On the Execution and Convergence of GMRES_BTF Method for Solving Sylvester Tensor Equations

Recently, Chen and Lu have handled the well-known generalized minimal residual based on tensor format (GMRES_BTF) for solving the Sylvester tensor equation. Nevertheless, the construction and convergence of the presented algorithm have not been discussed theoretically. This fact inspirits us to theoretically analyze the construction of the GMRES_BTF method. To this end, we introduce a new product between two tensors and elaborate some of its properties. The presented product can be exploited to illustrate that how the GMRES_BTF algorithm is implemented in the tensor format and to establish the convergence properties of the algorithm.