The relaxed gradient based iterative algorithm for solving matrix equations AiXBi=Fi

In this paper, we first present a relaxed gradient based iterative (RGI) algorithm for solving matrix equations A1XB1=F1 and A2XB2=F2. The idea is from (Niu et al., 2011; Xie and Ma, 2016) in which efficient algorithms were developed for solving the Sylvester matrix equations. Then the RGI algorithm is extended to solve the generalized linear matrix equations of the form AiXBi=Fi,(i=1,2,…,N). For any initial value, it is proved that the iterative solution converges to their true solution under some appropriate assumptions. Finally, a numerical example is given to illustrate that the introduced iterative algorithm is more efficient than gradient based maximal convergence rate iterative (GI) algorithm of Ding et al. (2010) in speed, elapsed time and iterative steps.