Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds

Abstract

A local convergence analysis of Inexact Newton's method with relative residual error tolerance for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, based on majorant principle, is presented in this paper. We prove that under local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q -linearly to a singularity of the vector field under consideration. Using this result we show that the inexact Newton method to find a zero of an analytic vector field can be implemented with a fixed relative residual error tolerance. In the absence of errors, our analysis retrieve the classical local theorem on the Newton method in Riemannian context.

A Global Version of the Newton Method for Finding a Singularity of the Nonsmooth Vector Fields on Riemannian Manifolds

“…AVVFs of dimensions 100, 400, 800, and 1600. The density of matrix A was set to 0.003, similar to that in [6]. This implies that only approximately 0.3% of the elements of A are non-null.…”

“…Recently, [12] proposed and analyzed a version of the Newton method for finding a singularity of a class of locally Lipschitz continuous vector field. For the smooth vector fields, much has already been done, see [1,6,17,19,20,30,36]. In, [7] was proposed a global version of the * Instituto de Matemática e Estatística, Universidade Federal de Goiás, CEP 74001-970 -Goiânia, GO, Brazil, E-mails: rodriguesfabiana@ufg.br.com, fabriciarodrigues@ufg.br.…”