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An efficient algorithm for solving general coupled matrix equations and its application. (English) Zbl 1208.65054

The authors present a method for the solution ${X}_{1},{X}_{2},\cdots ,{X}_{l},$ (here denoted by solution group of matrices) with ${X}_{j}\in {ℝ}^{{n}_{j}×{m}_{j}},j=1\left(1\right)l$, of the general coupled matrix equations

$\sum _{j=1}^{l}{A}_{ij}{X}_{j}{B}_{ij}={C}_{i},\phantom{\rule{1.em}{0ex}}i=1\left(1\right)l,$

for given ${A}_{ij}\in {ℝ}^{{p}_{i}×{n}_{j}}$, ${B}_{ij}\in {ℝ}^{{m}_{j}×{q}_{i}}$, and ${C}_{i}\in {ℝ}^{{p}_{i}×{q}_{i}}$, $i,j=1\left(1\right)l$.

Well-known special cases of these equations are the Sylvester and Lyapunov matrix equations that play e. g. a role in differential equations, control and stability theory. A large number of papers, which deal with these special cases including the applied methods, are quoted.

Unlike these methods in this paper an iterative algorithm is presented for the quite general coupled matrix equations that include several matrix equations extending them for the solution of large systems of linear equations used the conjugate gradient method. Assuming the consistence of the matrix equations, for any initial matrix group a solution group is found within finite iteration steps without roundoff errors. A least Frobenius norm solution can be derived choosing an appropriate initial matrix group. The authors prove that they can find the optimal approximation group in a Frobenius norm within the solution group set for any initial matrix group ${X}_{1}^{\left(1\right)},{X}_{2}^{\left(1\right)},\cdots ,{X}_{l}^{\left(1\right)}$.

The algorithm is validated using some simple numerical examples of coupled Sylvester equations taken from the literature. A comparison of the results with Jacobi and Gauss-Seidel methods created by [F. Ding and T. Chen, SIAM J. Control Optim. 44, No. 6, 2269–2284 (2006, Zbl 1115.65035)] for Sylvester equations shows that the proposed conjugate gradient algorithm is more efficient.

Additionally, the application of the proposed method to $\left(R,S\right)$-symmetric and $\left(R,S\right)$-skew symmetric matrices is analyzed.

##### MSC:
 65F30 Other matrix algorithms 65F10 Iterative methods for linear systems 15A24 Matrix equations and identities