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In this paper we shall present two new algorithms for solution of the diserete-time algebraic Riccati equation. These algorithms are related to Potter's and to Laub's methods, but are based on the solution of a generalized rather than an ordinary eigenvalue problem. The key feature of the new algorithms is that the system transition matrix need not be inverted. Thus, the numerical problems associated with an ill-conditioned transition matrix do not arise and, moreover, the algorithm is directly applicable to problems with a singular transition matrix. Such problems arise commonly in practice when a continuous-time system with time delays is sampled.