A class of low complexity intrinsic schemes for orthogonal integration

Numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is considered. Points on this manifold are represented as $n\times k$ matrices with orthonormal columns, of particular interest is the case when $n>>k$. Mainly two requirements are imposed on the integration schemes. First, they should have arithmetic complexity of order $nk^2$. Second, they should be intrinsic in the sense that they only require the ODE vector field to be defined on the Stiefel manifold, as opposed to for instance projection methods. The design of the methods makes use of retractions maps. Two algorithms are proposed, one where the retraction map is based on the $QR$ decomposition of a matrix, and one where it is based on the polar decomposition. Numerical experiments show that the new methods are superior to standard Lie group methods with respect to arithmetic complexity, and may be more reliable than projection methods, owing to their intrinsic nature.