On the convergence and optimization of the Baker-Campbell-Hausdorff formula

In this paper the problem of the convergence of the Baker-Campbell-Hausdorff series for Z = log(e(X)e(Y)) is revisited. We collect some previous results about the convergence domain and present a new estimate which improves all of them. We also provide a new expression of the truncated Lie presentation of the series up to sixth degree in X and Y requiring the minimum number of commutators. Numerical experiments suggest that a similar accuracy is reached with this approximation at a considerably reduced computational cost. (C) 2003 Elsevier Inc. All rights reserved.