Numerical Algorithms

, Volume 64, Issue 3, pp 455–480 | Cite as

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


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.


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

Mathematics Subject Classifications (2010)

65F10 65F30 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.School of Mathematics and Computing ScienceHunan University of Science and TechnologyXiangtanPeople’s Republic of China

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