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

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.