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

Abstract

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 X ∼ B - C Y ∼ D , E X ∼ F - G Y ∼ H ) = ( M ∼ , N ∼ ) , where M ∼ = M - AX 0 B + CY 0 D , N ∼ = N - EX 0 F + GY 0 H . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.