Research On Matrix Equation Iteration Method Under Submatrix Constraints

The matrix expansion problem is an inverse problem of a class of constraint matrices,which is a matrix equation problem with sub-matrix constraints.It plays a significant role in different areas of system identification,mechanics,control and engineering and is one of the important research topics in the field of computational mathematics.In this paper,we study the numerical solutions of the following problems.Problem ? Given A ? Cm×n,B ? Cm×n,X?Cp×q,S(?)Cnxn,find X?S,such that AX = B,X = X(p1:p2;q1:q2).where S = HCn×n,or S = AHCn×n,p2-p1 + 1 = p,q2-q1+1 = q.Problem ? Given X0 ? Cn×n,find X ? SE,such that where ?·? is Frobenius norm,SEis the solution set of problem ?.Problem ? Given A ? Cm×n,B?Cn×m,C? cm×m,X?Cp×q,S(?)Cn×n,find X ?S,such that AXB = C,X = X(p1:p2;q1:q2).where S = HCn×n,or S= AHCn×n,p2-p1 + 1 = p,q2-q1+1 =q.Problem ? Given X0 ?Cn×n,find X?SE such that where ?·? is Frobenius norm SE is the solution set of problem ?.When S is HCn×n,AHCn×n,firstly,the orthogonal projection idea is used to construct the orthogonal projection iterative algorithm of question ?.Secondly,the convergence of the algorithm is analyzed and the convergence estimate of the algorithm is obtained.The optimal solution of the problem can be obtained by slightly modifying the algorithm.The conjugate gradient iterative algorithm of problem ? and problem ? is constructed by using the idea of conjugate gradient.The convergence of the algorithm is proved and the optimal approximation solution of the corresponding problem is obtained.Finally,the validity of the numerical example is given.