【作者】 鲍安戈；

【导师】 徐秀斌；

【作者基本信息】 浙江师范大学 ， 应用数学， 2007， 硕士

【摘要】 我们在利用数学手段研究自然现象和社会现象,或解决工程技术问题时,常常会把不少实际问题归结为Banach空间中形如F(x)=0的非线性方程的求解。而求解非线性方程的一个最有效方法则是迭代法。因此迭代法的研究有着十分重要的科学价值和实际意义。这篇论文共分为五个部分。本文在第一章中给出了全文要用到的一些基本概念和记号,总结了证明各种迭代法收敛性的技巧以及几个著名迭代法的收敛条件。在第二章中,针对Sharma提出的一个三阶收敛复合型Newton-Steffensen迭代方法,在假设算子的一阶导数满足(K,p)-Holder连续条件时进行了讨论,得到了其半局部收敛定理,同时在相应条件下给出了其局部收敛定理。在第三章中,研究了复合型Newton-Steffensen迭代方法在算子的二阶导数满足Lipschitz连续条件时的半局部收敛性,并利用优函数技巧得到了其在该条件下的误差估计,最后给出了这个定理的一个应用。在第四章中,对于近年来出现的一类特殊高阶牛顿型迭代方法,我们对其由来给出了一个统一构造技巧,并用这一技巧在著名Halley迭代基础上构造了一个高阶Halley型迭代方法,并证得其局部收敛阶为五。最后通过几个数值例子将该方法与经典迭代法进行比较,结果表明新Halley型迭代法具有很好的可行性。在第五章中,通过优序列技巧,对第四章得到的新Halley型迭代方法在算子二阶导数满足广义Lipschitz条件时,给出了其半局部收敛定理。更多还原

【Abstract】 When we use mathematical methods to study natural and social phenomena or to solve engineering technique problems, we often regard the solutions of most practical problems as the ones of nonlinear equations in form ofF(x) = 0in Banach space. While iterative methods are the most efficient algorithms for solving nonlinear equations. So it is very important and meaningful to do the research of iterative methods.This thesis consists of five chapters. In Chapter 1, we give some basic definitions and notations which will be used throughout the whole thesis; summarize the techniques in proving iterative methods’ convergence theorems and the convergence conditions of several famous iterative methods.In Chapter 2, under the (K,p)-Ho|¨lder continuous condition of the first Fréchet derivative, we discuss the convergence property of a third order composite Newton-Steffensen method, which was introduced by Sharma , and obtain a semilocal convergence theorem for it. Meanwhile, a local convergence theorem is also given under the similar conditions.In Chapter 3, we study the semilocal convergence for the method appeared in Chapter 2 under Lipschitz continuous condition of the second Frechet derivative, and get the corresponding error estimation by majoriz-ing function technique. And also, an application of the theorem to a non- linear equation is given.In Chapter 4, we present a(?)united structural approach for a special type of high-order iterative methods which were proposed in recent years. By this means, we deduce a new high-order Halley type method based on the famous Halley’s method, and proves that its local convergence order is five. Finally, we give some numerical results with a number of function tests to show that the new method performs better in efficiency comparing to some classical methods.In Chapter 5, by using the majorizing function technique, we establish a semilocal convergence theorem for the method obtained in Chapter 4 under generalized Lipschitz continuous condition of the second Fréchet derivative.更多还原