A hybrid Newton's method for solving tensor square root problem

Tensor equations have been widely studied in recent years. In this paper, we proposed a hybrid Newton's method to solve tensor square root problem X & lowast;X=A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}*{\mathcal {X}}={\mathcal {A}}$$\end{document} via Einstein product. This algorithm makes use of tensor computations directly and combines the advantages of the Steepest descent method and the Newton's method, overcoming their disadvantages. The global convergence and the local quadratic convergence are obtained. Numerical results demonstrate that the hybrid Newton's method is competitive with the Newton's method in Duan (Appl Math Lett 98:57-62, 2019).