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

摘要

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 [X_1,Y_1], 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 [X_0,Y_0] 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 (A~XB - C~YD, E~XF - G~YH) = (~M,~N). where ~M =M - AX_0B + CY_0D, ~N = N - EX_0F + GY_0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.