Newton ’ s method on Riemannian manifolds : Smale ’ s point estimate theory under the γ - condition

Newton’s method and its variants are among the most efficient methods known for solving systems of non-linear equations when the functions involved are continuously differentiable. Besides its practical applications, Newton’s method is also a powerful theoretical tool. Therefore, it has been studied and used extensively. One of the famous results on Newton’s method is the well-known Kantorovich theorem (cf. Kantorovich & Akilov, 1982) which guarantees convergence of Newton’s sequence to a solution under very mild conditions. Another important result concerning Newton’s method is Smale’s point estimate theory (cf. Blum et al., 1997, Smale, 1981, 1986 and 1997). Newton’s method has been extended to finding numerically zeros of vector fields on Riemannian manifolds, see, e.g. Edelman et al., 1998; Gabay, 1982; Smith, 1993, 1994; Udriste, 1994. Recent research has focused on extensions of the Kantorovich theorem and Smale’s point estimate theory, see Ferreira & Svaiter, 2002; Dedieu et al., 2003. Here we are particularly interested in the work due to Dedieu et al. (2003). Let X be an analytic vector field on an analytic Riemannian manifold M . Let p ∈ M be such that DX (p)−1 exists, and define