一种带有近似Fréchet导数的单点打靶法用于计算Stiefel流形上的测地线
A single shooting method with approximate Fréchet derivative for computing geodesics on the Stiefel manifold
AI总结：
提出一种使用近似Fréchet导数的单点打靶法，用于计算Stiefel流形上的黎曼距离，数值实验表明该方法在精度和性能上具有竞争力。
摘要：
本文展示了如何使用打靶法（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) $（即所有$ n imes p $正交列矩阵的集合）上的黎曼距离。所提出的方法在经典打靶法求解边值问题的意义下是一种打靶法（参见Stoer和Bulirsch, 1991）。主要特点是，我们为打靶法中涉及的测地线的Fréchet导数提供了一个近似公式。数值实验证明了算法的准确性和性能。与现有解决同一问题的最先进算法的比较表明，我们的方法具有竞争力，并且在许多情况下甚至优于几种算法。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
