Gradient-based iterative algorithms for solving Sylvester tensor equations and the associated tensor nearness problems

Abstract

In this paper, an iterative algorithm is presented for solving Sylvester tensor equation $\mathscr{A}*_M\mathscr{X}+\mathscr{X}*_N\mathscr{C}=\mathscr{D}$, where $\mathscr{A}$, $\mathscr{C}$ and $\mathscr{D}$ are given tensors with appropriate sizes, and the symbol $*_N$ denotes the Einstein product. By this algorithm, the solvability of this tensor equation can be determined automatically, and the solution of which (when it is solvable) can be derived within finite iteration steps for any initial iteration tensors in the absence of roundoff errors. Particularly, the least F-norm solution of the aforementioned equation can be derived by choosing special initial iteration tensors. As application, we apply the proposed algorithm to the tensor nearness problem related to the Sylvester tensor equation mentioned above. It is proved that the solution to this problem can also be obtained within finite iteration steps by solving another Sylvester tensor equation. The performed numerical experiments show that the algorithm we propose here is promising.