Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds

The Newton method and its variations are the most efficient methods known for solving systems of nonlinear equations when they are continuously differentiable. Besides its practical applications, the Newton method is also a powerful theoretical tool. One of the famous results on the Newton method is the well-known Kantorovich’s theorem, which has the advantage that the Newton sequence converges to a solution under very mild conditions. Another important result on the Newton method is the Smale’s point estimate theory which was presented by Smale in his report written for the 20th International Conference of Mathematician. In this theory, the notion to be an approximation zero was introduced and the rule to judge an initial point of an approximation zero was provided, depending only on the information of the nonlinear operator at the initial point. Other results on the Newton method such as the estimates of the radii of convergence balls were given by Traub and Wozniakowski, and Wang independently. A big step in this direction was made recently by Wang, where the Kantorovich’s theorem and the Smale’s theory were unified and extended.