Robust Kantorovich's theorem on Newton's method under majorant condition in Riemannian Manifolds

TL;DR

This paper extends Kantorovich's theorem for Newton's method on Riemannian manifolds by introducing a robust, affine-invariant version that relaxes Lipschitz conditions using a majorant function, ensuring convergence and local uniqueness.

Contribution

It presents a new robust, affine-invariant Kantorovich's theorem on Riemannian manifolds that relaxes classical Lipschitz conditions with a majorant function, unifying previous results.

Findings

Establishes existence and local uniqueness of solutions.

Provides bounds for quadratic convergence.

Ensures convergence within a prescribed ball.

Abstract

A robust affine invariant version of Kantorovich's theorem on Newton's method, for finding a zero of a differentiable vector field defined on a complete Riemannian manifold, is presented in this paper. In the analysis presented, the classical Lipschitz condition is relaxed by using a general majorant function, which allow to establish existence and local uniqueness of the solution as well as unifying previously results pertaining Newton's method. The most important in our analysis is the robustness, namely, is given a prescribed ball, around the point satisfying Kantorovich's assumptions, ensuring convergence of the method for any starting point in this ball. Moreover, bounds for $Q$-quadratic convergence of the method which depend on the majorant function is obtained.