On the construction of numerical iterative schemes of any order of convergence for solving nonlinear systems of equations

TL;DR

This paper introduces a general methodology for creating high-order iterative schemes to solve nonlinear systems, along with formulas to determine their convergence order, demonstrated through numerical tests.

Contribution

It provides a novel, systematic approach for constructing iterative methods of any order of convergence for nonlinear systems.

Findings

Successfully constructed second and third order schemes

Derived formulas for convergence order of constructed schemes

Numerical tests confirm theoretical convergence orders

Abstract

This paper presents a methodology for constructing iterative schemes of any order of convergence for solving nonlinear systems of equations. It also provides formulas for the order of convergence of any iterative schemes constructed using the method proposed in this paper. A test case is conducted numerically for the second and third order of convergence using a computer algebra system called Maxima. The code used is listed at the end of the test case.