Wu et al. (Applied Mathematics and Computation 217(2011)8343-8353) constructed a gradient based iterative (GI) algorithm to find the solution to the complex conjugate and transpose matrix equation

and a sufficient condition for guaranteeing the convergence of GI algorithm was given for an arbitrary initial matrix. Zhang et al. (Journal of the Franklin Institute 354 (2017) 7585-7603) provided a new proof of GI method and the necessary and sufficient conditions was presented to guarantee that the proposed algorithm was convergent for an arbitrary initial matrix. In this paper, a relaxed gradient based iterative (RGI) algorithm is proposed to solve this complex conjugate and transpose matrix equation. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the introduced algorithm for any initial iterative matrix. Numerical results are given to verify the efficiency of the new method. Finally, the application in time-varying linear system of the presented algorithm is provided.

Wu等。（应用数学和计算217（2011）8343-8353）构造了一种基于梯度的迭代（GI）算法来查找复共轭和转置矩阵方程的解

为任意初始矩阵给出了保证GI算法收敛性的充分条件。张等。（Franklin Institute Journal 354（2017）7585-7603）提供了GI方法的新证明，并提出了必要和充分条件以保证所提出的算法对于任意初始矩阵都是收敛的。本文提出了一种基于松弛梯度的迭代（RGI）算法来求解该复共轭和转置矩阵方程。确定收敛因子的必要和充分条件，以保证对于任何初始迭代矩阵，引入算法的收敛性。数值结果证明了该方法的有效性。最后，给出了该算法在时变线性系统中的应用。