| The problem for solving equationF(x) = 0,where F:D C X→Y is nonlinear operation, is very important in view of theoretics and practice. Not only professional numerical researchers devote themselves to it, but also many engineers and mathematicians study it according to their own demands and interest. Some are the spokesmen among them, whose work reflects the features of the mathematics in their times.For example, in the seventeenth century-the period of foundation of differential calculus,Newton, who is one of founders of differential calculus, produced a new mathematical tool (now we call it Newton iterative method) for solving the nonlinear equation; Halley also raised an iterative method (now we call it Halley iteration) for the problem; hi the eighteenth century--the period of vigorous development of differential calculus, the partial sum of Euler series and Lagrange series formed two iterative families, which included a lot of iterative members. In the nineteenth century, when researchers began to pay attention to the analysis strictness in mathematics, Cauchy put forward major series technique, which was confirmed highly effective in applying it to the convergence analysis of iterations.There are three kinds of convergence theorems which related to iterative method, a) local convergence theorem, b) semilocal convergence theorem, c) global convergence theorem. The local convergence theorem is important because it shows the property of the iterative method near the solution, but the shortcoming is that its codition depends on the unknown solution. So it is necessary to seek the semilocal or global convergence theorem whose convergent condition doesn't depend on the unknown solution. In addition, the calculate efficiency of the iteration is also very important. So we also pay attention to the calculatiing cost of every iterative step, that is, we select the method also according to calculate efficiency.Various iterative methods have been produced since the seventeenth century. Among them, the most classical ones are Newton's method of two order convergence, Chebyshev's method of convergence order of three, Halley's method of convergence order of three, Newton's convex acceleration (or super-Halley) iteration method; and pratical King-Werner iteration and so on. In recent years, the fast development of computers has promoted the numerical analysis research. Defects of some classical methods appear in pratice. And they become more distinct in large scale computing problem. We try to seek deformations of classical methods.In chapter one of this paper, the deformed Chebyshev-Halley iteration family is produced on the real or complex number domain K. And some semilocal convergence theorems about the raised method are given under different kinds of conditions. Moreover, we apply the method to seek zeros of real and complex functions.In chapter two, we put forward an iteration (2.1.1) in which the first derivative and the first divided difference are only calculated in each step. The local and semilocal convergence theorems about the method are also given. In addition, we study semilocal convertgent theorem of another deformed Chebyshev-Halley iteration family-Jarratt type iteration family under the condition that the second Prechet derivative satisfies p-H61der continuity. And we point out that iteration (2.1.1) is much better than Jarratt-type method if the property of the operator isn't good enough. Moreover, we compare iteration (2.1.1) with other two iterations and two-step Newton iteration. All the theoretical conclusions are tested and verified by numerial examples.In chapter three, using the conception of divided difference in Banach spaces, we replace the first derivative in King-Werner iteration by the first divided difference of F and get the deformed King-Werner iteration. We also give the local and semilocal convergence theorems about the deformed method. This deformation improves the speed of convergence, especially for operators whose property isn't good enough. All the t...
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