Modified Scheme Of Newton's And Its Convergence Order

Newton method is a very important method to solve the nonlinear operator equation f(x)=0, the main object of this paper is to investigate modified scheme of Newton's iterative method. This paper is made up of four sections. In section one, some relervent theorems about Newton method are interpreted. In section two, Newton-Cotes formula and Gauss-Legendre formula are used to construct modified scheme of Newton's iterative method respectively in the case of using numerical integral formula with equidistant nodes and nonequidistant nodes, and the convergence order of the two modified schemes is 3. Therefore, we give a final conclusion for iteration of Newton's method constructed by integral formula. In section three, we investigate the solution of nonlinear operator equation F(x)=0 in Banach space. First, we generalize the trapezium formula about numerical integral of real function to the Bochner integral of nonlinear functional so that we obtain the trapezium formula of Bochner integralThen we use the formula to construct modified scheme of Newton's iterative method so as to obtain trapezium Newton's methodFuthermore , we proved its convergence underα-criterion of weak conditions by means of majorizing function. In section four, we prove a scheme of Newton's iterative method such asis optimal when a=d,b=0,c=(f~"(α))/(2!f~'(α))d and its convergence order is 3.