Newton's method for approximating zeros of vectorfields on Riemannian manifolds

摘要

We provide a local convergence analysis of Newton's method for pprox-imating zeros of vector fields on Riemannian manifolds. Using a combination ofthe radius and center-Lipschitz-type conditions on the covariant derivatives of thevector fields involved, we provide (under the same computational cost) a conver-gence analysis with following advantages over the corresponding one in (Li andWang in Sci. China Ser. 48:1465-1478, 2005): at least as: large radius of conver-gence; small ratio of convergence. As applications, we extend the applicability ofKantorovich's type result (Argyros in J. Comput. Appl. Math. 169:315-332, 2004;Argyros in J. Math. Anal. Appl. 298:374-397, 2004; Argyros in J. Appl. Math. Com-put. 25: 35-47, 2007; Argyros in J. Concr. Appl. Anal. 6(1):21-32, 2008; Argyros inConvergence and Applications f Newton-Type Iterations, 2008; Ferreira and Svaiterin J. Complex. 8:304-329, 2002; Kantorovich and Akilov in Functional Analysis inNormed Spaces, 1982), and the Smale gamma theory (Argyros in Convergence andApplications of Newton-Type Iterations, 2008; Smith in Fields Inst. Commun. vol. 3,pp. 113-146, 1994; Wang in IMA J. Numer. Anal. 20:123-134, 2000).