Iteration Method To The Matrix Equation AX+XB=C

Sylvester equation AX + XB = C plays an important role in many fields.For example,in control theory and in statistics,you will constantly meet such equations.The properties of the solutions of the equation and the methods of solving the equation have been studied extensively. And some excellent algorithms have been presented. The main purpose of this article is,after special splitting to the coefficient matrix,using the familiar iterative methods to solve the Sylvester equation. The example in the end of the article shows that,to the matrices that meet some special conditions,the three methods presented in this article are effective.In the article,we first change the Sylvester equation into (A In + Im BT)X = C,then we split the coefficient matrix A In + Im BT. In chapter 2,Jacobi iteration is discussed. In chapter 3,Gauss-Seidel iterative method is discussed. If the Sylvester equation coefficient matrices A,B are non-singular M-matrix,we use a splitting-up method which has a parameter in chapter 4. In the last chapter,an example will be given using the three algorithms in different precisions. The numbers of the iterative steps will also be given.