A unifying local convergence result for Newton's method in Riemannian manifolds
In the problem of singularities of a smooth vector field on a Riemannian manifold, an unified result for the existence and local uniqueness of the solution, and for local covergence of a Riemannian version of Newton's method is proposed. Different specializations of the main result recover Riemannian versions of the theorems of Kantorovich and \textit{S. Smale} [Proc. Symp. Honor G. S. Young, Laramie/Wyo. 1985, 185--196 (1986; Zbl 0613.65058)], and Euclidean self-contact theory of [\textit{Y. Nesterov} and \textit{A. Nemirovskii}, Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics. 13. Philadelphia, PA: SIAM (1994; Zbl 0824.90112)]. See also [\textit{J.-P. Dedieu, P. Priouret} and \textit{G. Malajovich}, IMA J. Numer. Anal. 23, No. 3, 395--419 (2003; Zbl 1047.65037)].
scientific article; zbMATH DE number 5314456