Numerical Implementation of Harmonic Polylogarithms to Weight w = 8

TL;DR

This paper presents a compact FORTRAN implementation (HPOLY.f) for numerically evaluating harmonic polylogarithms up to weight 8 with double-precision accuracy (~4.9e-15). It achieves this by representing HPLs in a chosen basis, mapping arguments to a core interval, and employing algebraic/argument relations plus a Bernoulli-improvement series to accelerate convergence. The authors provide extensive data and replacement files for basis reduction and argument transformations, enabling efficient numerical evaluation and straightforward verification against existing tools like Ginac. They also discuss cyclotomic HPLs, offering practical serial representations and guidance for applying the method to higher-loop calculations in phenomenology.

Abstract

We present the FORTRAN-code HPOLY.f for the numerical calculation of harmonic polylogarithms up to w = 8 at an absolute accuracy of $\sim 4.9 \cdot 10^{-15}$ or better. Using algebraic and argument relations the numerical representation can be limited to the range $x \in [0, \sqrt{2}-1]$. We provide replacement files to map all harmonic polylogarithms to a basis and the usual range of arguments $x \in ]-\infty,+\infty[$ to the above interval analytically. We also briefly comment on a numerical implementation of real valued cyclotomic harmonic polylogarithms.

Figures (2)

• Figure 1: The function H[{6,1},0,-1,x] in the region $x \in [-1,1]$.
• Figure 2: Real and imaginary part of the function $H[-1,1,0,x]$.