Two-Step Fifth-Order Efficient Jacobian-Free Iterative Method for Solving Nonlinear Systems

Alicia Cordero (), Javier G. Maimó, Antmel Rodríguez-Cabral and Juan R. Torregrosa
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Alicia Cordero: Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
Javier G. Maimó: Area de Ciencias Básicas y Ambientales, Instituto Tecnológico de Santo Domingo (INTEC), Av. Los Próceres, Gala, Santo Domingo 10602, Dominican Republic
Antmel Rodríguez-Cabral: Area de Ciencias Básicas y Ambientales, Instituto Tecnológico de Santo Domingo (INTEC), Av. Los Próceres, Gala, Santo Domingo 10602, Dominican Republic
Juan R. Torregrosa: Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain

Mathematics, 2024, vol. 12, issue 21, 1-20

Abstract: This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational overhead and simplifies the implementation process while maintaining high convergence rates. We demonstrate that this method achieves fifth-order convergence under specific parameter settings, with broad applicability across various types of nonlinear systems. The effectiveness of the proposed method is validated through a series of numerical experiments that confirm its superior performance in terms of accuracy and computational efficiency compared to existing methods.

Keywords: nonlinear systems; iterative processes; convergence order; efficiency (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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