Fast and flexible preconditioners for solving multilinear systems

This paper investigates a type of fast and flexible preconditioners to solve multilinear system $\mathcal{A}\textbf{x}^{m-1}=\textbf{b}$ with $\mathcal{M}$-tensor $\mathcal{A}$ and obtains some important convergent theorems about preconditioned Jacobi, Gauss-Seidel and SOR type iterative methods. The main results theoretically prove that the preconditioners can accelerate the convergence of iterations. Numerical examples are presented to reverify the efficiency of the proposed preconditioned methods.