Numerical solution of AXB=C for (R,S)-symmetric matrices

Let $R\in\mathbb{R}^{m\times m}$ and $S\in\mathbb{R}^{n\times n}$ be real nontrivial symmetric involutions; i.e., R=R ^T =R ^?1≠±I _m and S=S ^T =S ^?1≠±I _n . We say that $A\in \mathbb{R}^{m\times n}$ is (R,S)-symmetric ((R,S)-skew symmetric) if RAS=A (RAS=?A). Trench (Linear Algebra Appl. 389:23–31, 2004) has theoretically studied the minimization problems and the related approximation problems of matrix equation AZ=W for (R,S)-symmetric matrices, using their structure properties and Moore-Penrose inverse. In this paper, we extend and develop these research in a totally different way using iterative methods. We propose two algorithms based on the idea of the classical Conjugate Gradient method (CG) and Conjugate Gradient Least Squares method (CGLS), to solve the more general equation AXB=C for (R,S)-symmetric matrices X. Some numerical results confirm the efficiency of these algorithms. More importantly, some numerical stability analysis on the approximation problem is given combining with numerical examples, which is not given in the earlier papers.