【作者】 仝秋娟；

【导师】 陆全；

【作者基本信息】 西北工业大学 ， 计算数学， 2006， 硕士

【摘要】 随着计算机科学的迅速发展,国防科技和国民经济建设的许多领域不断提出许多大型和超大型的计算问题。对于此类问题,其相应的矩阵往往具有一些特殊的结构,因此,利用这些矩阵的特殊结构进行技巧性的处理,使矩阵的计算量降低一个数量级,这一问题的研究具有重要的理论意义和现实意义。 在工程计算问题中,有一类重要的问题就是线性方程组的求解问题,但有些线性方程组是不相容的。本文研究以Loewner型矩阵和对称Loewner型矩阵为系数矩阵的不相容线性方程组的极小范数最小二乘解的快速算法。对于m×n阶矩阵A,求以A为系数阵的线性方程组Ax=b的最小二乘解的一般方法是构造法方程组A^TAx=A^Tb,进而求解法方程组来实现的。特别地,当A的秩为n时,Ax=b的惟一极小范数最小二乘解为x₀=（A^TA）^-1A^Tb。但利用通常的方法求法方程组时,所需的运算量为O（mn²）+O（n³）,且若矩阵A本身病态时,构造法方程组后会更加病态。求方程组Ax=b的最小二乘解也常采用正交化法,这一方法虽然避免了构造法方程组,但所需的运算量会更大些。本文对于秩为n的m×n阶Loewner型或对称Loewner型矩阵L,通过构造特殊分块矩阵并研究其逆的三角分解或自身三角分解,进而得到了线性方程组Lx=b的极小范数最小二乘解快速算法,计算复杂度为O（mn）+O（n²）,而一般方法的计算复杂度为O（mn²）+O（n³）。 在工程计算问题中,由于矩阵的广义逆理论与计算在最优化理论、控制理论、计算数学、数理统计等领域中有着广泛的应用,也使得对于矩阵广义逆问题的研究显得尤为必要。本文对于秩为n的m×n阶Loewner型矩阵或对称Loewner型矩阵,通过构造特殊分块矩阵并研究其逆矩阵,进而得到它们的Moore-Penrose逆及其快速算法,所需运算量为O（mn）+O（n²）,而利用常规方法L⁺=（L^TL）^-1L^T求解时所需的运算量为O（mn²）+O（n³）。 本文的结构安排如下: 第一章主要以引言的形式概括了两类方程组和广义逆的历史及研究现状;更多还原

【Abstract】 With the speedy development of computer science, many large or sublarge computation problems are brought forward in fields as national defence and science and technology and national economy development. For these problems, the relevant matrices often have some special structure. Therefore, we may get on with these matrices technique by using their special structure, and then reduce a quantitative level of the multiplications, and the research have definite real meaning.There is a kind of important problem that is solving linear systems in the computing problems of engineering, but sometimes they are unsolvable. In this paper, fast algorithms are presented which compute the minimal norm least square solutions for linear systems such as the coefficients is Loewner-type matrix or symmetric Loewner-type matrix. For m×n matrix A , the normal method is that form anormal system A^T Ax = A^T b when we solute the system Ax = b, and then solute it.Specially, the minimal norm least square solution is x₀ = [A^Ta)^-1 A^Tb for Ax = b when the rank is n of A . But, when we solve the normal system with the common method, the multiplication is O\mn²)+ O（n³）, and if matrix A is morbid itself, it ismore morbid than before after forming normal system. We often use the orthogonalization method to solve the system Ax = b, although we don’t need forming the normal system, the multiplication is more than others of it. In this paper, for m× n Loewner-type matrix or symmetric Loewner-type matrix with full column rank, we form special block matrix firstly and research their triangular factorization or of its inverse, and then get the fast algorithm of the minimal norm least squaresolution .Its computation complexity is O（mn） + O（n²）, but that of usual algorithms is o（mn²）+o（n³）.The research for the problem of generalized inverse of matrix is very important due to the broad application of the theory and computation of matrix in many fields such as optimize theory, control theory, computational mathematics, mathematical statistics, etc. in computing problems of engineering. In this paper, for mxn Loewner-type matrix or symmetric Loewner-type matrix with full column rank, we form special block matrix firstly and research their inverse, and then get the fast algorithm of the Moore-Penrose inverse. Its computation complexity isO（mn） + O（n2）, but that of using r=（lTl）"’lT is o（mn2）+o（n’）.The paper is built as follows.In chapter 1, studying history and actuality of the two kinds linear system are given in introduction.In chapter 2, the basic theory of the algorithm is presented.In chapter 3, three fast algorithms of minimal norm least square solution for system whose coefficient is Loewner-type matrix and their numerical examples are given.In chapter 4, three fast algorithms of minimal norm least square solution for system whose coefficient is symmetric Loewner-type matrix and their numerical examples are given.In chapter 5, fast algorithm of Moore-Penrose inverse for Loewner-type matrix and its numerical examples are given.In chapter 6, fast algorithm of Moore-Penrose inverse for symmetric Loewner-type matrix and its numerical examples are given.In chapter 7, a unilateral inverse formula of Loewner-type nrntrix and symmetric Loewner-type matrix are given.更多还原