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Analytical results for the average sum rate achievable in the Multiple-Input Multiple-Output (MIMO) broadcast channel with algorithms relying on full channel state information at the transmitter are hard to obtain in practice. In the large system limit, when the number of transmit and receive antennas goes to infinity at a finite fixed ratio, however, the eigenvalues of many random matrices become deterministic and analytical expressions for the sum rate can be derived in some cases. In this paper we will present large system expressions for the sum rate for three sub-optimum algorithms, namely the Successive Encoding Successive Allocation Method (SESAM), Block Diagonalization and Block Diagonalization with Dirty Paper Coding. In case the large system limit of the sum rate does not exist, we derive lower bounds. By simulation results it is shown that the asymptotic results serve as a good approximation of the system performance with finite system parameters of reasonable size.