有关三对角矩阵的数值分析

摘要

3对角矩阵是1类很重要的特殊矩阵,在数学和物理学中有广泛的应用.文章将根据3对角矩阵的特征，用待定系数法求解3对角线性方程组的.数值解，并与常用的LU分解法从理论分析和数据实验两方面进行比较,结果表明,两者的时间复杂性前者稍差,而精度两者则相当,最后写出两者的C程序并运行结果.接下来用1种简单和容易实现的方法求出3对角矩阵的行列式，再利用其逆矩阵可以分解成两个很特殊的矩阵的乘积，给出1种算法实现3对角矩阵的逆的简便计算。
关键字：3对角矩阵；待定系数法；数值解；行列式；逆

Abstract
The tridiagonal matrix is a kind of matrix that with important special,it has widespread applications in mathematics and physics.In this paper,based on the characteristic of the tridiagonal matrix,the method of hypothetical coefficient is used for the numerical solution of tridiagonal system of linear equations,this method will be compared with the LU resolving
method through theory analysis and data experiment,compared the two methods,we will find the latter is better than the former in time complexity slightly ,but the precision is matched with each other,finally write the C procedures for the two methods and get results. The next part,an easy algorithm will be used to compute the determinant of the tridiagonal matrix.the inverse can be pided into two so special matrices that we can compute out the explicit inverse via an algorithm.
Keywords:tridiagonal matrix;numerical solution;determinant;inverse

目录

前言…………………………………………………………………………………………………………1
1 两类求解3对角方程组的数值方法……………………………………………………………………2
1．1 问题引入 ………………………………………………………………………………………2
1．2 待定系数法求解3对角方程组 ………………………………………………………………2
1．3 LU分解法求解3对角方程组…………………………………………………………………7
1. 4 算法性能分析 …………………………………………………………………………………9
2 关于3对角矩阵的行列式 ……………………………………………………………………………12
2．1 问题引入………………………………………………………………………………………12
2．2 方法提出………………………………………………………………………………………12
2．3 算法性能分析…………………………………………………………………………………13
3 3对角矩阵逆的数值解法 ……………………………………………………………………………15
3．1 问题引入………………………………………………………………………………………15
3．2 算法推导及实现 ……………………………………………………………………………15
3．3 程序与数值例子………………………………………………………………………………17
结论 ………………………………………………………………………………………………………20
参考文献 …………………………………………………………………………………………………20
致谢 ………………………………………………………………………………………………………21