A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

We consider the problem of finding a singularity of a differentiable vector field X defined on a complete Riemannian manifold. We prove a unified result for theexistence and local uniqueness of the solution, and for the local convergence of a Riemannian version of Newton's method. Our approach relies on Kantorovich's majorant principle: under suitable conditions, we construct an auxiliary scalar equation φ(r) = 0 which dominates the original equation X(p) = 0 in the sense that the Riemannian-Newton method for the latter inherits several features of the real Newton method applied to the former. The majorant φ is derived from an adequate radial parametrization of a Lipschitz-type continuity property of the covariant derivative of X, a technique inspired by the previous work of Zabrejko and Nguen on Newton's method in Banach spaces. We show how different specializations of the main result recover Riemannian versions of Kantorovich's theorem and Smale's α-theorem, and, at least partially, the Euclidean self-concordant theory of Nesterov and Nemirovskii. In the specific case of analytic vector fields, we improve recent developments inthis area by Dedieu et al. (IMA J. Numer. Anal., Vol. 23, 2003, pp. 395-419). Some Riemannian techniques used here were previously introduced by Ferreira and Svaiter (J. Complexity, Vol. 18, 2002, pp. 304-329) in the context of Kantorovich's theorem for vector fields with Lipschitz continuous covariant derivatives.