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

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.