On computing the eigenvalues of a symplectic pencil

The author presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete-time algebraic Riccati equation. The algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil. It requires about one-fourth the number of floating point operations that the QZ algorithm uses to compute the eigenvalues of the pencil directly. The proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by C. Van Loan (1984) for computing the eigenvalues of Hamiltonian matrices