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

Abstract

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 urgency 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 $F(tB)\in G$ if $B\in g$. Having fixed a basis of the Lie algebra, we write $F(tB)$ as a composition of exponentials of the basis elements pre-multiplied by suitable scalar functions.