Researching Of Fast Algorithms Of The Problem Connected With The Kind Of Loewner-type Matrix

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.