Newton's method on Riemannian manifolds: covariant alpha theory |
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Authors: | Dedieu Jean-Pierre; Priouret Pierre; Malajovich Gregorio |
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Institution: |
1 MIP. Département de Mathématique, Université Paul Sabatier, 31062 Toulouse cedex 04, France 2 Departamento de Matemática Aplicada, Universidade Federal de Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brazil
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Abstract: | In this paper, Smale's theory is generalized to the contextof intrinsic Newton iteration on geodesically complete analyticRiemannian and Hermitian manifolds. Results are valid for analyticmappings from a manifold to a linear space of the same dimension,or for analytic vector fields on the manifold. The invariant is defined by means of high-order covariant derivatives. Boundson the size of the basin of quadratic convergence are given.If the ambient manifold has negative sectional curvature, thosebounds depend on the curvature. A criterion of quadratic convergencefor Newton iteration from the information available at a pointis also given. |
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Keywords: | Newton iteration alpha-theory non-linear equations |
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