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An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [X1,Y1], the generalized 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 generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [X̂,Ŷ] to a given matrix pair [X0,Y0] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [X̃*,Ỹ*] of a new corresponding generalized coupled Sylvester matrix equation pair (AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), where M̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations