A Sylvester–Arnoldi type method for the generalized eigenvalue problem with two‐by‐two operator determinants
Copyright policy )A Sylvester–Arnoldi type method for the generalized eigenvalue problem with two‐by‐two operator determinants
SummaryIn 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 with 2 × 2 operator determinants of the form urn:x-wiley:nla:media:nla2005:nla2005-math-0001 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. Copyright © 2015 John Wiley & Sons, Ltd.
- KU Leuven Belgium
- University of Copenhagen Faculty of Science Denmark
- University of Ljubljana Slovenia
Numerical computation of eigenvalues and eigenvectors of matrices, Iterative numerical methods for linear systems, Eigenvalues, singular values, and eigenvectors, inverse iteration, Mathematics, Applied, Numerical & Computational Mathematics, separable boundary value problems, algorithms, Arnoldi method, 0102 Applied Mathematics, subspace iteration, 4901 Applied mathematics, Hopf bifurcation, Matlab, operator determinants, low-rank approximation, generalized eigenvalue problem, numerical examples, Science & Technology, 0103 Numerical and Computational Mathematics, Matrix equations and identities, Mathieu's system, Sylvester equation, Bartels-Stewart algorithm, Physical Sciences, 4903 Numerical and computational mathematics, Krylov-Schur iterations, Mathematics, two-parameter eigenvalue problem
Numerical computation of eigenvalues and eigenvectors of matrices, Iterative numerical methods for linear systems, Eigenvalues, singular values, and eigenvectors, inverse iteration, Mathematics, Applied, Numerical & Computational Mathematics, separable boundary value problems, algorithms, Arnoldi method, 0102 Applied Mathematics, subspace iteration, 4901 Applied mathematics, Hopf bifurcation, Matlab, operator determinants, low-rank approximation, generalized eigenvalue problem, numerical examples, Science & Technology, 0103 Numerical and Computational Mathematics, Matrix equations and identities, Mathieu's system, Sylvester equation, Bartels-Stewart algorithm, Physical Sciences, 4903 Numerical and computational mathematics, Krylov-Schur iterations, Mathematics, two-parameter eigenvalue problem
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