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 X p matrix with orthonormal columns. In recent papers algorithms of arithmetic complexity order np(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 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 X n matrices of rank 2p can be computed with O(np(2)) complexity.