Global and extended global Hessenberg processes for solving Sylvester tensor equation with low rank right-hand side

In this paper, we introduce two new schemes based on the global Hessenberg processes for computing approximate solutions to low-rank Sylvester tensor equations.  We first construct bases for the matrix and extended matrix Krylov subspaces by applying the global and extended global  Hessenberg processes. Then the initial problem is projected into the matrix or extended matrix Krylov subspaces with small dimensions. The reduced Sylvester tensor equation obtained by the projection methods can be solved by using a recursive blocked algorithm.   Furthermore, we present the upper bounds for the residual tensors without requiring the computation of the approximate solutions in any iteration. Finally, we illustrate the performance of the proposed methods with some numerical examples.

Keywords:

Low-rank Sylvester tensor equation , Global Hessenberg process , Extended Global Hessenberg process , CP decomposition