Iterative Methods Of Several Classes Of Matrix Equations

The constrained matrix equation problem has been widely used in electricity,vibration theory,structural design,biology,automatic control theory,finite element and multidimensional approximation problems and nonlinear optimal control,and so on.The constrained matrix equation problem has been one of the important topics in the field of numerical algebra in recent years.In this paper,the problem is as followsProbleml:GivenA,B,C?Rn×n,findX?Rn×n,such that AX+XB?C among them S???=CSR^n×n or S???CASRn×nProblem ?:Suppose that Problem ? is compatible and the solution set for it is SE,given X0?S,find X?SE,such that||X-X0|| = min||X-X0||Problem ?:Given A,B,C?Rm×n,find X ?S1,Y?S2,such that AX+YB = C among them S1???CSR^n×n or S1???CASRn×n;S2???CSR^m×m or S2???CASRm×m.Problem ?:Suppose that Problem ? is compatible and the solution set for it is SE,givenX0 ?S1,Y0?S2,find????SE,such that||X-X0||2+||Y-Y0||2=???[||X-X0||+||Y-Y0||2]If S is CSR^n×n,CASRn×n respectively,First,the iterative algorithm of Problem I is constructed by using the conjugate gradient and the matrix property.Second,the convergence of the algorithm is proved.At the same time,the algorithm is proved to converge to the minimal norm solution of the problem when the equation is compatible.Then the algorithm is modified slightly to get the corresponding optimal approximation.Finally,numerical examples are given to verify the validity of the algorithm.If[S1,S2]is[CSR^n×n,CSR^m×m],[CASn×n,CASRm×m]respectively,the conjugate gradient iterative algorithm of Problem III is constructed,the convergence of the algorithm is proved,the corresponding optimal approximation will be got,then numerical examples are given to verify the validity of the algorithm.