The reflexive least squares solutions of the matrix equation $A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=C$with a submatrix constraint.

In this paper, an efficient algorithm is presented for minimizing $\|A_1X_1B_1 + A_2X_2B_2+\cdots +A_lX_lB_l-C\|$ where $\|\cdot \|$ is the Frobenius norm, $X_i\in R^{n_i \times n_i}(i=1,2,\cdots ,l)$ is a reflexive matrix with a specified central principal submatrix $[x_{ij}]_{r\leq i,j\leq n_i-r}$. The algorithm produces suitable $[X_1,X_2,\cdots ,X_l]$ such that $\|A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l-C\|=\min $ within finite iteration steps in the absence of roundoff errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Given numerical examples show that the algorithm is efficient. [ABSTRACT FROM AUTHOR]