Convergence analysis of the global FOM and GMRES methods for solving matrix equations $AXB=C$ with SPD coefficients

In this paper‎, ‎we study convergence behavior of the global FOM (Gl-FOM) and global GMRES (Gl-GMRES) methods for solving the matrix equation $AXB=C$ where $A$ and $B$ are symmetric positive definite (SPD)‎. ‎We present some new theoretical results of these methods such as computable exact expressions and upper bounds for the norm of the error and residual‎. ‎In particular‎, ‎the obtained upper bounds for the Gl-FOM method help us to predict the behavior of the Frobenius norm of the Gl-FOM residual‎. ‎We also explore the worst-case convergence behavior of these methods‎. ‎Finally‎, ‎some numerical experiments are given to show the performance of the theoretical results‎.