On the global Krylov subspace methods for solving general coupled matrix equations

In the present paper, we propose the global full orthogonalization method (Gl-FOM) and global generalized minimum residual (Gl-GMRES) method for solving large and sparse general coupled matrix equations ∑j=1pAijXjBij=Ci,i=1,…,p, where Aij∈Rm×mAij∈Rm×m, Bij∈Rn×nBij∈Rn×n, Ci∈Rm×nCi∈Rm×n,i,j=1,2,…,pi,j=1,2,…,p, are given matrices and Xi∈Rm×nXi∈Rm×n, i=1,2,…,pi=1,2,…,p, are the unknown matrices. To do so, first, a new inner product and its corresponding matrix norm are defined. Then, using a linear operator equation and new matrix product, we demonstrate how to employ Gl-FOM and Gl-GMRES algorithms for solving general coupled matrix equations. Finally, some numerical experiments are given to illustrate the validity and applicability of the results obtained in this work.