### Solving coupled tensor equations via higher order LSQR methods

#### Abstract

Tensors have a wide application in control theory, data mining, chemistry, information sciences, documents analysis and medical engineering.

The material here is motivated by the development of the efficient

numerical methods for solving the coupled tensor

equations

$$

\left\{

\begin{array}{ll}

\mathcal{A}_1\ast_M\mathcal{X}\ast_N\mathcal{B}_1+\mathcal{C}_1\ast_M\mathcal{Y}\ast_N\mathcal{D}_1=\mathcal{E}_1,& \hbox{} \\\mathcal{A}_2\ast_M\mathcal{X}\ast_N\mathcal{B}_2+\mathcal{C}_2\ast_M\mathcal{Y}\ast_N\mathcal{D}_2=\mathcal{E}_2, & \hbox{}

\end{array}

\right.

$$

with Einstein product. We propose the tensor form of

the LSQR methods to find the solutions

$\mathcal{X}$ and $\mathcal{Y}$ of the coupled tensor

equations. Finally we give some numerical examples to illustrate that our proposed methods are able to accurately and efficiently find the solutions of tensor equations with Einstein product.

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