A modified block Hessenberg method for low-rank tensor Sylvester equation

This work focuses on iteratively solving the tensor Sylvester equation with low-rank right-hand sides. To solve such equations, we first introduce a modified version of the block Hessenberg process so that approximation subspaces contain some extra block information obtained by multiplying the initial block by the inverse of each coefficient matrix of the tensor Sylvester equation. Then, we apply a Galerkin-like condition to transform the original tensor Sylvester equation into a low-dimensional tensor form. The reduced problem is then solved using a blocked recursive algorithm based on Schur decomposition. Moreover, we reveal how to stop the iterations without the need to compute the approximate solution by calculating the residual norm or an upper bound. Eventually, some numerical examples are given to assess the efficiency and robustness of the suggested method.