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

摘要

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