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A matrix LSQR iterative method to solve matrix equation $AXB=C$. (English) Zbl 1195.65056
Summary: This paper is a matrix iterative method presented to compute the solutions of the matrix equation, $AXB=C$, with unknown matrix $X\in 𝒮$, where $𝒮$ is the constrained matrices set like symmetric, symmetric-$R$-symmetric and $\left(R,S\right)$-symmetric. By this iterative method, for any initial matrix ${X}_{0}\in S$, a solution ${X}^{*}$ can be obtained within finite iteration steps if exact arithmetics were used, and the solution ${X}^{*}$ with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution $\stackrel{^}{X}$, which is nearest to a given matrix $\stackrel{˜}{X}$ in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in earlier papers.
##### MSC:
 65F30 Other matrix algorithms 65F10 Iterative methods for linear systems