线性方程组的直接解法及matlab的实现

本科毕业论文

（ 2010 届）

题目线性方程组的直接解法及matlab的实现

学院数学与信息工程学院

专业数学与应用数学

班级2006级数学1 班

学号**********

学生姓名胡婷婷

指导教师王洁

完成日期2010年5月

摘要

随着科技技术的发展及人类对自然界的不断探索模拟.在自然科学和工程问题中的很多问题的解决常常归结为线性代数问题！

本文的主要内容是对线性方程组求解方法的探讨,主要介绍了四种求解线性方程组的方法,第一种是教科书上常见的消元法,我们称之为基本法.第二种方法是标准上三角形求解法,即将增广矩阵经过初等变换后化成标准上三角形,然后求解.它改进了一般教科书上的常见方法,与常见方法比较有如下优点:1)规范了自由未知量的选取;2)只用矩阵运算;3)减少了计算量.第三种方法是对特定的方程组（系数矩阵A为n阶对称正定矩阵,且A的顺序主子式均不为零.）的求解方法进行描述,并且为这种线性方程的求解提供了固定的公式化的方法.第四种方法是对现在实际问题中常常会遇到的系数矩阵为三对角矩阵的方程组的求解方法.同时给出这几种方法的数值解法（matlab程序）,由于运用电脑软件求解,所以必须考虑计算方法的时间、空间上的效率以及算法的数值稳定性问题,所以针对不同类型的线性方程组有不同的解法.但是,基本的方法可以归结为两大类,即直接法和迭代法.

关键词

高斯消去法;三角分解法;乔莱斯基分解法;追赶法

Abstract

Systems of linear equations are associated with many problems in engineering and scinence ,as well as with applications of mathematics to the social sciences and the quantitative study of business and economic problems.

The main content of this article is the method for solving linear equations, we introduce four methods for solving linear equations in this paper. The first is the elimination method which is commonly found in textbooks, and we call the Basic Law. The second method is Standard on the triangle Solution, that first change Augmented matrix into standards in primary triangle, and then solving. It improves the general textbook on common methods, compared with the common method has the following advantages:1) Specification of the free choice of unknowns; 2)Only matrix operations;3) Reduce the computation. The third method describes a way to solve a Specific equations(N coefficient matrix A is symmetric positive definite matrix, and A are not zero-order principal minor), And for this linear equation provides a fixed formulaic approach. The fourth method is to present practical problems often encountered in the coefficient matrix is tridiagonal matrix method for solving the equations. These methods are given numerical solution of (matlab program), As the use of computer software to solve, it is necessary to consider ways of computing time and space efficiency and numerical stability of algorithms, Therefore, different types of linear equations have a different solution. However, the basic method can be classified into two categories, namely direct methods and iterative methods.

Key words

Gaussian elimination; Triangular decomposition; Cholesky decomposition method;

Thomas algorithm

目录

1. 引言 (1)

2．相关知识 (2)

2.1 向量和矩阵 (2)

2.2 特殊矩阵 (3)

3.问题叙述 (3)

4．问题分析 (4)

4.1高斯分解法 (4)

4.2三角分解法 (6)

4.3乔莱斯基分解法 (6)

4.4追赶法 (7)

5. 举例说明与总结 (9)

5.1举例说明 (9)

5.1.1高斯分解的matlab程序方法 (9)

5.1.2三角分解法的matlab程序方法 (10)

5.1.3乔莱斯基分解法的matlab程序方法 (11)

5.1.4追赶法的matlab程序方法 (13)

5.2总结 (14)

参考文献 (16)

谢辞 (17)