A Novel Mass Cancellation Mechanism for Constructing an Optimal Sixteenth-Order Modified Newton Method

This study provides a comprehensive theoretical examination and rigorous numerical verification of the innovative modified Newtonian four-stage approach, designed to solve nonlinear equations of the form f(x)=0. The method achieves the optimal order of convergence P=16 using the minimum required number of functional evaluations, d=5 (four function evaluations and one first-derivative evaluation), resulting in a high Efficiency Index of EI≈1.741. The main idea is to use carefully designed filtering factors that perform a specialized "mass cancellation" of the error threshold down to the fifteenth rank. A broad numerical assessment is conducted using benchmark functions and real-world problems, enabling a comprehensive performance comparison with well-established optimal schemes, particularly King-Type methods of order 8 and 16. The obtained results unequivocally demonstrate that the P=16 method provides significantly faster convergence (achieving machine precision in a single iteration,N=1), exhibits higher computational efficiency (EI gain), and offers a more algebraically robust construction compared to classical and modern high-order optimal methods. The analysis identifies the strengths of the Mass Cancellation mechanism, offering guidance for its application in high-accuracy numerical computation. Keywords: Modified Newtonian method, nonlinear equations, four-stage iterative methods, optimal convergence order, efficiency index, mass cancellation technique, high-order root-finding methods.