This study presents an examination of local convergence for a third-order family of iterative algorithms based on the majorant principle for locating a singularity of a differentiable vector field defined on a complete Riemannian manifold. This study shows a crystal clear link between the vector field under consideration and the majorant function. Also Lipschitz continuity of the derivative has relaxed by the majorant function. Additionally, it enables us to determine the optimal convergence radius and the widest range for the uniqueness of the solution. Moreover, we provide an algorithm to find singularity of vector field K using Halley and super-Halley methods on two-dimensional sphere \(S^2\) . Numerical example has been provided in support of our theory.