Introducing memory to a family of multi-step multidimensional iterative methods with weight function

TL;DR

This paper develops a derivative-free multi-step iterative method with memory and weight functions to enhance convergence order, combining Steffensen's approach with dynamic parameter tuning and numerical analysis.

Contribution

It introduces a new family of derivative-free multi-step methods with adjustable convergence order and memory, incorporating weight functions and dynamic behavior.

Findings

Achieves convergence order 2m with m steps

Demonstrates improved convergence through memory incorporation

Provides numerical experiments illustrating dynamic behavior

Abstract

In this paper, we construct a derivative-free multi-step iterative scheme based on Steffensen's method. To avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence, we freeze the divided differences used from the second step and use a weight function on already evaluated operators. Therefore, we define a family of multi-step methods with convergence order 2m, where m is the number of steps, free of derivatives, with several parameters and with dynamic behaviour, in some cases, similar to Steffensen's method. In addition, we study how to increase the convergence order of the defined family by introducing memory in two different ways: using the usual divided differences and the Kurchatov divided differences. We perform some numerical experiments to see the behaviour of the proposed family and suggest different weight functions to visualize with dynamical planes in some cases the dynamical behaviour.