Preserving the order of convergence: Low-complexity Jacobian-free iterative schemes for solving nonlinear systems

In this paper, a new technique to construct a family of divided differences for designing derivative-free iterative methods for solving nonlinear systems is proposed. By using these divided differences any kind of iterative methods containing a Jacobian matrix in its iterative expression can be transformed into a “Jacobian-free” scheme preserving the order of convergence. This procedure is applied on different schemes, showing theoretically their order and error equation. Numerical experiments confirm the theoretical results and show the efficiency and performance of the new Jacobian-free schemes.