本文選題：(周期)三對角Toeplitz矩陣 + (周期)七對角矩陣�。� 參考：《陜西科技大學(xué)》2017年碩士論文

【摘要】：本文利用矩陣的LU分解法、線性方程的求解法、逆矩陣的定義、矩陣的擴(kuò)展法進(jìn)行求解幾類特殊對角矩陣的逆,依據(jù)從易到難的路線進(jìn)行推進(jìn),首先使用LU分解法求解形式簡單的三對角、周期三對角Toeplitz矩陣的逆,其次利用矩陣的擴(kuò)展法求解形式較為復(fù)雜的七對角、周期七對角矩陣的逆,在驗證以上方法均有效的情況下,最后再一次通過LU分解法對形式復(fù)雜的周期k-三對角矩陣和k-五對角矩陣進(jìn)行求逆。本文主要從以下四個方面進(jìn)行研究:一、求解三對角Toeplitz矩陣和周期三對角Toeplitz矩陣的逆的算法。該求解算法的思想為:根據(jù)三對角Toeplitz矩陣和周期三對角Toeplitz矩陣對應(yīng)的特殊構(gòu)造,使用矩陣的LU分解法,及其線性方程的解法進(jìn)行求逆。該算法的復(fù)雜度均基于O(n2),其中三對角Toeplitz矩陣的求逆算法的加減法復(fù)雜度為2n2-n-1,乘除法復(fù)雜度為3n2+n-3;周期三對角Toeplitz矩陣的求逆算法的加減法復(fù)雜度為2n2+3n-6,乘除法復(fù)雜度為3n2+9n-20.最后文中經(jīng)過數(shù)值例子驗證了算法的有效性和較強(qiáng)的穩(wěn)定性。二、求解七對角矩陣和周期七對角矩陣的逆的算法。該求解算法利用矩陣的擴(kuò)展法,將n×n七對角矩陣、n×n周期七對角矩陣擴(kuò)展為n×(n+3)型矩陣進(jìn)行求逆。該算法的復(fù)雜性較低,為O(n2),最后文中通過算法例子驗證了算法的實效性。三、求解周期k-三對角矩陣和k-五對角矩陣的逆的算法。該求解方法類似于一中的求解方法,均利用特殊矩陣所對應(yīng)的LU分解法,以及逆矩陣的定義進(jìn)行求解。該求解法對使用LU分解法求逆矩陣的辦法進(jìn)行了擴(kuò)展,并得到了理想型結(jié)果,其中周期k-三對角矩陣的求逆算法和k-五對角矩陣的求逆算法的復(fù)雜度均為O(n2).且該算法不需要對矩陣的各階順序主子式進(jìn)行任何條件的限制,同時還適用于計算機(jī)實現(xiàn)的代數(shù)系統(tǒng)。四、幾類特殊反對角矩陣的逆矩陣。在得到以上幾類特殊對角矩陣的逆矩陣的基礎(chǔ)上,利用原對角矩陣與其所對應(yīng)的反對角矩陣的性質(zhì),便可快速求解反對角矩陣的逆矩陣,本文以七對角矩陣和周期七對角矩陣為例,求解了反七對角矩陣和周期反七對角矩陣的逆矩陣。
[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.
【學(xué)位授予單位】：陜西科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O151.21

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