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.

受Ma等人（《富兰克林研究所学报》，2018年）的研究思路启发，本文采用松弛技术，将松弛因子引入基于梯度的迭代（gradient based iterative, GI）算法，构建基于松弛的迭代（relaxed based iterative, RGI）算法以求解广义耦合复共轭转置西尔维斯特矩阵方程。通过引入实表示与拉直操作，本文得到了RGI算法收敛性的充要条件。为实现该算法的高效应用，本文进一步推导得到了最优收敛参数及若干相关结论。此外，为解决高维计算难题，本文推导得到了计算复杂度更低的收敛充分条件。最后，通过数值算例验证了所构建迭代算法的有效性与优越性。