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

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