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Solutions of symmetry-constrained least-squares problems. (English) Zbl 1212.65181

Summary: Two new matrix-form iterative methods are presented to solve the least-squares problem: \[ \min _{X\in \mathcal S_{1}, Y\in \mathcal S_{2}}\| AXB+CYD-E\| \] and the matrix nearness problem: \[ \min _{[ X\:Y] \in S_{XY}}\| [ X\:Y] -[ \widetilde X\:\widetilde Y] \| \] where the matrices \(A\in\mathbb{R}^{p\times n_{1}}\), \(B\in \mathbb{R}^{n_{2}\times q}\), \(C\in\mathbb{R}^{p\times m_{1}}\), \(D\in\mathbb{R}^{m_{2}\times q}\), \(E\in \mathbb{R}^{p\times q}\), \(\widetilde X\in\mathbb{R}^{n_{1}\times n_{2}} \) and \(\widetilde Y\in \mathbb{R}^{m_{1}\times m_{2}}\) are given; \(\mathcal S _{1}\) and \(\mathcal S _{2}\) are the set of constraint matrices, such as the sets of symmetric, skew symmetric, bisymmetric and centrosymmetric matrices and \(S_{XY}\) is the solution pair set of the minimum residual problem. These new matrix-form iterative methods have also faster convergence rate and higher accuracy than the matrix-form iterative methods proposed by Z. Y. Peng and Y. X. Peng [Numer. Linear Algebra Appl. 13, No. 6, 473–485 (2006; Zbl 1174.65389)] for solving the linear matrix equation \(AXB+CYD=E\). Paige’s algorithms, which are based on the bidiagonalization procedure of Golub and Kahan, are used as the framework for deriving these new matrix-form iterative methods. Some numerical examples illustrate the efficiency of the new matrix-form iterative methods.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F10 Iterative numerical methods for linear systems

Citations:

Zbl 1174.65389
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References:

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