On Polar Decomposition of Tensors with Einstein Product and a Novel Iterative Parametric Method

Abstract

. This study aims to investigate the polar decomposition of tensors with the Einstein product for the first time. The polar decomposition of tensors can be computed using the singular value decomposition of the tensors with the Einstein product. In the following, some iterative methods for finding the polar decomposition of matrices have been developed into iterative methods to compute the polar decomposition of tensors. Then, we propose a novel parametric iterative method to find the polar decomposition of tensors. Under the obtained conditions, we prove that the proposed parametric method has the order of convergence four. In every iteration of the proposed method, only four Einstein products are required, while other iterative methods need to calculate multiple Einstein products and one tensor inversion in each iteration. Thus, the new method is superior in terms of efficiency index. Finally, the numerical comparisons performed among several well-known methods, show that the proposed method is remarkably efficient and accurate.