Numerical evaluation of multiple polylogarithms

TL;DR

This work develops comprehensive algorithms for the numerical evaluation of multiple polylogarithms with arbitrary complex arguments and unbounded weight, motivated by higher-order perturbative calculations in quantum field theory. It combines integral and series representations with trailing-zero removal, argument transformations, Bernoulli substitutions, and Hölder convolution to ensure fast, reliable convergence across many scales. The authors implement these methods in the GiNaC C++ framework, providing specialized routines for multiple zeta values, harmonic polylogarithms, and general multiple polylogarithms, and validate the implementation through extensive checks against known values and identities. The result is a scalable, high-precision numerical toolkit that supports complex multi-parameter polylogarithms essential for advanced particle-physics computations.

Abstract

Multiple polylogarithms appear in analytic calculations of higher order corrections in quantum field theory. In this article we study the numerical evaluation of multiple polylogarithms. We provide algorithms, which allow the evaluation for arbitrary complex arguments and without any restriction on the weight. We have implemented these algorithms with arbitrary precision arithmetic in C++ within the GiNaC framework.