# A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

Abstract : We consider the problem of finding a singularity of a vector field $X$ on a complete Riemannian manifold. In this regard we prove a unified result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of $X$. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area.
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https://hal.inria.fr/inria-00070622
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Submitted on : Friday, May 19, 2006 - 9:02:18 PM
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### Identifiers

• HAL Id : inria-00070622, version 1

### Citation

Felipe Alvarez, Jérôme Bolte, Julien Munier. A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds. [Research Report] RR-5381, INRIA. 2004, pp.27. ⟨inria-00070622⟩

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