Computationally Efficient Three-Step Derivative-Free Iterative Scheme for Nonlinear Algebraic and Transcendental Equations

Abstract

The solution of nonlinear algebraic and transcendental equations is fundamental in various fields of science and engineering, with applications from mathematical modeling to biological modeling, including population growth, blood rheology, and neurophysiology. Traditional methods, such as Newton’s method, often rely on derivatives and face challenges like slow convergence, high computational cost, and failure in cases where derivatives are difficult or impossible to compute. This research presents an efficient derivative-free iterative scheme constructed by Obadah Said Solaiman’s method and Newton method, in which derivatives are approximated by using forward difference and Pade approximation. The proposed scheme is designed to achieve rapid convergence while minimizing computational cost. Numerical experiments are conducted to compare the performance of the proposed scheme against existing methods, highlighting its superior efficiency and reliability in various test problems. The results show that the developed scheme is useful for solving nonlinear equations, particularly in scenarios where derivative computation is infeasible or computationally expensive.