A second-order convergent algorithm is presented which gives an approximation to the unique positive definite solution of the continuous-time algebraic Riccati equation (CARE). First the CARE is transformed into the discrete-time algebraic Riccati equation (DARE) using the transformation given by Hitz and Anderson (1972). Then the discrete-time doubling algorithm, whose initial values are expressed in forms suitable for computation, is applied to the DARE. Next, it is shown that this algorithm is convergent under the condition that the CARE has the unique positive definite solution. Finally an ‘inverse’ of the Hitz and Anderson (1972) transformation is presented, which transforms the DARE into the CARE. It is proved that the ‘inverse’ transformation preserves the stabilizability, controllability, detectability and observability of the DARE.