几类特殊矩阵求其逆的快速算法研究

发布时间：2018-05-26 13:15

本文选题：(周期)三对角Toeplitz矩阵 + (周期)七对角矩阵　；参考：《陕西科技大学》2017年硕士论文

【摘要】：本文利用矩阵的LU分解法、线性方程的求解法、逆矩阵的定义、矩阵的扩展法进行求解几类特殊对角矩阵的逆,依据从易到难的路线进行推进,首先使用LU分解法求解形式简单的三对角、周期三对角Toeplitz矩阵的逆,其次利用矩阵的扩展法求解形式较为复杂的七对角、周期七对角矩阵的逆,在验证以上方法均有效的情况下,最后再一次通过LU分解法对形式复杂的周期k-三对角矩阵和k-五对角矩阵进行求逆。本文主要从以下四个方面进行研究:一、求解三对角Toeplitz矩阵和周期三对角Toeplitz矩阵的逆的算法。该求解算法的思想为:根据三对角Toeplitz矩阵和周期三对角Toeplitz矩阵对应的特殊构造,使用矩阵的LU分解法,及其线性方程的解法进行求逆。该算法的复杂度均基于O(n2),其中三对角Toeplitz矩阵的求逆算法的加减法复杂度为2n2-n-1,乘除法复杂度为3n2+n-3;周期三对角Toeplitz矩阵的求逆算法的加减法复杂度为2n2+3n-6,乘除法复杂度为3n2+9n-20.最后文中经过数值例子验证了算法的有效性和较强的稳定性。二、求解七对角矩阵和周期七对角矩阵的逆的算法。该求解算法利用矩阵的扩展法,将n×n七对角矩阵、n×n周期七对角矩阵扩展为n×(n+3)型矩阵进行求逆。该算法的复杂性较低,为O(n2),最后文中通过算法例子验证了算法的实效性。三、求解周期k-三对角矩阵和k-五对角矩阵的逆的算法。该求解方法类似于一中的求解方法,均利用特殊矩阵所对应的LU分解法,以及逆矩阵的定义进行求解。该求解法对使用LU分解法求逆矩阵的办法进行了扩展,并得到了理想型结果,其中周期k-三对角矩阵的求逆算法和k-五对角矩阵的求逆算法的复杂度均为O(n2).且该算法不需要对矩阵的各阶顺序主子式进行任何条件的限制,同时还适用于计算机实现的代数系统。四、几类特殊反对角矩阵的逆矩阵。在得到以上几类特殊对角矩阵的逆矩阵的基础上,利用原对角矩阵与其所对应的反对角矩阵的性质,便可快速求解反对角矩阵的逆矩阵,本文以七对角矩阵和周期七对角矩阵为例,求解了反七对角矩阵和周期反七对角矩阵的逆矩阵。
[Abstract]:In this paper, the LU decomposition method of matrix, the solution method of linear equation, the definition of inverse matrix and the expansion method of matrix are used to solve the inverse of several kinds of special diagonal matrices, which are advanced according to the route from easy to difficult. Firstly, the LU decomposition method is used to solve the inverse of simple tridiagonal and periodic tridiagonal Toeplitz matrices, and then the expansion method of matrices is used to solve the inverse of complex forms of seven-diagonal and periodic seven-diagonal matrices. Under the condition that the above methods are effective, the complex periodic k- tridiagonal matrices and k- pentagonal matrices are inversed again by LU decomposition method. This paper mainly studies the following four aspects: first, the algorithm to solve the inverse of tridiagonal Toeplitz matrix and periodic tridiagonal Toeplitz matrix. The idea of the algorithm is as follows: according to the special construction of tridiagonal Toeplitz matrix and periodic tridiagonal Toeplitz matrix, the LU decomposition method of matrix and the solution of linear equation are used to solve the inverse problem. The complexity of the algorithm is based on Toeplitz matrix, where the complexity of the algorithm is 2n2-n-1, the complexity of multiplication and division is 3n2 n-3, the complexity of the inverse algorithm of periodic tridiagonal Toeplitz matrix is 2n2 3n-6, the complexity of multiplication and division is 3n2 9n-20, the complexity of the algorithm is 2n2-n-1, the complexity of multiplication and division is 3n2 n-3, the complexity of the algorithm is 2n2 3n-6 and the complexity of multiplication and division is 3n2 9n-20. Finally, a numerical example is given to verify the effectiveness and stability of the algorithm. Second, the algorithm for solving the inverse of the seven diagonal matrix and the periodic seven diagonal matrix. In this algorithm, n 脳 n 7 diagonal matrix is extended to n 脳 n periodic 7 diagonal matrix to n 脳 n 3) type matrix by using the expansion method of matrix. The complexity of the algorithm is low, which is called OFN _ 2. Finally, an example is given to verify the effectiveness of the algorithm. Third, the algorithm for solving the inverse of periodic k- tridiagonal matrix and k- pentagonal matrix. This method is similar to the solution method in one medium. It is solved by the LU decomposition method corresponding to a special matrix and the definition of inverse matrix. The method extends the method of solving inverse matrix by LU decomposition method, and obtains the ideal type result. The complexity of the inverse algorithm of periodic k- tridiagonal matrix and k- pentagonal matrix is OfN _ 2. Moreover, the algorithm does not need any restriction on every order of matrix, and it is also suitable for the algebraic system realized by computer. Four, several kinds of inverse matrices of special antiangular matrices. On the basis of obtaining the inverse matrices of some special diagonal matrices mentioned above, by using the properties of the original diagonal matrices and their corresponding anti-diagonal matrices, the inverse matrices of the anti-diagonal matrices can be solved quickly. In this paper, we take the seven diagonal matrix and the periodic seven diagonal matrix as examples to solve the inverse matrix of the anti 7 diagonal matrix and the period anti 7 diagonal matrix.
【学位授予单位】：陕西科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21

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