In this study, we provide a two-step derivative-free method with fourth-order of convergence to numerically solve the system \(\mathbb {K}_{l}(u)=0\) , where \(\mathbb {K}_{l}:\psi \subset \mathbb {R}^l \rightarrow \mathbb {R}^l\) is a Fr \(\grave{e}\) chet differentiable operator and \(\psi \) is a convex set. We use a weight function to minimize the computing cost per iteration, making our method more efficient and cost-effective. To demonstrate the comparison of efficiency and computational cost, we solve various numerical problems using our technique and some existing methods in the literature. Numerical testing on several different types of problems validate the resilience and efficiency of convergence behavior.
