The Iterative Method With Higher Order Convergence For Nonlinear Equation

With the development of science technology and the updating of electronic information technology, many problems about scientific research and engineering calculation can be constructed by the mathematical model. We can translate the practical problems into solve the root of mathematical equations（it contains linear or nonlinear algebra equations）.The main research of this paper is that the iterative methods to solve the root of nonlinear equations f（x）=0. This study includes three parts:Chapter one introduces the actual background of nonlinear equation, recalls the development situation of iterative method, and some basic conceptions.Chapter two proposes a new three-step with sixth-order iterative method based on Ostrowski’s fourth-order convergence iterative method and M.Grau’s sixth-order convergence iterative method. Per iteration of the new method requires three evaluations of the function and one evaluation of the first derivative, the efficiency index of new method is 6^1/4≈1.565. At the end of the chapter, the method is verified by numerical examples.Chapter three proposes two new families of three-step with eighth-order iterative methods. We introduce Hermite interpolation method, reduce the amount of calculation of derivative values by using the function fitting to approximate the original function. A new family of three-step with eighth-order iterative method is proposed based on W.Bi’s eighth-order convergence iterative method and X.Wang’s eighth-order convergence iterative method. The range of the new method is more extensive and contains X.Wang’s eighth-order iteration if taking some certain values of unknown quantities.Another eight-order iterative method is proposed by replacing the new derivative values with a new real-valued function. Per iteration of two new methods require three evaluations of the function and one evaluation of the first derivative, the efficiency index of two new methods is 8^1/4≈1.682. The convergences of two methods areverified by numerical examples.