The Study Of Iterative Algorithms For Solving Linear Matrix Equations

Iterative methods for solving linear matrix equations have been gaining popularity in many areas of scientific computing, such as electromagnetics, mechanics, theory of vibration, nonlinear programming, dynamic analysis, automatic control theory and others. Many developments have been received for linear matrix equations.In this dissertation, numerical methods for the special solutions of three kinds of linear matrix equations have been studied. Firstly, a finite iteration algorithm for the symmetric solution of linear matrix equations has been presented. Also, we proved that the required solution can be obtained within finite steps in the absence of roundoff errors. Numerical experiment has been given to verify the efficiency of algorithm. Secondly, two algorithms have been proposed for finding the generalized reflexive and generalized anti-reflexive solutions of equations AXB= C, respectively. The corresponding matrix nearness problem has been considered. By using the similar idea with the algorithm of Chapter 3, the iterative algorithm has been given for solving the generalized reflexive solution of the general linear systems of matrix equations. This dissertation includes five chapters, which is organized as follows:Firstly, the research background and research status are given, as well as the preliminary knowledge. Furthermore, the main contents of this paper are briefed.In the second chapter, a finite iteration algorithm for the symmetric solution of linear matrix equations has been presented.In Chapter 3, numerical methods for finding the generalized reflexive and generalized anti-reflexive solutions of equations AXB= C have been considered.In Chapter 4, an iterative algorithm for solving generalized reflexive solution of a general linear matrix equation was given.Finally, the research work of this dissertation is summarized and the possible research lines are discussed.