A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix

A fast method for computing all the eigenvalues of a Hamiltonian matrix M is given. The method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties. The algorithm requires about one-fourth the number of floating-point operations and one-half the space of the standard QR algorithm. The computed eigenvalues are shown to be the exact eigenvalues of a matrix M + E where ∥E∥ depends on the square root of the machine precision. The accuracy of a computed eigenvalue depends on both its condition and its magnitude, larger eigenvalues typically being more accurate.