Comparison of two numerical methods for Riemannian cubic polynomials on Stiefel manifolds
<|reference_start|>Comparison of two numerical methods for Riemannian cubic polynomials on Stiefel manifolds: In this paper we compare two numerical methods to integrate Riemannian cubic polynomials on the Stiefel manifold $\textbf{St}_{n,k}$. The first one is the adjusted de Casteljau algorithm, and the second one is a symplectic integrator constructed through discretization maps. In particular, we choose the cases of $n=3$ together with $k=1$ and $k=2$. The first case is diffeomorphic to the sphere and the quasi-geodesics appearing in the adjusted de Casteljau algorithm are actually geodesics. The second case is an example where we have a pure quasi-geodesic different from a geodesic. We provide a numerical comparison of both methods and discuss the obtained results to highlight the benefits of each method.<|reference_end|>
@article{simoes2024comparison,
    title={Comparison of two numerical methods for Riemannian cubic polynomials on
  Stiefel manifolds},
    author={Alexandre Anahory Simoes, Leonardo Colombo, F'atima Silva Leite},
    journal={arXiv preprint arXiv:2404.19407},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.19407},
    primaryClass={math.OC cs.NA cs.SY eess.SY math.NA}
}