An iterative method based on the average quadrature formula

In our research, a new third-order iterative method has been introduced. This method involves the creation of a new quadrature formula by averaging Simpson's 1/3 rd and Trapezoidal rules. The newly developed quadrature formula is then used to establish the new iterative scheme, which modifies the Newton-Raphson method. It has been demonstrated that the new iterative technique exhibits a convergence order of 3. Finally, examples have been provided to illustrate the effectiveness of the new process. The results indicate that the new approach finds the root of the nonlinear equation in fewer iterations compared to other methods, suggesting the potential superiority of our newly developed scheme.