On the implementation of Lie group methods on the Stiefel manifold

There are several applications in which one needs to integrate a system of ODEs whose solution is an $n\times p$ matrix with orthonormal columns. In recent papers algorithms of arithmetic complexity order $n\times p^2$ have been proposed. The class of Lie group integrators may seem like a worth while alternative for this class of problems, but it has not been clear how to implement such methods with $\mathcal{O}(np^2)$ complexity. In this paper we show how Lie group methods can be implemented in a computationally competitive way, by exploiting that analytic functions of $n\times n$ matrices of rank $2p$ can be computed with $\mathcal{O}(n p^2)$ complexity.