Title: A Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants
Authors: Meerbergen, Karl
Plestenjak, Bor
Issue Date: Aug-2014
Publisher: Department of Computer Science, KU Leuven
Series Title: TW Reports vol:TW653
Abstract: In various applications, for instance in the detection of a Hopf bifurcation or in solving separable boundary value problems using the two-parameter eigenvalue problem, one has to solve a generalized eigenvalue problem of the form

(B1 x A2 - A1 x B2) z = μ ( B1 x C2 - C1 x B2) z

where matrices are 2 × 2 operator determinants. We present efficient methods that can be used to compute a small subset of the eigenvalues. For full matrices of moderate size we propose either the standard implicitly restarted Arnoldi or Krylov--Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov-Schur method for the projected problems.
Publication status: published
KU Leuven publication type: IR
Appears in Collections:Numerical Analysis and Applied Mathematics Section

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