A new iterative multi-step method for solving nonlinear equation.

This study introduces an advanced iterative technique designed to solve nonlinear equations with simple roots efficiently. The newly developed algorithm achieves an impressive convergence order of sixteen, utilizing only five functional evaluations per iteration. By incorporating appropriate finite difference approximations, the method avoids the need for second derivatives, enhancing its computational efficiency and broad applicability. A detailed theoretical analysis of convergence is provided, highlighting the method's superior performance. Furthermore, numerical experiments are carried out to assess its reliability and effectiveness against established methods. The findings reveal that the proposed technique surpasses several renowned iterative methods in both accuracy and computational efficiency.•High-order convergence: The method demonstrates a sixteenth-order convergence rate, requiring just five function evaluations per iteration, which enhances computational efficiency.•Derivative-free approach: The algorithm employs finite difference approximations, eliminating the need for second derivative calculations and increasing its applicability to complex problems.•Numerical validation: Comprehensive numerical tests and comparative studies validate the exceptional accuracy and efficiency of the proposed iterative approach.