The present paper deals with the numerical solution of non-Hermitian positive definite tensor equation $\mathcal{A}{\ast }_N\mathcal{X}=\mathcal{B}$ under the Einstein product. Firstly, we extend the Hermitian and skew-Hermitian splitting (HSS) method to solve the tensor equation. Then we propose a new Hermitian splitting (NHS) method under some certain conditions, which is expected to converge faster than the HSS iteration. We also present the optimal parameters of both the HSS and NHS methods. Moreover, we apply the Smith technique to give two modified methods which can greatly accelerate the convergence rate. The performed numerical examples illustrate that the proposed methods are feasible and efficient.

本文研究了非Hermitian正定张量方程的数值解 $\mathcal{一种}{\ast }_{ñ}\mathcal{X}=\mathcal{乙}$根据爱因斯坦产品。首先，我们扩展了Hermitian和Skew-Hermitian分裂（HSS）方法来求解张量方程。然后，我们提出了在某些条件下的一种新的埃尔米特分裂（NHS）方法，该方法有望比HSS迭代更快地收敛。我们还介绍了HSS和NHS方法的最佳参数。此外，我们应用史密斯技术给出了两种可以大大加快收敛速度​​的改进方法。数值算例表明所提出的方法是可行和有效的。