A Sylvester–Arnoldi type method for the generalized eigenvalue problem with two‐by‐two operator determinants

Abstract

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.