离散型牛顿法在解非线性方程中的应用.doc

本科毕业设计(论文) 题目名称： 离散型牛顿法在解非线性方程中的应用 学 院： 数学学院 专业年级： 信息与计算科学2009级 学生姓名： 二O一三年 四 月 十七 日 摘 要 牛顿型方法是解非线性方程组的一类重要方法,在非线性方程组迭代解法的理论研究中占有十分重要的地位, 牛顿型方法是逐步线性化方法的典型代表, 牛顿法的收敛性理论及其研究方法,特别是Kahtopobnu的著名论文,对迭代的研究产生了深远的影响. 在通常情况下，非线性算子方程的解不能精确解出，而是用数值方法求其近似解.牛顿法是一种普遍适用的迭代法.它的计算格式简洁，程序简单，而且收敛速度快，适用范围广.多年来，众多学者对经典牛顿法提出多种改进方案，如：萨马斯基提出的修正牛顿法，阻尼牛顿法，拟牛顿法等各种变形.经典牛顿法尽管具有很多优点,但在处理某些不可微问题或导数难计算问题时会遇到一些困难,而离散型牛顿法可以在一定程度上弥补这方面的不足. 本文讨论了牛顿法及离散型牛顿法的半局部收敛性及大范围收敛性，并给出数值算例对此两种方法的执行情况. 关键词：非线性方程；牛顿法；离散型牛顿法；收敛性 Abstract The Newton method is an important method for the solution of nonlinear equations，Occupies a very important position in the theory group iterative method for solving nonlinear equations.The Newton method is a typical representative of successive linearization method, Newton method, convergence theory and research method, especially the famous paper Kahtopobnu, exerted a profound influence on the study of iteration. Generally speaking, we can not solve the nonlinear equations exactly. We always Give the approximate solution by using the numerical methods for nonlinear equations. Newton’s method is one of the most powerful and well-known iterative methods known to converge operator equation. In recent decades, scholars obtained many progresses of the classic Newton ’s method for solving nonlinear equations, Frozen- Newton method given by Samaski, damped Newton method, Quasi- Newton method and other forms. In this paper, we will give the convergence and convergence rate of the modified discrete Newton’s method, again. And numerical examples are given to verify the validity of the method. Moreover, using the modified discrete Newton’s method, we propose the modified continuous Newton’s method. We prove that it is convergence. Keywords: nonlinear equations; Newton’s method; Discrete Newton’s method; con