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Two Relaxed Gradient-Based Algorithms for the Hermitian and Skew-Hermitian Solutions of the Linear Matrix Equation AXB + CXD = F

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Abstract

In this paper, two relaxed gradient-based algorithms for solving the linear matrix equation \( AXB + CXD = F \) and finding the Hermitian and skew-Hermitian solutions are presented. We proved that the algorithms converge to the Hermitian and skew-Hermitian solutions. A sufficient condition is given to guarantee that the solutions given by the proposed algorithms converge to the Hermitian and skew-Hermitian solutions for any initial matrix. Two numerical examples are given to test its efficiency and accuracy compared with our proposed modification of the gradient-based iterative algorithm proposed in Ding et al. (Appl Math Comput 197(1):41–50, 2008).

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Acknowledgements

The authors wish to thank the anonymous referees for their valuable suggestions and comments.

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This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

    Ahmed M. E. Bayoumi

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  1. Ahmed M. E. Bayoumi
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Correspondence to Ahmed M. E. Bayoumi.

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Bayoumi, A.M.E. Two Relaxed Gradient-Based Algorithms for the Hermitian and Skew-Hermitian Solutions of the Linear Matrix Equation AXB + CXD = F. Iran J Sci Technol Trans Sci 43, 2343–2350 (2019). https://doi.org/10.1007/s40995-019-00694-5

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  • DOI: https://doi.org/10.1007/s40995-019-00694-5

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