Newton's method for nonlinear mappings into vector bundles
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912665343885312 |
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| author | Weigl, Laura Schiela, Anton |
| author_facet | Weigl, Laura Schiela, Anton |
| contents | We consider Newton's method for finding zeros of mappings from a manifold $\mathcal X$ into a vector bundle $\mathcal E$. In this setting a connection on $\mathcal E$ is required to render the Newton equation well defined, and a retraction on $\mathcal X$ is needed to compute a Newton update. We discuss local convergence in terms of suitable differentiability concepts, using a Banach space variant of a Riemannian distance. We also carry over an affine covariant damping strategy to our setting. Finally, we will illustrate our results by applying them to generalized non-symmetric eigenvalue problems and providing a numerical example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Newton's method for nonlinear mappings into vector bundles Weigl, Laura Schiela, Anton Differential Geometry Numerical Analysis 53-08, 58C15, 46T05, 49M15 We consider Newton's method for finding zeros of mappings from a manifold $\mathcal X$ into a vector bundle $\mathcal E$. In this setting a connection on $\mathcal E$ is required to render the Newton equation well defined, and a retraction on $\mathcal X$ is needed to compute a Newton update. We discuss local convergence in terms of suitable differentiability concepts, using a Banach space variant of a Riemannian distance. We also carry over an affine covariant damping strategy to our setting. Finally, we will illustrate our results by applying them to generalized non-symmetric eigenvalue problems and providing a numerical example. |
| title | Newton's method for nonlinear mappings into vector bundles |
| topic | Differential Geometry Numerical Analysis 53-08, 58C15, 46T05, 49M15 |
| url | https://arxiv.org/abs/2404.04073 |