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Kavcic proposed in (A. Kavcic, 2001) an algorithm that apparently finds the mutual-information-rate-maximizing parameters of a Markov source at the input to an indecomposable finite-state channel. In this paper we prove that the stationary points of this algorithm indeed correspond one-to-one to the critical points of the information-rate curve. Kavcic's algorithm can be considered as a generalized Blahut-Arimoto algorithm, as it includes as special cases the classical Blahut-Arimoto algorithm for discrete memoryless channels (DMCs) and the solution to finding the capacity-achieving input distribution for finite-state channels with no noise (C.E. Shannon, 1948).