Complex bifurcation from real paths
SIAM Journal on Applied Mathematics
Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
Stable solvers and block elimination for bordered systems
SIAM Journal on Matrix Analysis and Applications
Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Stability of travelling waves for a damped hyperbolic equation
Zeitschrift für Angewandte Mathematik und Physik (ZAMP)
Numerical methods for bifurcations of dynamical equilibria
Numerical methods for bifurcations of dynamical equilibria
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
Computing Connecting Orbits via an Improved Algorithm for Continuing Invariant Subspaces
SIAM Journal on Scientific Computing
The Quadratic Eigenvalue Problem
SIAM Review
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs
ACM Transactions on Mathematical Software (TOMS)
SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
The Conditioning of Linearizations of Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
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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 Thümmler, 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.