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

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 retrieves the classical local theorem on the Newton method in Riemannian context.