Kantorovich's theorem on Newton's method in Riemannian manifolds

Abstract

Newton's method for finding a zero of a vectorial function is a powerful theoretical and practical tool. One of the drawbacks of the classical convergence proof is that closeness to a non-singular zero must be supposed a priori. Kantorovich's theorem on Newton's method has the advantage of proving existence of a solution and convergence to it under very mild conditions. This theorem holds in Banach spaces. Newton's method has been extended to the problem of finding a singularity of a vectorial field in Riemannian manifold. We extend Kantorovich's theorem on Newton's method to Riemannian manifolds. © 2001 Elsevier Science (USA).