A new preconditioner of the tensor splitting iterative method for solving multi-linear systems with M-tensors

In this paper, we propose a new preconditioner of the tensor splitting iterative method for solving multi-linear systems with M-tensors. We theoretically show that the spectral radii of the preconditioner iterative tensor decrease as the parameters in the new preconditioners increase, if the preconditioned tensor is a strong M-tensor. Based on this, we give the comparison for spectral radii of preconditioned iterative tensors. Numerical examples are given to show our theoretical results and the efficiency of our new preconditioner. We also show the efficiency of our preconditioner used to solving higher order Markov chain.