Kantorovich's theorem on Newton's method in Riemannian Manifolds

Authors:
O. P. Ferreira;B. F. Svaiter
Affiliations:
IME/Universidade Federal de Goiás, Campus Samambaia, Caixa Postal 131, CEP 74001-970 Goiânia, GO, Brazil;Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Jardim Botânico, CEP 22460-320 Rio de Janeiro, RJ, Brazil
Venue:
Journal of Complexity
Year:
2002

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Abstract

Newton's method for finding a zero of a vectorial function is a powerful theoretical and practical tool. One of the drawbacks of the classical convergence proof is that closeness to a non-singular zero must be supposed a priori. Kantorovich's theorem on Newton's method has the advantage of proving existence of a solution and convergence to it under very mild conditions. This theorem holds in Banach spaces. Newton's method has been extended to the problem of finding a singularity of a vectorial field in Riemannian manifold. We extend Kantorovich's theorem on Newton's method to Riemannian manifolds.