Tensor Bi-CR Methods for Solutions of High Order Tensor Equation Accompanied by Einstein Product

Abstract

Tensors have a wide application in control systems, documents analysis, medical engineering, formulating an $n$-person noncooperative game and so on. It is the purpose of this paper to explore two efficient and novel algorithms for computing the solutions $\mathcal{X}$ and $\mathcal{Y}$ of the high order tensor equation $\mathcal{A}*_P\mathcal{X}*_Q\mathcal{B}+\mathcal{C}*_P\mathcal{Y}*_Q\mathcal{D}=\mathcal{H}$ with Einstein product. The algorithms are, respectively, based on the Hestenes-Stiefel (HS) and the Lanczos types of bi-conjugate residual (Bi-CR) algorithm. The theoretical results indicate that the algorithms terminate after finitely many iterations with any initial tensors. The resulting algorithms are easy to implement and simple to use. Finally, we present two numerical examples that confirm our analysis and illustrate the efficiency of the algorithms.