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Iterative solution to a class of complex matrix equations and its application in time-varying linear system

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Abstract

Wu et al. (Applied Mathematics and Computation 217(2011)8343-8353) constructed a gradient based iterative (GI) algorithm to find the solution to the complex conjugate and transpose matrix equation

$$\begin{aligned} A_{1}XB_{1}+A_{2}\overline{X}B_{2}+A_{3}X^{T}B_{3}+A_{4}X^{H}B_{4}=E \end{aligned}$$

and a sufficient condition for guaranteeing the convergence of GI algorithm was given for an arbitrary initial matrix. Zhang et al. (Journal of the Franklin Institute 354 (2017) 7585-7603) provided a new proof of GI method and the necessary and sufficient conditions was presented to guarantee that the proposed algorithm was convergent for an arbitrary initial matrix. In this paper, a relaxed gradient based iterative (RGI) algorithm is proposed to solve this complex conjugate and transpose matrix equation. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the introduced algorithm for any initial iterative matrix. Numerical results are given to verify the efficiency of the new method. Finally, the application in time-varying linear system of the presented algorithm is provided.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Jinan, Jinan, 250022, People’s Republic of China

    Wenli Wang & Caiqin Song

  2. Department of Mathematics and Statistics, University of Nevada, Reno, 89503, USA

    Caiqin Song

  3. School of of International Education, Guizhou Normal University, Guiyang, 550025, People’s Republic of China

    Shipu Ji

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  1. Wenli Wang
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  2. Caiqin Song
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Correspondence to Caiqin Song.

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This work is supported by Shandong Natural Science Foundation (No. ZR2020MA052, No. ZR2020MA055), Anhui Natural Science Foundation (No. 2008085MA12), the National NaturalScience Foundation of China (No.11501246 and No.11801216) and Shandong Natural Science Foundation (No. ZR2017BA010).

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Wang, W., Song, C. & Ji, S. Iterative solution to a class of complex matrix equations and its application in time-varying linear system. J. Appl. Math. Comput. 67, 317–341 (2021). https://doi.org/10.1007/s12190-020-01486-6

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  • DOI: https://doi.org/10.1007/s12190-020-01486-6

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