An Iterative Method for the Least-Squares Problems of a General Matrix Equation Subjects to Submatrix Constraints

Abstract

An iterative algorithm is proposed for solving the least-squares problem of a general matrix equation ${\sum }_{i=\mathrm{1}}^t\mathrm{‍}{M}_i{Z}_i{N}_i=F$, where $Z_i$ ($i=\mathrm{1,2},\dots ,t$) are to be determined centro-symmetric matrices with given central principal submatrices. For any initial iterative matrices, we show that the least-squares solution can be derived by this method within finite iteration steps in the absence of roundoff errors. Meanwhile, the unique optimal approximation solution pair for given matrices ${\tilde{Z}}_i$ can also be obtained by the least-norm least-squares solution of matrix equation ${\sum }_{i=\mathrm{1}}^t\mathrm{‍}{M}_i{\stackrel{-}{Z}}_i{N}_i=\stackrel{-}{F}$, in which ${\stackrel{-}{Z}}_i=Z_i-{\tilde{Z}}_i,\mathrm{}\stackrel{-}{F}=F-{\sum }_{i=\mathrm{1}}^t\mathrm{‍}{M}_i{\tilde{Z}}_i{N}_i$. The given numerical examples illustrate the efficiency of this algorithm.