The submatrix constraint problem of matrix equation [formula omitted

Abstract

We say that X = [ x ij ] i , j = 1 n is symmetric centrosymmetric if x ij = x ji and x n - 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 ∈ R t × n , B ∈ R n × s , C ∈ R t × m , D ∈ R m × s , E ∈ R t × s and X ∈ R n × n is symmetric centrosymmetric with a specified central submatrix [ x ij ] r ⩽ i , j ⩽ n - r , Y ∈ R m × m is symmetric with a specified central submatrix [ y ij ] 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.