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.


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


Zbl 1174.65389
Full Text: DOI


[1] Baksalary, Linear Algebra and its Applications 30 pp 141– (1980)
[2] Peng, Mathematics Theory and Applications 22 pp 99– (2002)
[3] Xu, Linear Algebra and its Applications 279 pp 93– (1980)
[4] Chu, Linear Algebra and its Applications 87 pp 83– (1987)
[5] He, Journal of Natural Science of Hunan Normal University 19 pp 17– (1996)
[6] Wang, Journal of Jilin Institute of Chemical Technology 20 pp 84– (2003)
[7] Peng, Numerical Linear Algebra with Applications 13 pp 473– (2006)
[8] Numerical Methods in Scientific Computing, vols. I–II. SIAM: Philadelphia, PA, U.S.A., 2006. Available from: http://www.mai.liu.se/akbjo/.
[9] Shim, SIAM Journal on Matrix Analysis and Applications 3 pp 8002– (2003)
[10] Liao, SIAM Journal on Matrix Analysis and Applications 27 pp 675– (2005)
[11] Paige, SIAM Journal on Numerical Analysis 11 pp 197– (1974)
[12] Golub, SIAM Journal on Numerical Analysis 2 pp 205– (1965)
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