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New matrix iterative methods for constraint solutions of the matrix equation $AXB=C$. (English) Zbl 1206.65145
Summary: Two new matrix iterative methods are presented to solve the matrix equation $AXB=C$, the minimum residual problem ${min}_{X\in 𝒮}\parallel AXB-C\parallel$ and the matrix nearness problem ${min}_{X\in {S}_{E}}\parallel X-{X}^{*}\parallel$, where $𝒮$ is the set of constraint matrices, such as symmetric, symmetric $R$-symmetric and $\left(R,S\right)$-symmetric, and ${S}_{E}$ is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in [Y.-B. Deng et al., Numer. Linear Algebra Appl. 13, No. 10, 801–823 (2006; Zbl 1174.65382); G.-X. Huang et al., J. Comput. Appl. Math. 212, No. 2, 231–244 (2008; Zbl 1146.65036); the author, Appl. Math. Comput. 170, No. 1, 711–723 (2005; Zbl 1081.65039); and Y. Lei and A. Liao, Appl. Math. Comput. 188, No. 1, 499–513 (2007; Zbl 1131.65038)]. Paige’s algorithm is used as the frame method for deriving these matrix iterative methods. Numerical examples are used to illustrate the efficiency of these new methods.
##### MSC:
 65F30 Other matrix algorithms