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In this paper, we study the capacity-achieving input covariance matrices for the multiuser multiple-input multiple-output (MIMO) uplink channel under jointly-correlated Rician fading when perfect channel state information (CSI) is known at the receiver, or CSIR while only statistical CSI at the transmitter, or CSIT, is available. The jointly-correlated MIMO channel (or the Weichselberger model) accounts for the correlation at two link ends and is shown to be highly accurate to model real channels. Classically, numerical techniques together with Monte-Carlo methods (named stochastic programming) are used to resolve the problem concerned but at a high computational cost. To tackle this, we derive the asymptotic sum-rate of the multiuser (MU) MIMO uplink channel in the large-system regime where the numbers of antennas at the transmitters and the receiver go to infinity with constant ratios. Several insights are gained from the analytic asymptotic sum-rate expression, based on which an efficient optimization algorithm is further proposed to obtain the capacity-achieving input covariance matrices. Simulation results demonstrate that even for a moderate number of antennas at each link, the new approach provides indistinguishable results as those obtained by the complex stochastic programming approach.