Local convergence of Newton's method under a majorant condition in Riemannian manifolds.

A local convergence analysis of Newton's method for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, based on the majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the vector field under consideration. It also allows us to obtain the optimal convergence radius and the biggest range for the uniqueness of the solution and to unify some previously unrelated results. [ABSTRACT FROM AUTHOR]