A fast and efficient Newton-Shultz-type iterative method for computing inverse and Moore-Penrose inverse of tensors

A fast and efficient Newton-Shultz-type iterative method  is presented to compute the inverse of an invertible tensor. Analysis of the convergence error shows that the proposed method has the sixth order convergence. It is shown that the proposed algorithm can be used for finding the Moore-Penrose inverse of tensors. Computational complexities of the algorithm is presented to support the theoretical aspects of the paper. Using the  new method, we obtain a new preconditioner to solve the multilinear  system $mathcal{A}ast_Nmathcal{X}=mathcal{B}$. The effectiveness and accuracy  of this method are re-verified by several numerical examples. Finally, some conclusions are given.