This work focuses on large scale multi-user MIMO systems in which the base station (BS) outfitted with M antennas communicates with K single antenna user equipments (UEs). In particular, we aim at designing the linear precoder and receiver that maximizes the minimum signal-to-interference-plus-noise ratio (SINR) subject to a given power constraint. To gain insights into the structure of the optimal precoder and receiver as well as to reduce the computational complexity for their implementation, we analyze the asymptotic regime where M and K grow large with a given ratio and make use of random matrix theory (RMT) tools to compute accurate approximations. Although simpler, the implementation of the asymptotic precoder and receiver requires fast inversions of large matrices in every coherence period. To overcome this issue, we apply the truncated polynomial expansion (TPE) technique to the precoding and receiving vector of each UE and make use of RMT to determine the optimal weighting coefficients on a per-UE basis that asymptotically solve the max-min SINR problem. Numerical results are used to show that the proposed TPE-based precoder and receiver almost achieve the same performance as the optimal ones while requiring a lower complexity.