On the convergence of conjugate direction algorithm for solving coupled Sylvester matrix equations

In this work, we obtain the matrix form of the conjugate direction (MCD) method for solving the coupled Sylvester matrix equations (CSMEs)∑i=1sAiXBi=C,∑j=1tDjXEj=F,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{array}{ll} \sum _{i=1}^s A_iXB_i=C, &{} \hbox {} \\ \sum _{j=1}^t D_jXE_j=F, &{} \hbox {} \end{array}\right. $$\end{document}which are defined in the domain of real numbers. We prove that the MCD method converges to the solution of the CSMEs for any initial guess within a finite number of iterations in the absence of round-off errors. Also we show that the MCD method can find the least Frobenius norm solution of the CSMEs with special initial guess. Finally three numerical examples show that the MCD method is efficient to solve some matrix equations.