Finite iterative algorithms for solving generalized coupled Sylvester systems-Part II: Two-sided and generalized coupled Sylvester matrix equations over reflexive solutions

In Part I of this article, we proposed a finite iterative algorithm for the one-sided and generalized coupled Sylvester matrix equations (AY − ZB, CY − ZD) = (E, F) and its optimal approximation problem over generalized reflexive matrices solutions. In Part II, an iterative algorithm is constructed to solve the two-sided and generalized coupled Sylvester matrix equations (AXB − CYD, EXF − GYH) = (M, N), which include Sylvester and Lyapunov matrix equations as special cases, over reflexive matrices X and Y. When the matrix equations are consistent, for any initial reflexive matrix pair [X1, Y1], the reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation reflexive solution pair [X^,Y^] to a given matrix pair [X0, Y0] in Frobenius norm can be derived by finding the least-norm reflexive solution pair [X∼∗,Y∼∗] of a new corresponding generalized coupled Sylvester matrix equations (AX∼B-CY∼D,EX∼F-GY∼H)=(M∼,N∼), where M∼=M-AX0B+CY0D,N∼=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.