An efficient algorithm for the submatrix constraint of the matrix equation \(A_1 X_1 B_1+A_2 X_2 B_2+{\cdots}+A_l X_l B_l =C\). (English) Zbl 1255.65076
Summary: Matrix \(A=(a_{ij}) \in \mathbb{R}^{n{\times}n}\) is said to be bisymmetric if \(a_{ij} =a_{ji} =a_{n+1-j, n+1-i}\) for all \(1\leq i, j \leq n\). In this paper, an efficient algorithm is presented for minimizing \(\|A_1 X_1 B_1+A_2 X_2 B_2+\cdots+A_l X_l B_l-C\|\), where \(\|\cdot\|\) is the Frobenius norm and \(X_i \in \mathbb{R}^{n_{i}\times n_{i}} (i=1, 2, \dots, l)\) is bisymmetric with a specified central principal submatrix \([x_{ij}]_{r\leq i, j\leq n_{i}-r}\). The algorithm produces suitable \([X_1, X_2, \dots , X_l]\) such that \(\|A_1 X_1 B_1+A_2 X_2 B_2+\cdots+A_l X_l B_l-C \|=\)min within finite iteration steps in the absence of roundoff errors. The results of given numerical experiments show that the algorithm has fast convergence rate.
MSC:
| 65F10 | Iterative numerical methods for linear systems |
| 65F30 | Other matrix algorithms (MSC2010) |
Keywords:
algorithm; matrix equation; central principal submatrix; bisymmetric matrix; least-squares solutionReferences:
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