In addition to conventional antenna-based array

the reconfigurable intelligent surface (RIS) holds promise as an alternative technology for manufacturing massive multi-input multi-output (MIMO) array for beyond 5G communications. This paper designs a fast algorithm to optimize the RIS-based MIMO precoder for maximizing the spectral efficiency

which includes the digital precoder and the RIS reflection phases. We evaluate the optimality of the algorithm by deriving an RIS channel capacity upper bound utilizing majorization theory. Our scheme can work for an RIS in both frequency flat and frequency selective channels

with either continuously or discretely tunable phases. The simulation results show that the proposed algorithm can achieve the capacity upper bound in some scenarios

which empirically proves its optimality. It is also shown that our algorithm is one-to-two orders of magnitude faster than the state-of-the-art methods in the literature.