Research On Constrained Matrix Equations And Their Preconditioning Techniques Of Iterative Methods

The constrained matrix equation problem is to find solution of a matrix equation or a system of matrix equations in a constrainer set.The study of it has been a hot topic in the filed of numerical algebra in rencent years.Actually,it has been widely used in automatics control theory,vibration theory,finite elements,nonlinear programming,and so on.This dissertation considers the polynomial preconditioning orthogonal projection iteration methods of the following problems. Problem 1 Given A, B∈R ^m×n ,S - R^n×n, find X∈Rn×n, such that AX = B. Where S are R n×n,SR n×n,ASR n×n respectively. Problem 2 Given A∈R^m×n, B∈R^p×q, D∈R ^m×q, find X∈R^n×p, such that AXB = D.The main works and results are as follows.1.For problem 1, when S are the matrix set satisfying some constraint conditions such as symmetric,skew-symmetric;Firstly, using the ideas of the singular values of matrix A and interpolation algorithms,the polynomial preconditioning matrix is con-structed. The new iteration method—the polynomial preconditioning orthogonal pro-jection iteration method is put forward by combining the orthogonal projection itera-tion method and the polynomial preconditioning matrix.Then, the analysis of conver-gence is given,which obtain the more accurate rate of new iteration method than the orthogonal projection iteration method. Futhermore, the numerical experiments are provided,which show that the effectiveness and feasibility of this new method.2.For problem 2, firstly, using the ideas of the singular values of matrix A , B and interpolation algorithms,the polynomial preconditioning matrix is constructed. The new iteration method—the polynomial preconditioning orthogonal projection iteration method is put forward by combining the orthogonal projection iteration method and the polynomial preconditioning matrix.Then, the analysis of convergence is given,which obtain the more accurate rate of new iteration method than the orthogonal projection iteration method. Futhermore, the numerical experiments are provided,which show that the effectiveness and feasibility of this new method.