Hermitian and skew-Hermitian splitting methods for solving a tensor equation

Abstract

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.

Keywords:

• Tensor equation
• Einstein product
• splitting methods
• Smith technique

2010 AMS Subject Classifications:

• 15A69
• 65F10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by National Natural Science Foundation of China [grant numbers 11971294 and 11571220].