Numerical Algorithms

, Volume 64, Issue 3, pp 455–480

# 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

• Zhuohua Peng
Original Paper

## Abstract

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.

## Keywords

Matrix equation Central principal submatrix Reflexive solutions Submatrix constraint Least squares solution

65F10 65F30

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