residual norm steepest descent based iterative algorithms for sylvester tensor equations

consider the following consistent sylvester tensor equation[mathscr{x}times_1 a +mathscr{x}times_2 b+mathscr{x}times_3 c=mathscr{d},]where the matrices $a,b, c$ and the tensor $mathscr{d}$ are given and $mathscr{x}$ is the unknown tensor. the current paper concerns with examining a simple and neat framework for accelerating the speed of convergence of the gradient-based iterative algorithm and its modified version for solving the mentioned sylvester tensor equation without setting the restriction of the existence of a unique solution. numerical experiments are reported which confirm the validity of the presented results.