Enhancing the Convergence Order from <em>p</em> to <em>p+3</em> in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices

Abstract

In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. Using this procedure, we achieve a remarkable increase in the order of convergence, raising it from p to p + 3 units, which results in a remarkable improvement in the overall performance. We have conducted comprehensive numerical tests in various scenarios to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes.