A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations

The present work proposes a relaxed gradient based iterative (RGI) algorithm to find the solutions of coupled Sylvester matrix equations AX+YB=C,DX+YE=F. It is proved that the proposed iterative method can obtain the solutions of the coupled Sylvester matrix equations for any initial matrices X0 and Y0. Next the RGI algorithm is extended to the generalized coupled Sylvester matrix equations of the form Ai1X1Bi1+Ai2X2Bi2+⋯+AipXpBip=Ci,(i=1,2,…,p). Then, we compare their convergence rate and find RGI is faster than GI, which has maximum convergence rate, under an appropriative positive number ω and the same convergence factor µ1 and µ2. Finally, a numerical example is included to demonstrate that the introduced iterative algorithm is more efficient than the gradient based iterative (GI) algorithm of (Ding and Chen 2006) in speed, elapsed time and iterative steps.