A single shooting method with approximate Fr\'{e}chet 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 Fr\'{e}chet 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.