The relaxed gradient based iterative algorithm for solving the generalized coupled complex conjugate and transpose Sylvester matrix equations

Inspired by the idea of Ma et al. (Journal of the Franklin Institute, 2018), we adopt relaxation technique and introduce relaxation factors into the gradient based iterative (GI) algorithm, and the relaxed based iterative (RGI) algorithm is established to solve the generalized coupled complex conjugate and transpose Sylvester matrix equations. By applying the real representation and straighten operation, we contain the sufficient and necessary condition for convergence of the RGI method. In order to effectively utilize this algorithm, we further derive the optimal convergence parameter and some related conclusions. Moreover, to overcome the high dimension calculation problem, a sufficient condition for convergence with less computational complexity is determined. Finally, numerical examples are reported to demonstrate the availability and superiority of the constructed iterative algorithm.