In this work we consider a class of delay eigenvalue problems that admit a
spectrum similar to that of a Hamiltonian matrix, in the sense that the
spectrum is symmetric with respect to both the real and imaginary axis. More
precisely, we present a method to iteratively approximate the eigenvalues of
such delay eigenvalue problems closest to a given purely real or imaginary
shift, while preserving the symmetries of the spectrum. To this end the
presented method exploits the equivalence between the considered delay
eigenvalue problem and the eigenvalue problem associated with a linear but
infinite-dimensional operator. To compute the eigenvalues closest to the given
shift, we apply a specifically chosen shift-invert transformation to this
linear operator and compute the eigenvalues with the largest modulus of the new
shifted and inverted operator using an (infinite) Arnoldi procedure. The
advantage of the chosen shift-invert transformation is that the spectrum of the
transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore,
it is proven that the Krylov space constructed by applying this operator,
satisfies an orthogonality property in terms of a specifically chosen bilinear
form. By taking this property into account during the orthogonalization
process, it is ensured that even in the presence of rounding errors, the
obtained approximation for, e.g., a simple, purely imaginary eigenvalue is
simple and purely imaginary. The presented work can thus be seen as an
extension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods for
Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils",
SIAM J. Sci. Comput. (22.6), 2001], to the considered class of delay eigenvalue
problems. Although the presented method is initially defined on function
spaces, it can be implemented using finite dimensional linear algebra
operations