Matrix form of the CGS method for solving general coupled matrix equations

This paper deals with the problem of solving the general coupled matrix equations ∑ j = 1 p A i j X j B i j = C i , i = 1 , 2 , … , p , (including several linear matrix equations as special cases) which plays important roles in system and control theory. Based on the conjugate gradients squared (CGS) method, a simple and efficient matrix algorithm is derived to solve the general coupled matrix equations. The derived iterative algorithm is illustrated by two numerical examples and is compared with other popular iterative solvers in use today.