基于稀疏矩阵的逆和行列式的计算文档.doc

基于稀疏矩阵的逆和行列式的计算 摘要：稀疏矩阵在现实中有很多应用，因此稀疏矩阵的计算近年来被广泛地研究.周期三对角矩阵和箭状矩阵作为两种常见的稀疏矩阵被广泛地应用在工程计算中。本文基于周期三对角矩阵和箭状矩阵的结构特点，根据矩阵分解的一些方法，着重讨论了一些周期三对角矩阵和箭状矩阵的求逆与行列式运算问题，并给出了相应的算法。本文首先根据周期三对角矩阵的特点研究了周期三对角矩阵的求逆算法，文中分别应用矩阵的递归方法、矩阵的分块技术来研究周期三对角矩阵的逆矩阵运算问题，并给出了相应于以上方法或技术的求逆新算法，同时也考虑一种求周期三对角矩阵的行列式算法，并通过一些数值例子验证了所给算法的可行性以及有效性。其次根据箭状矩阵的结构特点，利用矩阵的分块技术，给出了一个求解箭状矩阵的逆矩阵及行列式的算法，同时列举一个数值例子并给出了matlab程序，说明算法是有效可行的。 关键词：稀疏矩阵；周期三对角矩阵；箭状矩阵；逆矩阵 Based on sparse matrix inverse and determinant calculation Abstract: Sparse matrix computation has attracted extensive researches in recent years because it can have a lot of applications in reality .It is acknowledged that periodic tri-diagonal matrix and arrow matrix in sparse matrix have been widely applied in engineering calculation .Based on the structures of periodic tri-diagonal matrix and arrow matrix ,this paper focuses on the arithmetic of the inverse and determinant of periodic tri-diagonal matrix and arrow matrix ,and presents some new computational algorithms by using some methods of matrix decomposition . Firstly, it discusses the inverse of periodic tri-diagonal matrix on matrix structure of periodic tri-diagonal matrix. And some new computational algorithms are created by matrix recursion, blocking matrix technique of the inverse of periodic tri-diagonal matrix on its structure ,and then the feasibility and the effectiveness of these computational algorithms are verified by numerical examples in consideration of a simple algorithm of the determinant of periodic tri-diagonal matrix.Secondly, on the basis of the structure of arrow matrix ,an algorithm of the inverse and determinant of arrow matrix is presented by using the matrix blocking .At the same time ,the effectiveness and the feasibility are tested by numerical examples and the matlab algorithm . Key words: sparse matrix ;periodic tri-diagonal matrix ;arrow matrix ;inverse matrix 引言 在科学与工程计算中，经常会遇到稀疏矩阵的计算问题，例如关于三对角矩阵、周期三对角矩阵和箭状矩阵的行列式计算、确定逆矩阵、线性方程组求解问题。它们在矩阵分析理论、样条插值函数确定、微分方程的差分法