Iterative Methods For Solving Structural Linear Systems Aχ=b And Sylvester Matrix Equations

The content of this paper consists of two parts:part one is how to solve the linear systems Ax=b iteratively,which coefficient matrices are centrosymmetric matrices; part two pays attention to solving the Lyapunov matrix equations and Sylvester matrix equations in control theory by numcrical methods.For the iterative solutions of the linear systems Ax=b,wc consider to solve a special linear system which coefficient matrix is a centrosymmetric matrix.In this paper we construct several centrosymmetric splittings according to the special construction of the centrosymmetric matrices.Compared to the Jacobi and Gauss-Seidel methods,the iterative methods based on these centrosymmetric splittings have faster convergence and less cost of computation and store.Here we mainly investigate the iterative methods for the linear systems with a centrosymmetric M-matrix and a centrosymmctric H-matrix, the other cases are also been studied.In the analyse and design of control systems,the solutions of matrix equations and matrix inequations play an important role,and have caused many attentions in control and mathematical fields.In this paper we give two iterative methods for solving the matrix equations:1) solving the discrete Lyapunov equation matrix AXA~T-X+Q=0 and continuous Lyapunov matrix equation AX+XA~T+Q=0 by using Kronecker Products iterative method;2) solving the Sylvester matrix equations AX+XB=C and AXB+X=C by using the gradient iterative methods based on the matrix splittings.