On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods
For researchers in numerical analysis, this work solves a specific bottleneck in preserving convergence order when using divided differences in derivative-free iterative methods.
The paper develops a new inverse first-order divided difference operator for multivariate functions that preserves the local convergence order of iterative methods, addressing a failure of classical operators. Two derivative-free variants of Ostrowski's method are analyzed, and numerical tests confirm the preserved order.
A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms builded up from Ostrowski's method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowski's method is given. In order to preserve the local order of convergence, any divided difference operator is not valid. Two counterexamples of computation of a classical divided difference operator without preserving the order are presented. A new divided difference operator solving this problem is proposed. Furthermore, a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.