Contents Online

# Annals of Mathematical Sciences and Applications

## Volume 7 (2022)

### Number 1

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

Pages: 3 – 19

DOI: https://dx.doi.org/10.4310/AMSA.2022.v7.n1.a1

#### Author

#### Abstract

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.

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