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Kantorovich’s theorem on Newton’s method in Riemannian manifolds. (English) Zbl 1003.65057
This paper is concerned with the problem of finding a singularity of a vector field in a Riemannian manifold. The authors present an extension of Kantorovich’s theorem on Newton’s method for this problem in finite dimensional Riemannian manifolds.

MSC:
65J15 Numerical solutions to equations with nonlinear operators
58K45 Singularities of vector fields, topological aspects
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[1] Appel, J.; De Pascale, E.; Lysenko, J.V.; Zabrejko, P.P., New results on newton – kantorovich approximations with applications to nonlinear integral equations, Numer. funct. anal. optim., 18, 1-17, (1997) · Zbl 0881.65049
[2] Argyros, I.K., A unified approach for constructing fast tow-step Newton-like methods, Monatsh. math., 119, 1-22, (1995) · Zbl 0817.65042
[3] Cauchy, A., Sur la détermination des racines d’une équation algébrique ou transcendente, Œuvres complétes (II), (1899), Gauthier-Villars Paris, p. 573-609
[4] da Cruz Neto, J.X., Métodos geodésicos na progamação matemática, (1995), COPPE/UFRJ Rio de Janeiro
[5] Dennis, J.E., On the Kantorovich hypothesis for Newton’s method, SIAM J. numer. anal., 6, 493-507, (1969) · Zbl 0221.65098
[6] do Carmo, M.P., Riemannian geometry, (1992), Birkhäuser Boston
[7] Edelman, A.; Arias, T.A.; Smith, T., The geometry of algorithms with orthogonality constraints, SIAM J. matrix anal. appl., 20, 303-353, (1998) · Zbl 0928.65050
[8] Gabay, D., Minimizing a differentiable function over a differential manifold, J. optim. theory appl., 37, 177-219, (1982) · Zbl 0458.90060
[9] Hadeler, K.P., Shadowing orbits and Kantorovich’s theorem, Numer. math., 73, 65-73, (1996) · Zbl 0862.58017
[10] Hildebrand, F.B., Introduction to numerical analysis, (1956), McGraw-Hill New York · Zbl 0070.12401
[11] Kantorovich, L.V., On Newton’s method for functional equations, Dokl. akad. nauk. SSSR, 59, 1237-1240, (1984)
[12] Kantorovich, L.V., Further applications of Newton’s method, Vestnik leningr. univ., 2, 68-103, (1957) · Zbl 0091.11502
[13] Kantorovich, L.V.; Akilov, G.P., Functional analysis in normed spaces, (1964), Pergamon Oxford · Zbl 0127.06104
[14] Moser, J., A new techniques for the construction of solutions of nonlinear differential equations, Proc. natl. acad. sci. USA, 47, 1824-1831, (1961) · Zbl 0104.30503
[15] Nash, J., The embedding problem for Riemannian manifolds, Ann. math., 63, 20-63, (1956) · Zbl 0070.38603
[16] Ortega, J.M., The newton – kantorovich theorem, Amer. math. monthly, 75, 658-660, (1968) · Zbl 0183.43004
[17] Ortega, J.M.; Rheimboldt, W.C., Interactive solution of nonlinear equations in several variables, (1970), Academic Press New York
[18] C. Runge, Separation und Approximation der Wurzeln von Gleichungen, in Enzyklopädie der Mathematichen Wissenschaften, Vol. 1, pp. 405-449, Teubner, Leipzig.
[19] Shub, M.; Smale, S., Complexity of Bezout’s theorem. I. geometric aspects, J. amer. math. soc., 6, 459-499, (1993) · Zbl 0821.65035
[20] Smale, S., Newton method estimates form data at one point, (), 185-196
[21] Smith, S.T., Optimization techniques on Riemannian manifolds, Fields institute communications, (1994), American Mathematical Society Providence, p. 113-146 · Zbl 0816.49032
[22] Smith, S.T., Geometric optimization method for adaptive filtering, (1993), Harvard University Cambridge
[23] Synge, J.L.; Schild, A., Tensor calculus, (1949), Dover New York · Zbl 0038.32301
[24] Udriste, C., Convex functions and optimization methods on Riemannian manifolds, Mathematics and its applications, 297, (1994), Kluwer Academic Dordrecht
[25] Yamamoto, T., A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions, Numer. math., 49, 203-220, (1986) · Zbl 0607.65033
[26] Wayne, C.E., An introduction to KAM theory. dynamical systems and probabilistic methods in partial differential equation, Lectures in applied mathematics, 31, (1996), American Mathematical Society Providence
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