数学与应用数学毕业论文设计计算机科学中的重要数学思想—迭代

本科毕业论文（设计）

题 目： 计算机科学中的重要数学思想—迭代法 学生姓名： 某某 学号：A8888888

院（系）： 数学科学学院 专业：数学与应用数学 入学时间： 二○○六 年 九 月 导师姓名： 李某某 职称/学位： 教授 导师所在单位： 数学科学学院 毕业论文（设计）提交时间： 二○一○ 年 五 月

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计算机科学中的重要数学思想—迭代法

摘要

本文将探讨数学中重要的思想方法—迭代的实际应用。主要介绍几种常见问题的迭代方法如：求解非线性方程f(x)=0的不动点迭代、二分法、牛顿迭代法，还有求解线性方程组及非线性方程组的各种迭代法。并对各种迭代法的收敛性进行讨论和比较，讨论各种迭代法的优缺点。在分析结果的基础上我们可以看出迭代法的实际应用性很强，对于计算机上的应用尤为突出。研究结果告诉我们：在具体的应用中要根据实际情况来选择不一样的迭代法，也可以将几种方法结合来运用。

关键词

迭代；收敛性；牛顿法；雅克比迭代；塞德尔迭代；非线性方程；（非）线性方程组。

An important Mathematics thought from the computer

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science---- Iteration Method

Abstract

In this paper, we will discuss an important mathematics thought of iteration method and discuss its realistic application First, I will introduce a lot of iteration method from some natural question ，for example ：the Fixed-point iteration ，the Bisection Method ，Newton method ，and some iteration method of solving linear equation set or not linear equation set。Second ， we will discuss and compare the astringency of those methods 。We will find out the advantages and disadvantages of several methods for iteration。From analysis the result of iteration method ，we can find it has lots of action in reality，particularly in computer science。According to the analysis result，we get that we use iteration method must base on reality，and also we can combine different method to deal with some problem around us。

Keywords：astringency；Newton Method；Jacobi iteration；Gauss-Seidel iteration；nonlinear equation；linear or nonlinear equation set。

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目 录

第一章 迭代法·····································4 1.1 迭代法简介··································4 1.2 运用迭代法的前提准备························4 第二章 不动点迭代法·······························5 2.1 不动点的寻找································5 2.2 绝对误差与相对误差··························6 第三章 二分法·····································8 3.1 波尔查诺二分法······························8 3.2 试值法的收敛性······························9 第四章 牛顿-拉夫森法与割线法······················11 4.1 求根的斜率法································11 4.2 收敛速度与缺陷······························13 4.3 割线法与加速收敛····························16 第五章 求线性方程组的迭代法·······················19 5.1 雅克比迭代··································19 5.2 高斯-塞德尔迭代与收敛性·····················20 第六章 非线性方程组的迭代法·······················24 6.1 理论与广义微分······························24 6.2 接近不动点处的收敛性························25 6.3 赛德尔迭代法与牛顿法························26 结束语·············································29 主要参考文献·······································30 致谢···············································31

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