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.

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.

This paper presents new optimal eighth-order families with weight functions for solving nonlinear systems, obtained as a generalization of the first optimal eighth-order CTT8 method introduced by Cordero, Torregrosa and Triguero-Navarro. The proposed schemes are constructed by combining a Newton-type predictor with high-order correction steps whose weight functions are suitably chosen to preserve optimal convergence while keeping a low computational cost. To the best of our knowledge, this work introduces the first family of optimal eighth-order methods for nonlinear systems, in the sense of the Cordero–Torregrosa conjecture, developed through a weight-function technique. A complete local convergence analysis is carried out under standard smoothness assumptions, proving eighth-order convergence for nondegenerate solutions. The computational efficiency of the proposed methods is also studied and compared with several existing high-order iterative schemes. Numerical experiments on nonlinear systems of different dimensions confirm the theoretical order of convergence and show the robustness of the new families. In addition, a Fredholm integral equation is solved, followed by a semilinear elliptic Dirichlet problem, further illustrating the reliability and computational performance of the proposed weight-function-based methods.

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 burden 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.

This paper presents new optimal eighth-order families with weight functions for solving nonlinear systems, obtained as a generalization of the first optimal eighth-order CTT8 method introduced by Cordero, Torregrosa and Triguero-Navarro. The proposed schemes are constructed by combining a Newton-type predictor with high-order correction steps whose weight functions are suitably chosen to preserve optimal convergence while keeping a low computational cost. To the best of our knowledge, this work introduces the first family of optimal eighth-order methods for nonlinear systems, in the sense of the Cordero Torregrosa conjecture, developed through a weight-function technique. A complete local convergence analysis is carried out under standard smoothness assumptions, proving eighth-order convergence for nondegenerate solutions. The computational efficiency of the proposed methods is also studied and compared with several existing high-order iterative schemes. Numerical experiments on nonlinear systems of different dimensions confirm the theoretical order of convergence and show the robustness of the new families. In addition, a Fredholm integral equation is solved, followed by a semilinear elliptic Dirichlet problem, further illustrating the reliability and computational performance of the proposed weight-function-based methods.

We establish the foundations of a theory of genuinely vectorial iterative methods for coupled nonlinear systems. Within classical local convergence analysis, we define the GV_p property through a bound guaranteeing local order at least p for every problem in a declared admissible class, under suitable smoothness and simple-root assumptions. We propose the Leonardo–Ureña certificate as a sufficient criterion for establishing this property. By ensuring class-wide cancellation of all vectorial error terms of degree below p and controlling the local remainder, the certificate guarantees a minimum local convergence order that remains valid when particular problems exhibit higher order. We organize convergence claims by distinguishing scalar, separable, trajectory-dependent, directional, and class-guaranteed orders, and introduce the structural order profile and coupling-order defect. Rooted composition trees preserve multilinear nesting and stage substitutions. We prove that polynomial scalarization is injective through degree three and loses injectivity from degree four onward, while norm-dependent weights require additional directional analysis. Finite-jet realizations provide explicit counterexamples and separate sharpness witnesses. We generalize the established CM4 scheme into a parameterized two-step family in which CM4 is recovered for α=0. We apply the proposed certification framework to establish the GV₄ property of this family through a composition-preserving proof and obtain a class-wide eighth-order lower bound for the Jacobian-defect method JD8. Explicit coupled realizations demonstrate the sharpness of both bounds. A complementary coupled counterexample excludes universal eighth-order convergence for WFM1–WFM5. A numerical assessment of 19 published methods also reveals cases in which the stated orders are not reproduced on the coupled problems considered, highlighting the relevance of the structural assumptions supporting convergence claims. Finally, we introduce EDOL as a directional diagnostic and two intra-iterative EOLI formulations: an error-based estimator requiring a reference root and a residual-based estimator evaluable along a trajectory without knowing the root. We establish their consistency under stated conditions and use numerical comparisons as evidence complementary to the analytical guarantees.