Restarted global FOM and GMRES algorithms for the Stein-like matrix equation [formula omitted

Abstract

In this paper, we propose the restarted global full orthogonalization method (Gl-FOM) and global generalized minimum residual (Gl-GMRES) method to solve the Stein-like matrix equation X+M(X)=C with M(X)=AXB,M(X)=AX⊤B,M(X)=AX¯B or M(X)=AXHB, respectively, where X is an unknown matrix to be solved. First, by using a real inner product in complex matrix spaces, a generalized global Arnoldi process is given. Then we demonstrate how to employ the restarted Gl-FOM and Gl-GMRES algorithms for solving the Stein-like matrix equation. The proposed algorithms take advantage of the shifted structure of the matrix equation and are implemented by the original coefficient matrices. Finally, some numerical examples are given to illustrate the effectiveness with comparison to some existing methods.