In Time Division Duplex reciprocity-based massive MIMO, it is essential to compute the downlink precoding matrix over all OFDM resource blocks within a small fraction of the uplink-downlink slot duration. Because of this harsh computation latency constraint, early implementations of massive MIMO considered the simple Conjugate Beamforming (ConjBF) precoding method. On the other hand, it is well-known that in the regime of a large but finite number of antennas, the Regularized Zero-Forcing (RZF) precoding is generally much more effective than ConjBF. In order to close the gap between ConjBF and RZF, while meeting the latency constraint, truncated polynomial expansion (TPE) methods have been proposed. In this paper, we present a novel TPE method that outperforms previously proposed methods in the non-symmetric case of users with different channel correlations, subject to the condition that the covariance matrices of the user channel vectors can be approximated, for a large number of antennas, by a family of matrices with common eigenvectors. This condition is met, for example, by uniform linear and uniform planar arrays in far-field conditions. The proposed method is computationally simple and lends itself to classical power allocation optimization such as min-sum power and max-min rate . We provide a detailed analysis of the computation latency vs computation resources, specifically targeted to a highly parallel FPGA hardware architecture. We conclude that the proposed TPE method can effectively close the performance gap between ConjBF and RZF with computation latency of less than one LTE OFDM symbol, as assumed in Marzetta’s work on massive MIMO.