An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications

TL;DR

This paper introduces a fast algorithm for computing the Baker-Campbell-Hausdorff series in free Lie algebras, leveraging rooted tree structures, with applications to matrix convergence analysis.

Contribution

A novel, efficient algorithm for generating the BCH series up to degree 20 using a rooted tree approach, significantly reducing computation time.

Findings

BCH series computed up to degree 20 in under 15 minutes

The algorithm uses a Lie algebraic structure of labeled rooted trees

Provides an optimal convergence domain for matrix cases

Abstract

We provide a new algorithm for generating the Baker--Campbell--Hausdorff (BCH) series $Z = \log(\e^X \e^Y)$ in an arbitrary generalized Hall basis of the free Lie algebra $\mathcal{L}(X,Y)$ generated by $X$ and $Y$. It is based on the close relationship of $\mathcal{L}(X,Y)$ with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree 20 (111013 independent elements in $\mathcal{L}(X,Y)$) takes less than 15 minutes on a personal computer and requires 1.5 GBytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when $X$ and $Y$ are real or complex matrices.