Modified conjugate gradient method for obtaining the minimum-norm solution of the generalized coupled Sylvester-conjugate matrix equations

In this study, we consider the iteration solutions of the generalized coupled Sylvester-conjugate matrix equations: A1X+B1Y=D1X¯E1+F1,A2Y+B2X=D2Y¯E2+F2, where X¯ and Y¯ denote the conjugation of X and Y, respectively. We propose a modified conjugate gradient method and give the convergence analysis based on the premise that the coupled matrix equations are consistent. The convergence theorem shows that a solution (X*, Y*) can be obtained within finite iterative steps in the absence of round-off error for any initial value. Furthermore, we provide a method for choosing the initial matrices to obtain the minimum-norm solution of the problem. Finally, some numerical examples are given to demonstrate the behavior of the algorithms considered.