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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.

MSC:
65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities
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