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IMA Journal of Numerical Analysis 2003 23(3):395-419; doi:10.1093/imanum/23.3.395
© 2003 by Institute of Mathematics and its Applications
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Newton's method on Riemannian manifolds: covariant alpha theory

Jean-Pierre Dedieu1, Pierre Priouret1 and Gregorio Malajovich2

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

In this paper, Smale's {alpha} theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant {gamma} is defined by means of high-order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given.

Key Words: Newton iteration; alpha-theory; non-linear equations

Received 10 June 2002. Revised 8 January 2003.


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