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An efficient iterative method for solving the matrix equation \(AXB + CYD = E\). (English) Zbl 1174.65389

Summary: This paper presents an iterative method for solving the matrix equation \(AXB + CYD = E\) with real matrices \(X\) and \(Y\). By this iterative method, the solvability of the matrix equation can be determined automatically. When the matrix equation is consistent, then, for any initial matrix pair \([X_0, Y_0]\), a solution pair can be obtained within finite iteration steps in the absence of round-off errors, and the least norm solution pair can be obtained by choosing a special kind of initial matrix pair. Furthermore, the optimal approximation solution pair to a given matrix pair \([\bar {X},\bar {Y}]\) in a Frobenius norm can be obtained by finding the least norm solution pair of a new matrix equation \(A\tilde {X}B + C\tilde {Y}D =\tilde {E}\), where \(\tilde {E}= E - A\bar {X}B - C\bar {Y}D\). The given numerical examples show that the iterative method is efficient.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
65F20 Numerical solutions to overdetermined systems, pseudoinverses
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