A Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants.

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 with 2 × 2 operator determinants of the form (B1 ⊗ A2 - A1 ⊗ B2)z = µ B1 ⊗ C2 - C1 ⊗ B2)z. 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. [ABSTRACT FROM AUTHOR]