A Structure Preserving Shift-Invert Infinite Arnoldi Algorithm for a Class of Delay Eigenvalue Problems with Hamiltonian Symmetry

Abstract

This work considers 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 axes. We propose a method to iteratively approximate the eigenvalues closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end, our 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 subspace 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, SIAM J. Sci. Comput., 22 (2001), pp. 1905–1925] 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. The performance of the resulting numerical algorithm is verified for two example problems: the first example illustrates the advantage of the proposed approach in preserving purely imaginary eigenvalues when working in finite precision, while the second one demonstrates its applicability to a large scale problem.