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

In this work we consider a class of non-linear 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 non-linear 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 non-linear 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 in 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 non-linear eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite dimensional linear algebra operations.

## Authors

• 3 publications
• 5 publications
• ### Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems

When computing the eigenstructure of matrix pencils associated with the ...
05/10/2020 ∙ by Volker Mehrmann, et al. ∙ 0

• ### Spectrum-based stability analysis and stabilization of a class of time-periodic time delay systems

We develop an eigenvalue-based approach for the stability assessment and...
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• ### Eigenvalues of the non-backtracking operator detached from the bulk

We describe the non-backtracking spectrum of a stochastic block model wi...
07/12/2019 ∙ by Simon Coste, et al. ∙ 0

• ### A rational Even-IRA algorithm for the solution of T-even polynomial eigenvalue problems

In this work we present a rational Krylov subspace method for solving re...
09/03/2020 ∙ by Peter Benner, et al. ∙ 0

• ### Fast optical absorption spectra calculations for periodic solid state systems

We present a method to construct an efficient approximation to the bare ...
07/05/2019 ∙ by F. Henneke, et al. ∙ 0

• ### Electromagnetic Stekloff eigenvalues: approximation analysis

We continue the work of [Camano, Lackner, Monk, SIAM J. Math. Anal., Vol...
09/02/2019 ∙ by Martin Halla, et al. ∙ 0

• ### Simplified Eigenvalue Analysis for Turbomachinery Aerodynamics with Cyclic Symmetry

Eigenvalue analysis is widely used for linear instability analysis in bo...
12/18/2019 ∙ by Shenren Xu, et al. ∙ 0

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