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.

翻译：本文展示了如何利用打靶法——一种经典的求解边值问题的数值算法——来计算Stiefel流形 $ \mathrm{St}(n,p) $（即具有标准正交列的 $ n \times p $ 矩阵集合）上的黎曼距离。所提出的方法在求解边值问题的经典打靶法意义下是一种打靶法；参见例如Stoer和Bulirsch（1991）。其主要特点在于，我们为打靶法中涉及的测地线之弗雷歇导数提供了一个近似公式。数值实验验证了算法的精度与性能。与现有求解同一问题的最先进算法进行比较表明，我们的方法具有竞争力，并且在许多情况下甚至优于若干算法。