摘要很多领域都有涉及到非线性方程组，例如天气预报，石油地质勘探，电力系统计算等，甚至商业领域也有非线性优化问题，这些问题要从本质上解决就是求出非线性方程组的解.但是目前已知的数值解法并不完善，选择不同的方法，有着不同的收敛速度和计算量，而收敛速度和计算量影响着计算效率，所以数值解法的研究十分重要.58513

本篇论文首先简单介绍了非线性方程组的几种经典数值解法，如Newton法、区间迭代法、不动点迭代法等，并通过几个数值例子，对一些经典的迭代型数值解法在收敛速度、计算量等方面进行对比分析，得出这些算法的优缺点，并研究了Newton法的改进算法.然后研究了同伦延拓法，最后给出了基于单纯形法的萤火虫算法及改进遗传算法来求解非线性方程组.

毕业论文关键词：Newton法；区间迭代法；不动点迭代法；遗传算法；延拓法

Abstract  Many fields related to nonlinear equations, for example, weather forecasts and even life in the field of business, petroleum geological exploration, computing power system also has non-linear optimization problems that solve nonlinear equations is obtained essentially from the solution. However, currently known numerical method is not perfect, choose different methods and have different convergence speed and calculation, and the convergence rate and calculate the amount of influence the calculation of efficiency, so the numerical solution of the research is very important.

     This paper first introduces some classical numerical solution of nonlinear equations, such as the Newton method, interval iterative method, fixed point iteration method, and through several numerical examples, some of the classic iterative numerical solution convergence the speed, the amount of calculation and other aspects of comparative analysis, the advantages and disadvantages of these algorithms, and to study the improved algorithm Newton method. Then we study the homotopy continuation method. Finally, the algorithm based on firefly and improved simplex method of genetic algorithm to solve nonlinear equations.

Keywords:Newton method; interval iterative method; fixed point iterative method; genetic algorithm; continuation method  

 目录

第一章 绪论 1

1.1 选题背景和意义 1

1.2 研究现状 1

1.3 本文研究的主要内容 1

第二章 几种迭代法的分析及改进 2

2.1 牛顿型迭代法 3

2.1.1 牛顿法简介 3

2.1.2 三种牛顿法的比较 5

2.1.3 九阶牛顿型迭代法 12

2.2 区间迭代法简介 17

2.2.1 区间与区间运算定义 17

2.2.2 区间运算 17

2.2.3 区间向量 18

2.2.4 区间迭代法 18

2.3不动点迭代法 19

    2.4 迭代型数值解法小结 21

2.4.1实例分析 21

2.4.2总结 22

第三章 同伦延拓法及延拓法 23

3.1延拓法及同伦延拓法 23

3.2 实例 27

3.3小结