Elsevier

Journal of Complexity

Volume 24, Issue 3, June 2008, Pages 423-451
Journal of Complexity

Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory

https://doi.org/10.1016/j.jco.2007.12.003Get rights and content
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Abstract

One kind of the L-average Lipschitz condition is introduced to covariant derivatives of sections on Riemannian manifolds. A convergence criterion of Newton's method and the radii of the uniqueness balls of the singular points for sections on Riemannian manifolds, which is independent of the curvatures, are established under the assumption that the covariant derivatives of the sections satisfy this kind of the L-average Lipschitz condition. Some applications to special cases including Kantorovich's condition and the γ-condition as well as Smale's α-theory are provided. In particular, the result due to Ferreira and Svaiter [Kantorovich's Theorem on Newton's method in Riemannian manifolds, J. Complexity 18 (2002) 304–329] is extended while the results due to Dedieu Priouret, Malajovich [Newton's method on Riemannian manifolds: covariant alpha theory, IMA J. Numer. Anal. 23 (2003) 395–419] are improved significantly. Moreover, the corresponding results due to Alvarez, Bolter, Munier [A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math. to appear] for vector fields and mappings on Riemannian manifolds are also extended.

Keywords

Newton's method
Riemannian manifold
Section
Convergence criterion
Approximate singular point

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Supported in part by the National Natural Science Foundation of China (Grants 10671175; 10731060) and Program for New Century Excellent Talents in University.