Development and analysis of an efficient Jacobian-free method for systems of nonlinear equations

A multi-step derivative-free iterative technique is developed by extending the well-known Traub-Steffensen iteration for solving the systems of nonlinear equations. Keeping in mind the computational aspects, the general idea to construct the scheme is to utilize the single inverse operator per iteration. In fact, these type of techniques are hardly found in literature. Under the standard assumption, the proposed technique is found to possess the fifth order of convergence. In order to demonstrate the computational complexity, the efficiency index is computed and further compared with the efficiency of existing methods of similar nature. The complexity analysis suggests that the developed method is computationally more efficient than their existing counterparts. Furthermore, the performance of method is examined numerically through locating the solutions to a variety of systems of nonlinear equations. Numerical results regarding accuracy, convergence behavior and elapsed CPU time confirm the efficient behavior of the proposed technique.