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Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. (English) Zbl 1143.65035
By extending the well-known Jacobi and Gauss-Seidel iterations for $\mathrm{𝐀𝐱}=𝐛$, the iterative solutions of matrix equations $\mathrm{𝐀𝐗𝐁}=𝐅$ and generalized Sylvester matrix equations $\mathrm{𝐀𝐗𝐁}+\mathrm{𝐗𝐁}=𝐅$ (including the Sylvester equation $\mathrm{𝐀𝐗}+\mathrm{𝐗𝐁}=𝐅$ as a special case) are studied, and a gradient based and a least-squares based iterative algorithms for the solution is proved. It is proved that the iterative solution always converges to the exact solution for any initial values. The basic idea is to regard the unknown matrix $𝐗$ to be solved as the parameters of a system to be identified, and to obtain the iterative solutions by applying the hierarchical identification principle. Finally, the algorithms are tested and their effectiveness is shown using a numerical example.
##### MSC:
 65F30 Other matrix algorithms