A gradient based iterative solutions for Sylvester tensor equations (Q474659)

Summary: This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by \textit{F. Ding} and \textit{T. Chen} [IEEE Trans. Autom. Control 50, No. 8, 1216--1221 (2005; \url{doi:10.1109/TAC.2005.852558)}], and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.