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

TL;DR

This paper introduces a new divided difference operator for functions of multiple variables, ensuring the preservation of local convergence order in derivative-free algorithms based on Ostrowski's method for solving nonlinear systems.

Contribution

It develops a novel inverse first-order divided difference operator that maintains the convergence order, along with generalized derivative-free algorithms and numerical validation.

Findings

New divided difference operator preserves local convergence order.

Counterexamples show classical operators fail to preserve order.

Numerical experiments confirm the theoretical convergence order.

Abstract

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.