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

TL;DR

This paper investigates methods for approximating the matrix exponential within Lie groups and Lie algebras, focusing on ensuring that approximations stay within the group, which is crucial for geometric integration on manifolds.

Contribution

It introduces a framework for constructing approximants of the matrix exponential in Lie algebra settings, ensuring they remain within the Lie group, advancing geometric integration techniques.

Findings

Developed a composition-based approximation method

Ensured approximants lie within the Lie group

Applicable to geometric integration on manifolds

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.