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A unifying local convergence result for Newton’s method in Riemannian manifolds. (English) Zbl 1147.58008
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 S. Smale [Proc. Symp. Honor G. S. Young, Laramie/Wyo. 1985, 185–196 (1986; Zbl 0613.65058)], and Euclidean self-contact theory of [Y. Nesterov and A. Nemirovskii, Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics. 13. Philadelphia, PA: SIAM (1994; Zbl 0824.90112)]. See also [J.-P. Dedieu, P. Priouret and G. Malajovich, IMA J. Numer. Anal. 23, No. 3, 395–419 (2003; Zbl 1047.65037)].
MSC:
58C15Implicit function theorems and global Newton methods on manifolds
49M15Newton-type methods in calculus of variations
90C48Programming in abstract spaces
90C53Methods of quasi-Newton type