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 BAx = lambda Ex that arises in linear lossless dissipative Hamiltonian descriptor systems, with B being skew-symmetric and A(T)E = E(T)A. We rewrite the problem as BAE(-1)y = lambda y to preserve the symmetry of A(T)E and convert the problem into the equivalent T-Hamiltonian eigenvalue problem Hz = lambda z. Furthermore, T-symplectic URV decomposition and a corresponding periodic QR (PQR) method are proposed to compute all eigenvalues of H. The structurepreserving 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.