Abstract- In this study, a numerical method for determining the root of non-linear algebraic and transcendental equations is developed. Here, a two-step derivative free method for solving non-linear algebraic and transcendental equations is provided. This approach has an order four accuracy with three functional evaluations and no derivative. The generated scheme has an efficiency index of 1.587. The novelty is the use of Traub’s method as a first and second step.  Numerical examples, real-world problems, and a study of dynamics are utilized to demonstrate the performance of the provided scheme and to compare it to other approaches of the same order that are accessible in the literature. For numerical results and basin of attraction, the software MATLAB, Mathematica, and Maple are used.

Keywords: Non-linear Equation, Convergence Analysis, Efficiency Index, Basin of attraction, Derivative free