On finding strong approximate inverses for tensors

This article investigates a fast and highly efficient algorithm to find the strong approximation inverse of an invertible tensor. The convergence analysis shows that the proposed method is of ten order of convergence using only six tensor–tensor multiplications per iteration. Also, we obtain a bound for the perturbation error in each iteration. We show that the proposed algorithm can be used for finding the Moore–Penrose and outer inverses of tensors. We obtain the relationship between the singular values of an arbitrary tensor 𝒜 and eigenvalues of the 𝒜∗⋆N𝒜 . We give the computational complexity of our algorithm and prove the theoretical aspects of the article. The generalized Moore–Penrose inverse of tensors is defined. As an application, we use the iteration obtained by the algorithm as preconditioning of the Krylov subspace methods to solve the multilinear system 𝒜⋆N𝒳=ℬ . Several numerical experiments are proposed to show the effectiveness and accuracy of the method. Finally, we give some concluding remarks.