User manual for bch, a program for the fast computation of the Baker-Campbell-Hausdorff and similar series

TL;DR

This paper introduces bch, a fast C program for computing Baker-Campbell-Hausdorff series efficiently in various bases, enabling high-degree calculations with minimal resources.

Contribution

The paper presents a new software tool that significantly speeds up BCH series computations up to degree 30 using optimized basis representations.

Findings

BCH series coefficients up to degree 20 computed in under half a second.

Coefficients up to degree 30 computed in 55 hours with 5.5GB memory.

Efficient implementation in the Lyndon basis enhances computational performance.

Abstract

This manual describes bch, an efficient program written in the C programming language for the fast computation of the Baker-Campbell-Hausdorff (BCH) and similar Lie series. The Lie series can be represented in the Lyndon basis, in the classical Hall basis, or in the right-normed basis of E.S. Chibrikov. In the Lyndon basis, which proves to be particularly efficient for this purpose, the computation of 111013 coefficients for the BCH series up to terms of degree 20 takes less than half a second on an ordinary personal computer and requires negligible 11MB of memory. Up to terms of degree 30, which is the maximum degree the program can handle, the computation of 74248451 coefficients takes 55 hours but still requires only a modest 5.5GB of memory.