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(ee ) 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 20 (111013 independent elements in 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. Departament de Matemàtiques, Universitat Jaume I, E-12071 Castellón, Spain. Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, EHU/UPV, Donostia/San Sebastián, Spain.