Iterative Methods for Solving Sylvester Transpose Tensor Equation A⋆NX⋆MB+C⋆MXT⋆ND=E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$~\mathcal A\star _N\mathcal X\star _M\mathcal {B}+\mathcal {C}\star _M\mathcal X^T\star _N\mathcal {D}=\mathcal {E}$$\end{document}

In recent years, solving tensor equations has attracted the attention of mathematicians in applied mathematics. This paper investigated the gradient-based and gradient-based least-squares iterative algorithms to solve the Sylvester transpose tensor equation A⋆NX⋆MB+C⋆MXT⋆ND=E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal A\star _N\mathcal X\star _M\mathcal {B}+\mathcal {C}\star _M\mathcal X^T\star _N\mathcal {D}=\mathcal {E}$$\end{document}. These algorithms use tensor computations with no matricizations involved which includes the Sylvester transpose matrix equation as special case. The first algorithm is applied when the tensor equation is consistent. Error convergence analysis of the proposed methods has been discussed. For inconsistent Sylvester transpose tensor equation, the gradient-based least-squares iterative method is presented. Modified versions of these algorithms are obtained by little changes. Also, it is showed that for any initial tensor, a solution of related problems can be obtained within finite iteration steps in the absence of round-off errors. In addition, the computational cost of the methods is obtained. The effectiveness of these procedures are illustrated by several numerical examples. Finally, some concluding remarks are given.