A mean Newton's method for simple roots

In this work, we study the convergence behavior of a modified Newton's method based on Simpson integral rule. The convergence properties of this method for solving equations which have simple roots have been discussed and it has been shown that it converges cubically to simple roots. And the values of the corresponding asymptotic error constants of convergence are determined. Theoretical results have been verified on the relevant numerical problems. A comparison of the efficiency of this method with other mean-based Newton's methods, based on the arithmetic, harmonic means and geometric means, is also included.