-scale applications. However, if good starting guesses are available for the parameter and the purely imaginary eigenvalue at the Hopf point, then efficient algorithms are available. In this paper, we propose a method for obtaining such good starting guesses, based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem (possibly arising after a linearization process). The problem is formulated as an inexact inverse iteration method that requires the solution of a sequence of Lyapunov equations with low rank right-hand sides. It is this last fact that makes the method feasible for large systems. The power of our method is tested on four numerical examples.