A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

Abstract

In this study, based on an iterative method to solve nonlinear equations, a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed. Numerical comparisons of the proposed method with other methods show that the average number of iterations, number of the Einstein products, and CPU time of our method are considerably less than other methods. In some applications, partial and fractional differential equations that lead to sparse matrices are considered as prototypes. We use the iterates obtained by the method as a preconditioner, based on tensor form to solve the multilinear system A * N X = B. Finally, several practical numerical examples are also given to display the accuracy and efficiency of the new method. The presented results show that the proposed method is very robust for obtaining the Moore-Penrose inverse of tensors.