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 main feature is that we provide neat, explicit expressions for the Jacobians. To the author's knowledge, this is the first time some explicit formulas are given for the Jacobians involved in the shooting methods to find the distance between two given points on the Stiefel manifold. This allows us to perform a preliminary analysis for the single shooting method. Numerical experiments demonstrate the algorithms in terms of accuracy and performance. Finally, we showcase three example applications in summary statistics, shape analysis, and model order reduction.

翻译：本文展示了如何使用打靶方法（一种求解边值问题的经典数值算法）来计算Stiefel流形 $\mathrm{St}(n,p)$（即列正交的 $ n \times p $ 矩阵集合）上的黎曼距离。主要特点是，我们提供了简洁且显式的雅可比表达式。据作者所知，这是首次给出Stiefel流形上两点间距离打靶方法所涉及雅可比矩阵的显式公式，从而能够对单打靶方法进行初步分析。数值实验展示了算法在精度和性能方面的表现。最后，我们展示了三个示例应用：汇总统计、形状分析和模型降阶。