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**Weighted least squares solutions to general coupled Sylvester matrix equations.**
*(English)*
Zbl 1161.65034

The paper is devoted to the weighted least squares solutions problem for general coupled Sylvester matrix equations. By adopting the gradient search principle in optimization theory, the authors provide a general gradient based iterative algorithm to solve this problem. The method used in the paper is quite different from the well known one. Especially, the authors introduce a necessary and sufficient condition to guarantee the convergence of the proposed algorithm. Based on this necessary and sufficient condition, a sufficient but wieldy condition can be induced. In deriving some of the results, they discover that a related result in the literature is in fact incorrect. Moreover, they suggest a method to choose the optimal step size in the algorithm, such that the proposed iteration converges fastest. They also provide several numerical examples to show the effectiveness of the proposed approach.

The paper is organized as follows. The problem formulation and some preliminary results are given in the beginning. The main results of the paper are shown next. Then, numerical examples are provided to illustrate the effectiveness of the proposed algorithm. Some concluding remarks are finally given.

The paper will be useful for all students, specialists and researchers, working in the area of weighted least squares solutions for coupled Sylvester matrix equations. It also is suitable for practitioners who wish to brush up these fundamental concepts.

The paper is organized as follows. The problem formulation and some preliminary results are given in the beginning. The main results of the paper are shown next. Then, numerical examples are provided to illustrate the effectiveness of the proposed algorithm. Some concluding remarks are finally given.

The paper will be useful for all students, specialists and researchers, working in the area of weighted least squares solutions for coupled Sylvester matrix equations. It also is suitable for practitioners who wish to brush up these fundamental concepts.

Reviewer: Tzvetan Semerdjiev (Sofia)

### MSC:

65F30 | Other matrix algorithms (MSC2010) |

15A24 | Matrix equations and identities |

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

### Keywords:

weighted least squares; weighted generalized inverses; gradient based iterative algorithms; maximal convergence rate; coupled Sylvester matrix equations; convergence; algorithm; numerical examples### Software:

KELLEY
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\textit{B. Zhou} et al., J. Comput. Appl. Math. 224, No. 2, 759--776 (2009; Zbl 1161.65034)

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