On global Hessenberg based methods for solving Sylvester matrix equations

In the first part of this paper, we investigate the use of Hessenberg-based methods for solving the Sylvester matrix equation $AX+XB=C$. To achieve this goal, the Sylvester form of the global generalized Hessenberg process is presented. Using this process, different methods based on a Petrov–Galerkin or on a minimal norm condition are derived. In the second part, we focus on the SGl-CMRH method which is based on the Sylvester form of the Hessenberg process with pivoting strategy combined with a minimal norm condition. In order to accelerate the SGl-CMRH method, a preconditioned framework of this method is also considered. It includes both fixed and flexible variants of the SGl-CMRH method. Moreover, the connection between the flexible preconditioned SGl-CMRH method and the fixed one is studied and some upper bounds for the residual norm are obtained. In particular, application of the obtained theoretical results is investigated for the special case of solving linear systems of equations with several right-hand sides. Finally, some numerical experiments are given in order to evaluate the effectiveness of the proposed methods.