Solving nonlinear equations by a derivative-free form of the King's family with memory

TL;DR

This paper introduces an eighth-order derivative-free iterative method with memory for solving nonlinear equations, which is optimized for high efficiency and can be accelerated to twelfth order without additional function evaluations.

Contribution

It develops a new derivative-free method based on King's family with memory, achieving higher convergence order via a novel parameter adjustment using Newton's interpolation.

Findings

Achieves eighth-order convergence with four function evaluations per iteration.

Can be accelerated to twelfth-order convergence without extra function evaluations.

Demonstrates superior performance in high precision numerical experiments.

Abstract

In this paper, we present an iterative three-point method with memory based on the family of King's methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newton's interpolation polynomial of fourth degree. The order of convergence is increased from 8 to 12 without any extra function evaluation. Consequently, this method, possesses a high computational efficiency. Finally, a numerical comparison of the proposed method with related methods shows its effectiveness and performance in high precision computations.