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

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.