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Finite iterative method for solving coupled Sylvester-transpose matrix equations

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Abstract

This paper is concerned with solutions to the so-called coupled Sylvester-transpose matrix equations, which include the generalized Sylvester matrix equation and Lyapunov matrix equation as special cases. By extending the idea of conjugate gradient method, an iterative algorithm is constructed to solve this kind of coupled matrix equations. When the considered matrix equations are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, a numerical example is given to illustrate that the algorithm is effective.

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Acknowledgements

The authors are very grateful to the anonymous reviewers and the editor for their helpful comments and suggestions which have helped us in improving the quality of this paper.

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Authors and Affiliations

  1. College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao, 266590, P.R. China

    Caiqin Song

  2. School of Mathematics, Shandong University, Jinan, 250100, P.R. China

    Caiqin Song & Jun-e Feng

  3. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, P.R. China

    Xiaodong Wang

  4. College of Mathematics Science, Liaocheng University, Liaocheng, 252059, P.R. China

    Jianli Zhao

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  1. Caiqin Song
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  2. Jun-e Feng
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  3. Xiaodong Wang
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  4. Jianli Zhao
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Corresponding author

Correspondence to Xiaodong Wang.

Additional information

This project is granted financial support from Postdoctoral Science Foundation of China (no. 2013M541900), NNSF 61174141 of China, Research Awards Young and Middle-Aged Scientists of Shandong Province (BS2011SF009, BS2011DX019), and Excellent Youth Foundation of Shandong’s Natural Scientific Committee (JQ201219).

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Song, C., Feng, Je., Wang, X. et al. Finite iterative method for solving coupled Sylvester-transpose matrix equations. J. Appl. Math. Comput. 46, 351–372 (2014). https://doi.org/10.1007/s12190-014-0753-x

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  • DOI: https://doi.org/10.1007/s12190-014-0753-x

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