Developing CRS iterative methods for periodic Sylvester matrix equation

In this paper, by applying Kronecker product and vectorization operator, we extend two mathematical equivalent forms of the conjugate residual squared (CRS) method to solve the periodic Sylvester matrix equation AjXjBj+CjXj+1Dj=Ejfor j=1,2,…,λ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} A_{j} X_{j} B_{j} + C_{j} X_{j+1} D_{j} = E_{j} \quad \text{for } j=1,2, \ldots ,\lambda . \end{aligned}$$ \end{document} We give some numerical examples to compare the accuracy and efficiency of the matrix CRS iterative methods with other methods in the literature. Numerical results validate that the proposed methods are superior to some existing methods and that equivalent mathematical methods can show different numerical performance.