Enhancing Efficiency of Newton-Raphson Method with Chaotic Mappings for Nonlinear Equation Solving

One of the most powerful methods for solving nonlinear equations is the Newton-Raphson algorithm. Although this algorithm is highly efficient, it faces two challenges: sensitivity to the starting point and the possibility of getting stuck in a loop for the solution sequence. In this paper, by adding a small disruptive term to the Newton recursive relation, we practically generate a chaotic pseudo-random sequence, thus addressing these two challenges. The effectiveness of the proposed version has been numerically demonstrated on several nonlinear equations. The results show that the improved version is not sensitive to the initial conditions and escapes from the loop due to the generation of a random sequence, while maintaining an acceptable execution time due to the use of a deterministic system in generating the random sequence.