An algorithm to solve continuous-time algebraic Riccati equations through the Hamiltonian Schur form is developed. It is an adaption for Hamiltonian matrices of an unsymmetric Jacobi method of Eberlein [15]. It uses unitary symplectic similarity transformations and preserves the Hamiltonian structure of the matrix, Each iteration step needs only local information about the current matrix, thus admitting efficient parallel implementations on certain parallel architectures, Convergence performance of the algorithm is compared with the Hamiltonian-Jacobi algorithm of Byers [12], The numerical experiments suggest that the method presented here converges considerably faster for non-Hermitian Hamiltonian matrices than Byers’ Hamiltonian-Jacobi algorithm, Besides that, numerical experiments suggest that for the method presented here, the number of iterations needed for convergence can be predicted by a simple function of the matrix size.