Methods for the approximation of the matrix exponential in a Lie-algebraic setting

Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater interest recently, within the context of geometric integration and discretization methods on manifolds based on the use of Lie-group actions. In the present paper we study the approximation of the matrix exponential in a particular context: given a Lie group G and its Lie algebra g, we seek approximants F(tB) of exp(tB) such that F(tB) is an element of G if B is an element of g. Having fixed a basis V-1,...,V-d of g, we write F(tB) as a composition of exponentials of the type exp(alpha(i)(t)V-i), where alpha(i) for i = 1,2,..., d are scalar functions. In this manner it becomes possible to increase the order of the approximation without increasing the number of exponentials to evaluate and multiply together. We study order conditions and implementation details and conclude the paper with some numerical experiments.