Graphical Insights into Dynamical Behavior: Advanced Iterative Methods for Root Finding in Computational Mathematics

This paper introduces a novel family of Newton-like iterative methods ranging from fourth to seventh order, specifically designed for the efficient determination of simple roots in nonlinear equations. Outperforming existing methods, our proposed techniques exhibit superior stability and expanded basin sizes. Their unprecedented reliability and broad applicability position them as valuable assets for addressing real-world challenges in diverse domains such as Mathematical Modeling, Biomathematics, and Thermodynamics. To illuminate the dynamic behavior of these methods, we employ the graphical tool 'Basin of Attraction'. Through comprehensive analysis, our research demonstrates a significant enhancement in accuracy and convergence rates, presenting a modest yet impactful contribution to the realm of computational mathematics. Mathematics Subject Classification: 65H04, 65H05