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 \in R^{n\times n} , C \in R^{n\times n} , B \in R^{s\times s} and  D \in R^{s\times s} are given matrices and X \in R^{s\times s}  is the unknown matrix. We present a global conjugate gradient (GL-CG) algo- rithm 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 experi- ments are given to illustrate the efficiency of the method.