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**Gradient based and least squares based iterative algorithms for matrix equations \(AXB + CX^{T}D = F\).**
*(English)*
Zbl 1210.65097

A gradient based iterative algorithm and a least squares based iterative algorithm are developed and presented for the solution of the matrix equation \(AXB + CX^{T}D = F\). The hierarchical identification principle is applied to the matrix equation in order to decompose the system under consideration into two subsystems and to derive the iterative algorithms by extending the iterative methods for solving \(Ax = b\) and \(AXB = F\). Further analysis shows that when the matrix equation has a unique solution, under the sense of least squares, the iterative solution converges to the exact solution for any initial values. A numerical example is used to verify the proposed methods.

Reviewer: Vasilis Dimitriou (Chania)

### MSC:

65F30 | Other matrix algorithms (MSC2010) |

65F10 | Iterative numerical methods for linear systems |

15A24 | Matrix equations and identities |

### Keywords:

iterative algorithm; gradient search; least squares; Lyapunov matrix equations; Sylvester matrix equations; numerical example
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\textit{L. Xie} et al., Appl. Math. Comput. 217, No. 5, 2191--2199 (2010; Zbl 1210.65097)

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