Global conjugate gradient method for solving large general Sylvester matrix equation

In this paper, an iterative method is proposed for solving large general Sylvester matrix equation AXB+CXD = E, where A 2 Rn×n,C 2 Rn×n,B 2 Rs×s and D 2 Rs×s are given matrices and X 2 Rn×s is the unknown matrix. We present a global conjugate gradient (GL-CG) algorithm for solving linear system of equations with multiple right-hand sides. By defining a linear matrix operator and imposing some conditions on this operator, we demonstrate how to employ the GL-CG algorithm for solving large general Sylvester matrix equation. Finally, some numerical experiments are given to illustrate the efficiency of the method.