A new iterative three-point method for solving systems of nonlinear equations

TL;DR

This paper introduces a novel sixth-order iterative three-point method for solving systems of nonlinear equations, demonstrating its efficiency and convergence properties through theoretical analysis and numerical experiments.

Contribution

The paper develops a new sixth-order three-point iterative method for nonlinear systems, extending scalar methods and validating its effectiveness through analysis and numerical tests.

Findings

Achieves sixth-order convergence theoretically and computationally.

Outperforms existing third-point methods of lower order.

Demonstrates higher efficiency in numerical experiments.

Abstract

A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that the new method also has a sixth order of convergence. It is confirmed that the theoretical order of convergence coincides with the computational order of convergence by the numerical solution of two problems. Finally, its computational efficiency is calculated and subsequently compared with that of other three-point methods of fifth and sixth order convergence that also solve systems of non-linear equations.