Extended Newton-Traub Method of Order Six

In various scientific and engineering fields, many applications can be simplified as the task of solving equations or systems of equations within a carefully chosen abstract space. However, analytical solutions for such problems are often difficult or even impossible to find. As a result, iterative methods are widely employed to obtain the desired solutions. This article focuses on introducing a highly efficient three-step iterative method that exhibits sixth order convergence. The analysis presented here thoroughly explores the local and semi-local convergence properties, taking into account the continuity conditions imposed on the operators present in the method. The innovative methodology described in this article is not limited to specific methods but can be extended to a broader range of approaches that involve the utilization of inverse operations on linear operators or matrices.