MATLAB在求非线性方程近似解中的应用

范文编号:XXLW116 范文字数:8177,页数:21

摘要

 在实际工程和科学计算中，经常会遇到求解高次代数方程或超越方程问题，我们把这些方程统称为非线性方程。在非线性方程中，除了二次、三次、四次代数方程外，求解其他的方程不但没有一般的公式，而且若只依据方程本身来判别是否有根及根的个数是很困难的。因此，我们需要寻求非线性方程根的比较精确的近似解。但是如果我们直接用在大学数学中学习的几种传统的方法求解不仅难度较大而且需要做大量繁杂的计算，本课题旨在利用MATLAB数学软件,通过传统的方程求解思路，编写出对应的MATLAB程序来求解。这里主要有二分法、简单迭代法、牛顿迭代法三种解题思路，编写出程序后，再将这三种方法进行比较，判断其优劣。研究结果表明利用MATLAB数学软件可以省略大量繁杂的计算，并使求解的精确度大大提高，且三种方法中牛顿迭代法收敛最快。
 关键词：  MATLAB  非线性方程 二分法  简单迭代法  牛顿迭代法  程序
Abstract
 In practical engineering and scientific calculations, you often encounter the problem of the solution of higher order algebraic equations or transcendental equations, we put these equations are collectively referred to as non-linear equations. In non-linear equations, in addition to the quadratic, cubic and quartic equations, solving other equations not general formula, but if the only basis for the equation itself to identify whether there is a root and the number of roots are very difficult. Therefore, we need to find the roots of nonlinear equations more accurate approximate solution. But if we direct use the traditional method learned in College mathematics for solving not only difficult but also need to do a lot of complicated calculation, This topic is designed to use MATLAB mathematics software, through the traditional thinking, writing for the corresponding MATLAB programs to solve. Here mainly have dichotomy, simple iterative method, Iterative method Newton three thoughts, write a program, then compared these three methods to determine their advantages and disadvantages. The results show that the use of MATLAB software can omit a lot of complicated mathematical calculations, and make greatly improve the accuracy of the solution, and Iterative method Newton is the fastest convergence in the three methods.
 Key words: MATLAB    Nonlinear equations    Dichotomy   Simple iterative method Iterative method Newton   Program
 
目  录
中文摘要 i
英文摘要 ii
目录 iii
第一章  绪论 ... 1
 1.1   研究动机与目的 1
1.2   研究背景 1
1.3   研究方法 2
1.4   范文内容概述 2
第二章  MATLAB简介  4
 2.1   MATLAB发展史 4
 2.2   MATLAB的特点 5
第三章  解题思想及程序编写 6
3.1   二分法 6
            3.1.1   二分法的基本原理 6
            3.1.2   二分法的MATLAB实现 7
 3.2   简单迭代法 8
           3.2.1   简单迭代法的基本原理 8
           3.2.2   简单迭代法的MATLAB实现 9
 3.3   牛顿迭代法 10
 3.3.1   牛顿迭代法的基本原理 10
           3.3.2   牛顿迭代法的MATLAB实现 11
第四章  三种方法之间的比较 13
  4.1   二分法实例求解 13
4.2   简单迭代法实例求解 14
4.3   牛顿迭代法实例求解 14
4.4   列表比较 15
4.5   小结 15
第五章  结论 16
致谢 17
参考文献 18

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