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

2013/09/18 by T. Bittencourt, Bittencourt, T., O. P. Ferreira +1
Engineering · Mathematics · #49M15 #Advanced Numerical Analysis Techniques #Advanced Optimization Algorithms Research #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1309.4734

openalex publication_date 2013/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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