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This paper proposes an efficient numerical algorithm to compute the optimal input distribution that maximizes the sum capacity of a Gaussian multiple-access channel with vector inputs and a vector output. The numerical algorithm has an iterative water-filling interpretation. The algorithm converges from any starting point, and it reaches within 1/2 nats per user per output dimension from the sum capacity after just one iteration. The characterization of sum capacity also allows an upper bound and a lower bound for the entire capacity region to be derived.