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A development of an inverse first-order divided difference operator forfunctions of several variables is presented. Two generalized derivative-freealgorithms builded up from Ostrowski's method for solving systems of nonlinearequations are written and analyzed. A direct computation of the local order ofconvergence for these variants of Ostrowski's method is given. In order topreserve the local order of convergence, any divided difference operator is notvalid. Two counterexamples of computation of a classical divided differenceoperator without preserving the order are presented. A new divided differenceoperator solving this problem is proposed. Furthermore, a computation thatapproximates the order of convergence is generated for the examples and itconfirms in a numerical way that the order of the methods is well deduced.

On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods