Iterative Methods With Higher-order Convergence For Solving Nonlinear Equations And Analysis Convergence

Nonlinear problems is one of the main fields in modern mathematical research. The iterative method is an excellent method to solve nonlinear operator equations f(x) = 0 in Banach space. Following the development in research of Mathematics and the appearance and consummation of mainframe computer, several kinds of nonlinear problems have been regarded by many scientists and engineers sooner or later. They have more and more interest in these questions, especially, some key questions in engineering calculation of recent physics and science are depended on the solution of nonlinear equation. In numerical scientific computation, especially for complicated scientific and engineering 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. For the past several decades, the rapid advancement of computers has effectively promoted the research on numerical analysis. Strictly examined by practices, some classical methods are proved to have some drawbacks which remain especially evident in the solving of some practical problems with mass calculational scales. The computational efficiency plays a crucial role in large scale calculation, which calls on us to avoid applying computational methods with low efficiency. Thus, while considering the convergence order of computational method, we also take computational cost in every step into consideration particularly. Therefore, the research on iterative method with high efficiency means a lot in terms of both scientific research and practical application.This paper consists of four chapters. In chapter 1, we summarize mainly about the development of iterative methods during the past several centuries and introduce several representative iterative formulae and some basic theories on relevant iterative methods. For the past several decades, professional numerical researchers have kept coming up with various new iterative methods, most of which, in fact, are the modification and deformation of classical iterative method according actual necessaries. Thus, a series of classical iterative methods such as the Newton method and so on is the starting point in our discussion on the new methods. Mathematicians have conducted profound researches in this regard, producing large numbers of papers and literatures, from which we have profuse theoretical fruit and skills to borrow.In the second chapter, we present a family of iterative methods with higher-order convergence for solving nonlinear equations, and analysis and prove the convergence by basic method of mathematical analysis. The higher-order convergence we discussed here is different from third or fourth order iterative method with fixed convergence order in the opening chapter, but refers to those whose concrete formula and convergence order change under different circumstances. Theoretically, the more complex the condition is, the more complex the form is, and the higher the convergence is. Take Euler and Halley iterative methods as examples, they have high convergence order, while entailing the calculation of derivative of higher order. The new method we put forward here which needn't evaluate second or higher derivative of the function but first derivative can have higher-order convergence. Compared with other methods, our has lower computational complexity. In particular, it presents an obvious advantage regarding the problems on n-dimensions space. Besides, we display some examples further indicating that our method can have higher convergence rate without the computation of the higher-order derivative of the function.In the third chapter, we combine the Newton and other iterative methods, producing two families of new iterative methods. Here, we concern our efforts on the introduction of its construction technique and discussion on its convergence. Combination means that we formulate a new iterative method using two same or different iterative method. Generally speaking, the convergence of combined iterative method can be improved, but its computational cost also increases accordingly. However, the convergence order of the combined iterative method we produced here through only adding one evaluation of the function at another iterated point. Then we use this combined technique to construct several actual iterative methods, and compare them with other methods by applying them in some calculational problems, indicating that our methods has higher computa- tional efficiency.In the fourth chapter, we introduce a deformed Jarratt method. We needn't computing the inverse of first derivative in the new method. Using the majorant function, we discuss the convergence analysis of the deformed method under the condition of Kantorovich, and come up with convergence theorem. We also prove that the new method need not calculate any inverse of derivative value. Applied in the practical problems, this can promote the efficiency greatly.