Chebyshev–Halley's method on Riemannian manifolds

This paper is concerned with the problem of finding singular points of vector fields on Riemannian manifolds. We, using non trivial techniques and results of differential geometry, will exhibit a new method defined on complete Riemannian manifolds, which generalizes the important Newton and Chebyshev–Halley's methods. Moreover, a characterization of the convergence under Kantorovich-type conditions and error estimates are also given in this study. Using the method introduced in this paper we give an algorithm which will allow us to find singular points of a vector field on the two-dimensional sphere S². Finally, in order to illustrate our method and its relevance, we develop an example which allows us to find a singularity of a vector field on S², such a singularity is not possible to find it using the classical numerical methods on R³.