A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker–Campbell–Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.