Research On The Iterative Algorithm For Several Kinds Of Constrained Matrix Equation

The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set, and it is widely used in many fields such as structural design, parameter identification, automatics control, finite elements theory, linear optimal control theory, and so on. The research of this problem involves two aspects:one is the solvability in theory, that is to find the neces-sary and sufficient conditions for the solvability of the problem, the other one is constructing algorithms in problem solving, that is to solve the problem in the algorithm. Iterative algorithm of the constrained matrix equation is one of the important methods in algorithm implementation (the other one is called direct algorithm).Based on the conjugate gradient method which is presented to solve system of linear equations in numerical linear algebra, this dissertation constructs some new algorithms, analyzes the convergence and practicability of the algorithms, and carries out the numerical examples to solve several kinds of constrained matrix equations. This is an exciting development of the algorithms for find the solution of the constrained matrix equation problem.The main achievements are as follows:1. Reflexive and anti-reflexive solutions and its optimal approximation con-strained solutions of the linear matrix equation pair AXB=E,CXD=F. Firstly, we transform the constrained problem equivalently to an unconstrained problem by eigenvalue decomposition, then new iterative algorithms are con-structed to solve these unconstrained matrix equations and the solution of the constrained problem is reconstructed. The convergence and the practicability of the iterative algorithms are analyzed, and numerical examples are given to illustrate that the algorithms are effective.2. By adjusting the residual matrices of the conjugate direction, we derive some new algorithms for the general solutions, the symmetric and anti-symmetric solutions, the centro-symmetric and centro-skew symmetric solutions, the gener-alized reflexive and generalized anti-reflexive solutions and its optimal approx-imation constrained solutions to the matrix equations A1X1B1+A2X2B2E, C1X1D1+C2X2D2=F. The convergence and the practicability of the it-erative algorithms are analyzed, and numerical examples are given to illustrate that the algorithms are effective.3. Based on the above work, we present new algorithms for the general solu-tions, the generalized reflexive and generalized anti-reflexive solutions and its op-timal approximation constrained solutions to the matrix equations The convergence and the practicability of the iterative algo-rithms are analyzed. This provides a new effective algorithm for the solutions to constrained matrix equation problems.4. If the aboved constrained matrix equation is consistent, the algorithms will converge to the lest-norm solution of the equation, and if the equation is not consistent, the algorithm will converge to the optimal approximation solution of the equation. The research process indicates that the algorithm has stronger portability, moreover, the residual matrices are sparse matrices which is better for improving the convergence rates.This dissertation is supported by the National Natural Science Foundation of China (11171205).