Methods for the approximation of the matrix exponential in a Lie-algebraic setting
Elena Celledoni, Arieh Iserles
TL;DR
The paper tackles efficient, invariant-preserving approximation of the matrix exponential in Lie-group settings for geometric integration. It introduces canonical coordinates of the second kind to express $\exp(tB)$ as a product of exponentials of Lie-algebra basis elements, derives order conditions via adjoint-action polynomials, and leverages sparse bases to achieve $O(n^3)$- or even near-linear-cost implementations for structured problems. Time-symmetric composition (Yošida) is developed to attain higher-order accuracy without increasing the number of exponentials, with explicit BCH-based corrections computed for key algebras such as $so(n)$, $sl(n)$, and $so(3,1)$. Numerical experiments validate the approach on full and sparse $so(50)$ cases and a KdV-reduction ODE, showing substantial speedups over traditional expm-based methods while maintaining high accuracy. The methods enable scalable, invariant-preserving exponentials for ODEs on manifolds and have practical impact in geometric integration and related applications.
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.