In this paper we present a numerical approach to the closed-loop spectrum of a discrete-time algebraic Riccati equation. The concerned symplectic pencil N - lambdaL is proven to be equivalent to the pencil P - lambdaQ, where P is skew -symmetric and Q is symmetric. Then, the HR process, which can be viewed as a generalization of the QR method, is applied to compute the eigenvalue of P - lambdaQ. Some numerical examples are included.