An iterative algorithm for computing probability distributions and rates for lossless source coding with compressed side information is presented.  The algorithm is based primarily on the Blahut-Arimoto algorithm for computing channel capacity and the Blahut algorithm for computing the rate-distortion function.  The objective is reformulated as maximizing the mutual information between the variable to be compressed (X) and an auxiliary random variable (U) subject to an upper bound on the mutual information between the side information (Y) and U and subject to U-Y-X being a Markov chain in that  order.  The results build on the partial characterization presented by Marco and Effros in 2006.  Examples and numerical issues are addressed.  Possible extensions to other problems whose rates tradeoff two mutual informations are also discussed.