Two New Predictor-Corrector Iterative Methods with Third- and Ninth-Order Convergence for Solving Nonlinear Equations

In this paper, we suggest and analyze two new predictor-corrector iterative methods with third and ninth-order convergence for solving nonlinear equations. The first method is a development of [M. A. Noor, K. I. Noor and K. Aftab, Some New Iterative Methods for Solving Nonlinear Equations, World Applied Science Journal, 20(6),(2012):870-874.] based on the trapezoidal integration rule and the centroid mean. The second method is an improvement of the first new proposed method by using the technique of updating the solution. The order of convergence and corresponding error equations of new proposed methods are proved. Several numerical examples are given to illustrate the efficiency and performance of these new methods and compared them with the Newton's method and other relevant iterative methods. Keywords: Nonlinear equations, Predictor–corrector methods, Trapezoidal integral rule, Centroid mean, Technique of updating the solution; Order of convergence