A convergence analysis of a family of third order iterative methods in Riemannian manifold

In this paper we use a family of third order iterative methods to find singular points on complete Riemannian manifolds. This family of root-finding iterative methods generalizes the important Newton, Chebyshev, Halley, super-Halley and c-type methods, as particular cases. Moreover, the convergence analysis under Kantorovich-type conditions and error estimates are also given. Finally, we generate an example that allows us to find singular points on \(\textbf{S}^2\) ${\mathbf{S}}^2$ , which is impossible to find using classical numerical iterative methods on \(\mathbb {R}^3\) ${\mathbb{R}}^3$ .