Work

Generalized Traub Family for Solving Nonlinear Systems: Fourth-Order Optimal Method and Dynamical Analysis

Year: 2026

Type: article

Abstract: A novel two-stage procedure for approximating solutions of nonlinear systems is introduced. The scheme employs two evaluations of the vector function F together with a single Jacobian computation, followed by the resolution of two linear subproblems that share an identical coefficient matrix. This structure reduces the computational cost and enhances the adaptability of the method with respect to existing alternatives. The design of the algorithm is motivated by criteria relating efficiency to the total number of functional evaluations, ensuring that the resulting strategy achieves the optimal convergence order permitted within this framework. A proof of the local convergence order is provided, and its accuracy is supported by a series of experiments on distinct nonlinear models, including problems arising from differential equations. The numerical evidence confirms that the developed technique reaches the theoretical convergence rate and performs favorably when compared with other methods of equal order. Moreover, we examine the dynamical features of the related parametric variant, offering additional understanding of its stability properties and iterative behavior.

Source: Mathematics

DOI: 10.3390/math14071161

Authors: Alicia Cordero, Miguel A. Leonardo Sepúlveda, Juan R. Torregrosa, Antmel Rodríguez Cabral, María P. Vassileva

Institutions: Universitat Politècnica de València, Instituto Superior de Formación Docente Salomé Ureña, Universidad Acción Pro-Educación y Cultura, Universidad Autónoma de Santo Domingo, Instituto Tecnológico de Santo Domingo

Language: English

Topic: Iterative Methods for Nonlinear Equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: University of Agricultural Sciences, Dharwad