A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

Abstract.

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order $\sqrt{\varepsilon}$, where $\varepsilon$ is the machine precision, the new method computes the eigenvalues to full possible accuracy.