TY - JOUR
T1 - On Polar Decomposition of Tensors with Einstein Product and a Novel Iterative Parametric Method
AU - Erfanifar , Raziyeh
AU - Hajarian , Masoud
AU - Sayevand , Khosro
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 69
EP - 92
PY - 2024
DA - 2024/02
SN - 17
DO - http://doi.org/10.4208/nmtma.OA-2023-0065
UR - https://global-sci.org/intro/article_detail/nmtma/22911.html
KW - Iterative methods, Einstein product, polar decomposition of a tensor, polar factor, order of convergence.
AB - <p style="text-align: justify;">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.</p>