Solving The Iterative Method Of Nonlinear Equations

Tthe iterative method is the most important and convenient one for solving equations in the form off(x) = 0in C(orR) space. So far, we have had the following representative iterative methods: the well-known second-order Newton iterative;practical 1-order deformed Newton iterative with lagged and with the advanced derivative evaluttion; third-order Halley's iterative, Chebyshev's iterative, Super-Halley's iterative(or the convex acceleration of Newton iterative), etc. .This dissertation consists of there chapters, In the first chapter, we discussed the important of solving nolinear equation in science research. In chapter two, we compares various iteration with Efficiency Index denoted by E= logp/C, and point out that the Traub iterativeis better than others in some ways.In the chapter there, we give a following convergence theorem and error estimation for the Traub iterative under new Kantorovitch-ostrowski type condition use the information of higher derivatives at initial points. Theorem2.1. Given0x0,x-1,x-2 D X. aconvex subsett of X, and f(x) have first-and second-order derivatives on D,and t is the smaller positive root of the equationthen, the sequence {xn} starting from X0,x-1,x-2 denned by Traub iterative con-verges to x, the unique solution of f(x) in O(x0,t) O(x0,t). Theorem 2.2 Under the assumption of Theorem 2.1, we have:and t,t (0 < t< t) are two positive zeros of the (x). And Fn defined byCompared with the corresponding study [20], the convergence determination is established under one global condition, instead of two, on the function.