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

Discretization methods for ordinary differential equations basedon the use of matrix exponentials have been known for decades.This set of ideas has come off age and acquired greater interestrecently, within the context of geometric integration and discretizationmethods on manifolds based on the use of Lie-group actions. In the present paper we study the approximation of the matrixexponential in a particular context: given a Lie group G andits Lie algebra g, we seek approximants F(t B) of exp(t B) suchthat F(t B) G if B g. Having fixed a basis V₁, ..., V_d ofg, we write F(t B) as a composition of exponentials of the typeexp(_i (t) V_i), where _i for i = 1, 2, ..., d are scalar functions.In this manner it becomes possible to increase the order ofthe approximation without increasing the number of exponentialsto evaluate and multiply together. We study order conditionsand implementation details and conclude the paper with somenumerical experiments. Received 24 March 1999. Accepted 22 November 1999.