Bernard Deconinck, Matthias Heil, Alexander Bobenko, Mark van Hoeij, and Markus Schmies.
The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.
Download the paper as ps.gz file or as pdf file. See this NSF nugget on this topic. Maple implementation.