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A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils. (English) Zbl 0889.65036
The paper presents a new method to compute the eigenvalues of real Hamiltonians or symplectic pencils. The method is constructed in such a way that it applies to Hamiltonian matrices, symplectic matrices, Hamiltonian pencils and symplectic pencils. It is structure preserving, backward stable and needs \(O(n^3)\) floating point operations. The main elements of this approach are a new matrix decomposition which can be viewed as a symplectic URV decomposition, a periodic Schur decomposition for a product of two or four matrices, and the generalized Cayley transformation which allows a unified treatment of Hamiltonian and symplectic problems.
The paper is organized as follows. One introduces the notations and reviews some basic results, then one develops the theoretical basis for the new algorithm, and the procedure is described, with error analysis. A numerical example is displayed.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Software:
LAPACK
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