A new iterative method to compute nonlinear equations

The aim of this paper is to construct a new efficient iterative method to solve nonlinear equations. The new method is based on the proposals of Abbasbandy on improving the order of accuracy of Newton–Raphson method [S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation 145 (2003) 887–893] and on the proposals of Babolian and Biazar on improving the order of accuracy of Adomian’s decomposition method [E. Babolian, J. Biazar, On the order of convergence of Adomian method, Applied Mathematics and Computation 130 (2002) 383–387]. The convergence of the new scheme is proved and at least the cubic order of convergence is established. Several examples are presented and compared to other methods, showing the accuracy and fast convergence of this new method. Also, it is shown in this paper, that the modified Adomian’s method developed by Babolian and Biazar to solve nonlinear equations [E. Babolian, J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation 132 (2002) 167–172] should be slightly modified, due to the fact that convergence of Adomian’s method does not ensure convergence of the modified method. An example illustrates this fact, which, unlike what is claimed by the authors, does not converge with their method, but with a simple different choice of the zero component becomes convergent.