Linear Matrix Equation Group Iterative Solution

In control theory and other engineering fields, Lyapunov matrix equationsAX + XB = F , AXB + CXD = F and A_SB_S = F are need to be solved. Asstability is an important character in control theory , it denotes that system can conserve reserved working conditions and resist effects of all kinds of disadvantage factors. Matrix equation AX + XB = F has important meaning in system stability and spot devices. In constant studies of ordinary differential equations and block and Rung-kwtta approach of solving ordinary differential equations, matrix equation AX + XB = F is mentioned. Solving matrix equations AX + XB - F is key to the computation of observe design. Furthermore, in disturbance studies of generalized eigenvalues problem and computational solutions of ordinary differential equations, matrix equation AXB + CXD = F is often need to be solved.In this paper, iterative method in groups for solving these three matrix equations is studied when the equation has a unique solution. The main contributions are described as follows:First of all, iterative method in groups Jacobi , JGS , Q-JGS, B-Jacobi and B-JGS iterative method in groups for solving matrix equations AX + XB = F is constructed. According to its converging conditions of Jacobi and JGS iterative methods of linear algebraic equation, we obtain a sufficient condition and a necessary and sufficient condition which Jacobi iterative method converge and two sufficient conditions which JGS iterative method converge, moreover, we discuss two sufficient conditions which JGS , Q-JGS , B-Jacobi and B-JGS iterative method converge.Then, iterative method in groups Jacobi, JGS, Q-JGS, B-Jacobi and B-JGS iterative method in groups for solving matrix equations AXB + CXD - F is constructed. We derive a sufficient condition and a necessary and sufficient condition which Jacobi iterative method converges and one sufficient condition which JGS iterative method converges .Furthermore, we discuss a sufficient condition which JGS, Q-JGS, B-Jacobi and B-JGS iterative method for solving matrix equations AXB + CXD = F converge. These results are extended to solve generalizedmatrix equation A_SXB_S =F . Finally, these iterative methods in groups arecompared by numerical examples.