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有關三對角矩陣的數值分析
摘要
3對角矩陣是1類很重要的特殊矩陣,在數學和物理學中有廣泛的應用.文章將根據3對角矩陣的特征，用待定系數法求解3對角線性方程組的數值解，并與常用的LU分解法從理論分析和數據實驗兩方面進行比較,結果表明,兩者的時間復雜性前者稍差,而精度兩者則相當,最后寫出兩者的C程序并運行結果.接下來用1種簡單和容易實現的方法求出3對角矩陣的行列式，再利用其逆矩陣可以分解成兩個很特殊的矩陣的乘積，給出1種算法實現3對角矩陣的逆的簡便計算。
關鍵字：3對角矩陣；待定系數法；數值解；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 divided into two so special matrices that we can compute out the explicit inverse via an algorithm.
Keywords:tridiagonal matrix;numerical solution;determinant;inverse
目錄
前言…………………………………………………………………………………………………………1
1 兩類求解3對角方程組的數值方法……………………………………………………………………2
1．1 問題引入 ………………………………………………………………………………………2
1．2 待定系數法求解3對角方程組 ………………………………………………………………2
1．3 LU分解法求解3對角方程組…………………………………………………………………7
1. 4 算法性能分析 …………………………………………………………………………………9
2 關于3對角矩陣的行列式 ……………………………………………………………………………12
2．1 問題引入………………………………………………………………………………………12
2．2 方法提出………………………………………………………………………………………12
2．3 算法性能分析…………………………………………………………………………………13
3 3對角矩陣逆的數值解法 ……………………………………………………………………………15
3．1 問題引入………………………………………………………………………………………15
3．2 算法推導及實現 ……………………………………………………………………………15
3．3 程序與數值例子………………………………………………………………………………17
結論 ………………………………………………………………………………………………………20
參考文獻 …………………………………………………………………………………………………20
致謝 ………………………………………………………………………………………………………21
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