Dynamical Study via Möbius Conjugacy of a New Third-Order Uniparametric Family for Simple Root-Finding

This article studies the complex dynamics of a parametric family of third-order iterative methods for solving nonlinear equations with simple roots. The fixed and critical points of the corresponding rational operators are determined analytically, and the parameter space is explored for representative real and complex values of \(A\). The regions of convergence and divergence in the parameter space are shown, highlighting how the choice of \(A\) affects stability and basins of attraction. Furthermore, numerical experiments confirm the high efficiency and accuracy of the proposed methods when compared to classical schemes, such as Newton's and Halley's. A key theoretical contribution is the proof that this family's dynamical behavior is equivalent to that of the Gander, Chun, Jarratt-Gander, and Traub-Gander classes, thereby unifying their dynamical analysis.