On the approximation of mth power divided differences preserving the local order of convergence

摘要

In this paper, some extensions are presented for a new technique to construct a family of divided differences that has been proposed previously. By applying a new definition for local order of convergence and presenting a comprehensive analysis, the relation between local order of convergence and R-order and Q-order is investigated. We also propose a new technique to approximate the elements of the developed divided differences. The divided differences are approximated by using the new scheme and are replaced in some iterative methods. Numerical experiments show that new derivative-free iterative methods obtained in this way are with high order of convergence. Numerical results confirm the theoretical results and indicate the efficiency and robustness of the new Jacobian-free methods.