A method for solving the spectral problem of Hamiltonian matrices with application to the algebraic Riccati equation

In this paper an effective iterative method for computing the eigenvalues and eigenvectors of a real Hamiltonian matrix is described and its applicability discussed. The method is an adaptation for Hamiltonian matrices of the methods for computing eigenvalues of real matrices due to Veselić and Voevodin. It uses symplectic similarity transformations and preserves the Hamiltonian structure of the matrix. Out method can be used for solving algebraic Riccati equation. The method is tested numerically and a comparison with the performance of other numerical algorithms is presented.