On a Uniparametric Class of Sixth-Order Multiple-Root Finders Using Rational Weighting

This investigation provides a comprehensive analytical framework for the topological morphology and global convergence dynamics governing a specific family of sixth-order iterative schemes designed for nonlinear equations with multiple roots. By invoking a Möbius conjugacy transformation upon the specialized polynomial class f(z)=((z−p)(z−q))m, we project the iterative sequence onto the Riemann sphere C^, effectively recasting the algorithm as a discrete complex dynamic system. The core of this study lies in the bifurcation analysis of the associated parameter space. We meticulously chart the stability manifolds, tracing the evolution of critical orbits to distinguish between regions of predictable convergence and those characterized by chaotic instability. By examining the iterative methods generated by these rational endomorphisms, the research unveils the intricate fractal boundaries that delineate the basin of attraction, offering a profound insight into the structural robustness of higher-order methods. In the dynamical plane, the geometry of the basins of attraction is scrutinized to evaluate the robustness of the numerical flow and its sensitivity to the configuration of weight functions. By analyzing the fractal complexity of the boundaries within these basins, we provide a detailed characterization of the iterative morphology and its global reliability. The analytical findings are supported by high-resolution graphical representations and comparative numerical data, illustrating the superior performance and structural integrity of the proposed methods in solving nonlinear problems.