First optimal vectorial eighth-order iterative scheme for solving non-linear systems

We introduce a novel iterative method achieving eighth-order convergence, establishing its optimality for solving non-linear systems. A rigorous analysis of convergence order is presented, complemented by investigations into both efficiency indices and the complex dynamics of the proposed method. To assess its performance, extensive numerical experiments are conducted, facilitating comparative analysis with established methods from the literature. • The first eighth-order optimal iterative method for solving nonlinear systems, has been designed. • A comprehensive analysis of the complex dynamics of the proposed method is conducted, providing its strong stability. • The efficiency study states the lowest computational cost in front of several known schemes of the same or higher order of convergence. • A series of numerical experiments are performed to evaluate the behavior of the procedure, with outstanding results. • The concept of optimality is a vectorial natural extension of Kung-Traub's conjecture, proposed by the authors.