| The algorithm problem of solving nonlinear equation in Banach spaceF(x) = 0has been studied by many numerical scientists. One of the main algorithms is iterative method, so now, the research of iterative methods become the hardcore of finding solution to all kinds of nonlinear problems. Whether the nonlinear problems will be solved well or not is directly affected by the choice of iterative methods. So it is very important and meaningful to do the research of iterative methods.There are the well-known second-order Newton's iteration, third-order Halley's iteration, Chebyshev's iteration ,Super-Halley's iteration and their deformations, and so on. This dissertation consists of five chapters.It mainly makes an analysis on the convergence of a family of iterations with cubic order which can avoid the computation of the second Frechet-derivative under Kantorovich conditions.In Chapter 1,we summarize several iterative methods and their convergence condition .At the same time ,we also present the techniques in proving the iterative method's convergence theorem..In Chapter 2, we study the convergence of deformed Chebyshev method under γ— condition by means of the majorant method.Moreover,we find this method not only can avoid the computation of the second Frechet-derivative but also has the convergence of cubic order.In Chapter 3,we derive the second-order-derivative-free iterations with two parameters from the third-order iterations with one parameter to approximate the roots of nondifferentiable equations in Banach space.We also provided a existence-uniqueness theorem by using a new system of recurrence relationsIn Chapter 4, For the family of iterations which mention in the chaper 3.,we give a result on the existence of a unique solution for the nonlinear equation by using a techniquebased on a new system of recurrence relations under the same Lipschitz condition as for Newton's method.In Chapter 5, we give two numerical cases. |
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