Optimizing Nonlinear Root Finding: A Novel Fourth Order Derivative-Free Iterative Method with Maximum Efficiency Index and Comparative Analysis

Key Points

• This fourth order iterative method finds roots of nonlinear equations efficiently, demonstrating a significant advancement in root-finding techniques.
• Maximum efficiency is achieved with only three functional evaluations needed per iteration, enabling rapid convergence in numerical solutions.
• The method is analyzed against existing techniques, including the Traub-Ostrowski and modified King's methods, highlighting its comparative advantages.
• Results indicate that this novel approach provides improved accuracy and efficiency, which may transform practices in numerical analysis.