Dynamics of a Uniparametric Sixth-Order Family for Multiple Roots with Rational-Polynomial Weight Functions

In this study, we have performed a comprehensive dynamical assessment of a sixth-order parametric family of iterative methods for computing multiple roots of nonlinear equations. While existing literature has primarily focused on establishing the theoretical order of convergence, the present work extends this understanding by mapping the stability landscape of the proposed family. Furthermore, the integration of parameter space characterization with fractal visualizations of basins of attraction has elucidated the intricate relationship between the rational map’s fixed points and its global convergence properties. The methodology presented herein offers a robust framework for researchers to evaluate and optimize iterative schemes, ensuring both high computational efficiency and predictable numerical stability in practical root-finding applications.