Finite iterative algorithms for solving generalized coupled Sylvester systems – Part I: One-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this two-part article, finite iterative methods are proposed for solving one-sided (or two-sided) and generalized coupled Sylvester matrix equations and the corresponding optimal approximation problem over generalized reflexive solutions (or reflexive solutions). In part I, an iterative algorithm is constructed to solve one-sided and coupled Sylvester matrix equations (AY − ZB, CY − ZD) = (E, F) over generalized reflexive matrices Y and Z. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [Y1, Z1], 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 solution pair can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [Y^,Z^] to a given matrix pair [Y0, Z0] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [Y∼∗,Z∼∗] of two new corresponding generalized coupled Sylvester matrix equations (AY∼-Z∼B,CY∼-Z∼D)=(E∼,F∼), where E∼=E-AY0+Z0B,F∼=F-CY0+Z0D. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.