A single shooting method with approximate Fréchet derivative for computing geodesics on the Stiefel manifold

Abstract

This paper shows how to use the shooting method, a classical numerical algorithm for solving boundary value problems, to compute the Riemannian distance on the Stiefel manifold $ \mathrm{St}(n,p) $, the set of $ n \times p $ matrices with orthonormal columns. The proposed method is a shooting method in the sense of the classical shooting methods for solving boundary value problems; see, e.g., Stoer and Bulirsch, 1991. The main feature is that we provide an approximate formula for the Fréchet derivative of the geodesic involved in our shooting method. Numerical experiments demonstrate the algorithms' accuracy and performance. Comparisons with existing state-of-the-art algorithms for solving the same problem show that our method is competitive and even beats several algorithms in many cases.