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

Abstract

We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) series Z=log(eXeY) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y) generated by X and Y. It is based on the close relationship of L(X,Y) with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree of 20 [111 013 independent elements in L(X,Y)] takes less than 15min on a personal computer and requires 1.5Gbytes 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.