A structure-preserving algorithm for the linear lossless dissipative Hamiltonian eigenvalue problem

In this paper, we propose a structure-preserving algorithm for computing all eigenvalues of the generalized eigenvalue problem $BA\mathrm{x} =\lambda E \mathrm{x}$ that arises in linear lossless dissipative Hamiltonian descriptor systems, with $B$ being skew-symmetric and $A^\top E = E^\top A$. We rewrite the problem as $BAE^{-1} \mathrm{y} = \lambda \mathrm{y}$ to preserve the symmetry of $A^\top E$ and convert the problem into the equivalent $\top$-Hamiltonian eigenvalue problem $\mathscr{H} \mathrm{z} = \lambda \mathrm{z}$. Furthermore, $\top$-symplectic URV decomposition and a corresponding periodic QR (PQR) method are proposed to compute all eigenvalues of $\mathscr{H}$. The structure-preserving property ensures that the computed eigenvalues appear pairwise, in the form$ (\lambda , -\lambda)$, as they should. Numerical experiments show that the computed eigenvalues are more accurate and strictly paired than those of the classical QZ method, while the residuals of the eigenpairs are comparable.

Keywords

structure-preserving algorithm, $\top$-Hamiltonian eigenvalue problem, $\top$-symplectic URV decomposition, periodic QR.

Full Text (PDF format)

The author was supported in part by the National Natural Science Foundation of China (NSFC) 11971105.

Received 12 January 2022

Accepted 21 January 2022

Published 7 April 2022