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In this work we consider a class of delay eigenvalue problems that admit aspectrum similar to that of a Hamiltonian matrix, in the sense that thespectrum is symmetric with respect to both the real and imaginary axis. Moreprecisely, we present a method to iteratively approximate the eigenvalues ofsuch delay eigenvalue problems closest to a given purely real or imaginaryshift, while preserving the symmetries of the spectrum. To this end thepresented method exploits the equivalence between the considered delayeigenvalue problem and the eigenvalue problem associated with a linear butinfinite-dimensional operator. To compute the eigenvalues closest to the givenshift, we apply a specifically chosen shift-invert transformation to thislinear operator and compute the eigenvalues with the largest modulus of the newshifted and inverted operator using an (infinite) Arnoldi procedure. Theadvantage of the chosen shift-invert transformation is that the spectrum of thetransformed 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 bilinearform. By taking this property into account during the orthogonalizationprocess, it is ensured that even in the presence of rounding errors, theobtained approximation for, e.g., a simple, purely imaginary eigenvalue issimple and purely imaginary. The presented work can thus be seen as anextension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods forComputing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils",SIAM J. Sci. Comput. (22.6), 2001], to the considered class of delay eigenvalueproblems. Although the presented method is initially defined on functionspaces, it can be implemented using finite dimensional linear algebraoperations.

A structure preserving shift-invert infinite Arnoldi algorithm for a class of delay eigenvalue problems with Hamiltonian symmetry