Computations in a free Lie algebra

Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker-Campbell-Hausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witt's formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides good bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new MATLAB toolbox 'DIFFMAN'.