New Three Step Steffensen Simpson Method with Seventh Order Convergence For Solving Nonlinear Equations Using Halley Method

In this paper, we suggested and analyzed new higher- order iterative methods for solving of nonlinear equations of the form . This method based on a Halley iterative method and the combination of mid – point with Simpson quadrature formulas and using predictor – corrector technique. The new method is also based on steffensen’s method. The new method is called STEFFENSEN SIMPSON method ( ). The convergence analysis of our methods are discussed. It is established that has convergence order seven. We have compared this Steffensen Simpson method ( ) with some other iterative method which shows that this new method ( ) is a robust one. Numerical method for finding approximate solutions of the nonlinear equation are being developed by using several different techniques including Taylor series, quadrature formulas, decomposition technique and variational iteration technique. Numerical tests show that the new three step iterative method  is comparable vary fast convergent than the well – known existing methods and provides better results.