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 is about four times faster than the standard Q-R algorithm. The computed eigenvalues are shown to be the exact eigenvalues of a matrix M+E where $\Vert E \Vert$ depends on the square root of the machine precision. The accuracy of a computed eigenvalue depends on its condition and its magnitude, larger eigenvalues typically being more accurate.