Iterative algorithms for the minimum-norm solution and the least-squares solution of the linear matrix equations A1XB1 + C1XTD1 = M1, A2XB2 + C2XTD2 = M2

In this paper, two iterative algorithms are proposed to solve the linear matrix equations A(1)XB(1) + (C1XD1)-D-T = M-1, A(2)XB(2) + (C2XD2)-D-T = M-2. When the matrix equations are consistent, by the first algorithm, a solution X-psi can be obtained within finite iterative steps in the absence of roundoff-error for any initial value, furthermore, the minimum-norm solution can be got by choosing a special kind of initial matrix. Additionally, the unique optimal approximation solution to a given matrix X-0 can be derived by finding the minimum-norm solution of a new matrix equations A(1)(X) over tildeB(1) + C-1(X) over tilde D-T(1) = M-1, A(2)(X) over tildeB(2) + C-2(X) over tilde D-T(2) = M-2. When the matrix equations are inconsistent, we present the second algorithm to find the least-squares solution with the minimum-norm. Finally, two numerical examples are tested by MATLAB, the results show that these iterative algorithms are efficient. (c) 2011 Elsevier Inc. All rights reserved.