We compare three pseudospectral algorithms for mean-field polymer self-consistent field theory (SCFT) simulations and beyond mean-field field-theoretic simulations (FTS) using the complex Langevin (CL) sampling technique. In agreement with a study by Stasiak and Matsen, we find that for SCFT the fourth-order algorithm developed by Ranjan, Qin, and Morse usually outperforms the other pseudospectral algorithms. In contrast, for CL simulations we find that the second-order algorithm adapted to SCFT by Rasmussen and co-workers often outperforms the fourth-order methods not only in computational speed but also, surprisingly, in accuracy at a fixed contour resolution. This suggests an intricate coupling between pseudospectral schemes for chain contour integration and pseudotime stepping methods in CL simulations. Finally, the nominal stability of the algorithms is examined for the case of homogeneous fields.