A single shooting method with approximate Fr\'echet derivative for computing geodesics on the Stiefel manifold
<|reference_start|>A single shooting method with approximate Fr\'echet derivative for computing geodesics on the Stiefel manifold: 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.<|reference_end|>
@article{sutti2024a,
    title={A single shooting method with approximate Fr\'{e}chet derivative for
  computing geodesics on the Stiefel manifold},
    author={Marco Sutti},
    journal={arXiv preprint arXiv:2404.04089},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.04089},
    primaryClass={math.NA cs.NA}
}