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

摘要

In this study, we consider the iteration solutions of the generalized coupled Sylvester-conjugate matrix equations: A_1X + B_1Y = D_1XE_1 + F_1, A_2Y + B_2X = D_2YE_2 + F_2, 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~*, V~*) 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.