Comparison of two numerical methods for Riemannian cubic polynomials on Stiefel manifolds
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929331684507648 |
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| author | Simoes, Alexandre Anahory Colombo, Leonardo Leite, Fátima Silva |
| author_facet | Simoes, Alexandre Anahory Colombo, Leonardo Leite, Fátima Silva |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19407 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison of two numerical methods for Riemannian cubic polynomials on Stiefel manifolds Simoes, Alexandre Anahory Colombo, Leonardo Leite, Fátima Silva Optimization and Control Numerical Analysis Systems and Control 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. |
| title | Comparison of two numerical methods for Riemannian cubic polynomials on Stiefel manifolds |
| topic | Optimization and Control Numerical Analysis Systems and Control |
| url | https://arxiv.org/abs/2404.19407 |