2016届毕业生 毕业论文
题 目： 非线性方程求重根方法研究 院系名称： 理学院 专业班级： 学生姓名： 学 号： 指导教师： 教师职称：
2016年05月20日
摘 要
随着科学技术的发展，在现代科学和工程技术中，经常会遇到大量而复杂的数学计算问题。这些问题常常归结为非线性方程求根的问题。求解非线性方程的单根已经具有了比较成熟和丰富的构造技术手段。例如，其中在工程和其他领域的科学计算中的广泛应用迭代算法，它从某个初始点出发，由迭代格式生成一种收敛于方程根的序列。这些方法在面对非线性方程单根的时候可以很好的解决问题，然而这些方法在求解非线性方程的重根时，构造的算法显得相当的复杂甚至是无效的。举一个简单的例子就是平时我们经常研究的经典的牛顿迭代法。它对方程的单根二阶收敛，但是对于于方程的重根只能线性收敛，并且收敛速度变慢。因此非线性方程重根的高阶，尤其是最优解的迭代格式如何构造是一项具有挑战性的工作。直到现在，这方面的研究成果还不是很丰富。目前绝大多数求重根的最优阶迭代算法都是利用方程重根的重数信息来构造迭代格式。对于各种求非线性方程求重根这一问题，国内的许多数学界的前辈对此从不同的方面展开了研究，并在不同方面取得了一定的成果。全文共分为三章
第一章概述了相关的基础理论知识，主要介绍了非线性方程求根的研究背景和及研究现状，着重介绍了迭代法的相关知识，探讨了几种求非线性方程的解的方法，论述了各个解法的优缺点。
第二章主要介绍了迭代法在非线性方程求重根的情形下的应用，给出了几种新的修正迭代格式，从各个思路对非线性方程求重根进行了探讨，并且了解了一些其他求非线性方程重根的方法。 第三章是总结了全文主要的讨论内容。
关键词: 非线性 二分法 迭代 收敛 迭代加速 牛顿法 修正牛顿法 重根 阶乘法
I
Title Nonlinear equation root method and study
Abstract
With the development of science and technology,people often encounter large and complicated mathematics problems in the modern science and engineering. These questions often come down to the problem of nonlinear equation for the root. To solve the nonlinear equation of single has mature technology and rich structure. For example, one in the field of engineering and other scientific computing is widely used in the iterative algorithm，It starting from an initial point, generated by the iterative format a sequence converges to equation root.These methods when he faced the nonlinear equation of single can well solve the problem, however, these methods in solving the nonlinear equations of roots, the structure of the algorithm is quite complex and even invalid.A simple example of this is we often study at ordinary times the classic Newton iteration method.It to the equation of single second order convergence, but for the equation of double root only linear convergence, and slow convergence speed.So the roots of the high-order nonlinear equation, especially iterative format how to construct the optimal solution is a challenging job.Until now, the research achievements are not very rich.At present, most of the multiple roots optimal order iterative algorithm is using heavy equation root of multiplicity information to construct the iterative format.For a variety of heavy to nonlinear equations for the root of this problem, the predecessor of many domestic to this from different aspects, and has obtained certain achievements in different aspects.Full text is divided into three chapters
The first chapter summarizes the related basic theoretical knowledge, mainly introduced the research background of nonlinear equation for the root and and the research status, introduces the iterative method of related knowledge, discusses several ways to the solution of nonlinear equations, the advantages and disadvantages of each method are discussed.
The second chapter mainly introduces the iterative method in nonlinear equations
II
roots under the situation of the application, several new modified iterative format is given, from different way of thinking are discussed in this paper, the roots of nonlinear equations for heavy and learning some other nonlinear equation root method.
The third chapter summarizes the full text is the main discussion.
Keywords:
method
Nonlinear dichotomy iterations convergence an iterative
acceleration Newton's method modified Newton method multiple root factorial
III
目 录
1 非线性方程求根的基本方法 ........................................... 1
1.1 非线性方程求根 ................................................ 1 1.2 迭代法的基本思想 .............................................. 2 1.3 二分法 ........................................................ 2
1.4 不动点迭代法 .................................................. 3 1.5 迭代法的收敛性 ................................................ 4 1.6 迭代法的收敛速度 .............................................. 6 1.7 迭代加速收敛的方法 ............................................ 7
1.7.1 Aitken加速方法 .......................................... 7 1.7.2 Steffensen迭代方法 ...................................... 8 1.8 Newton法 ..................................................... 9
1.8.1 Newton法及其收敛性 ...................................... 9 1.8.2 简化牛顿法及牛顿下山法 ................................. 10 2 非线性方程求重根方法研究 .......................................... 12 2.1 牛顿法在非线性方程求重根时的情形 .............................. 12
13 2.1.1 已知根的重数m ........................................ 14 2.1.2 未知根的重数m ........................................ 2.2 牛顿法在非线性方程求重根的情形下的一些改进方法 ............... 15
2.2.1 无约束优化技术中的牛顿法 ............................... 15
.............. 18 2.2.2 Aitken加速外推下的修正牛顿法求重根重数
20 2.3 一些其他的求非线性方程重根的方法 .......................... 总 结 ............................................................... 22
参 考 文 献 ......................................................... 23
IV