The submatrix constraint problem of matrix equation AXB+CYD=E

摘要

We say that X=[xij]i,j=1n is symmetric centrosymmetric if xij=xji and xn-j+1,n-i+1,1⩽i,j⩽n. In this paper we present an efficient algorithm for minimizing ‖AXB+CYD-E‖ where ‖·‖ is the Frobenius norm, A∈Rt×n,B∈Rn×s,C∈Rt×m,D∈Rm×s,E∈Rt×s and X∈Rn×n is symmetric centrosymmetric with a specified central submatrix [xij]r⩽i,j⩽n-r,Y∈Rm×m is symmetric with a specified central submatrix [yij]1⩽i,j⩽p. Our algorithm produces suitable X and Y such that AXB+CYD=E in finitely many steps, if such X and Y exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.