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.
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Research supported by: (1) National Natural Science Foundation of China (11071062), (2) Natural Science Foundation of Hunan Province (10JJ3065), (3) Scientific Research Fund of Hunan Provincial Education Department of China (10A033) and (4) Scientific Research Fund of Hunan Science and Technology Department (2012FJ3048).
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Peng, Z. 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. Numer Algor 64, 455–480 (2013). https://doi.org/10.1007/s11075-012-9674-7
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DOI: https://doi.org/10.1007/s11075-012-9674-7
Keywords
- Matrix equation
- Central principal submatrix
- Reflexive solutions
- Submatrix constraint
- Least squares solution