Increasing in three units the order of convergence of iterative methods for solving nonlinear systems

In this paper, we present a technique aimed at the implementation of high accuracy iterative procedures in nonlinear systems. The main result of this research is a general proof that guarantees an increase in the order of convergence to order p + 3 units for any method of order p. This methodology has as its main objective not only this increase, but also the optimization in the use of resources by requiring only one additional functional evaluation per iteration, thus keeping a tight control over the computational cost. Although there are previous proposals in the literature that approach this goal, in this work we consider such approaches as particular cases of our general proposal. This approach encompasses many methods described by other authors and can be applied to any p-order scheme to increase it by three units. Numerical evaluations, supported by a rigorous theoretical framework, validate the robustness of the proposal, highlighting its advantage over conventional techniques and demonstrating its effectiveness in the face of numerical challenges of considerable magnitude. Summing up, with this research, we offer a substantial contribution to the field of numerical methods, proposing a paradigm that consistently enhances the convergence order of iterative methods.