Numerical Implementation of Harmonic Polylogarithms to Weight w = 8

TL;DR

This paper introduces a FORTRAN implementation for accurately computing harmonic polylogarithms up to weight 8, facilitating their use in complex calculations with high precision.

Contribution

It provides a numerical code for harmonic polylogarithms up to weight 8, including argument range reduction and basis mapping, enhancing computational efficiency and accuracy.

Findings

Achieves an accuracy of ~4.9 x 10^{-15} in calculations.

Provides argument transformation tools for harmonic polylogarithms.

Includes brief discussion on cyclotomic harmonic polylogarithms.

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.