Newton's method on Riemannian manifolds: Smale's point estimate theory under the -condition

Abstract

The γ-conditions for vector fields on Riemannian manifolds are introduced. The γ-theory and the α-theory for Newton's method on Riemannian manifolds are established under the γ-conditions. Applications to analytic vector fields are provided and the results due to Dedieu et al. (2003, IMA J. Numer. Anal., 23, 395–419) are improved.