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Volume 17, Issue 1
On Polar Decomposition of Tensors with Einstein Product and a Novel Iterative Parametric Method

Raziyeh Erfanifar, Masoud Hajarian & Khosro Sayevand

Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 69-92.

Published online: 2024-02

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  • 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.

  • AMS Subject Headings

15A24, 65F10, 65F30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-17-69, author = {Erfanifar , RaziyehHajarian , Masoud and Sayevand , Khosro}, title = {On Polar Decomposition of Tensors with Einstein Product and a Novel Iterative Parametric Method}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {1}, pages = {69--92}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2023-0065}, url = {http://global-sci.org/intro/article_detail/nmtma/22911.html} }
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 -

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.

Raziyeh Erfanifar, Masoud Hajarian & Khosro Sayevand. (2024). On Polar Decomposition of Tensors with Einstein Product and a Novel Iterative Parametric Method. Numerical Mathematics: Theory, Methods and Applications. 17 (1). 69-92. doi:10.4208/nmtma.OA-2023-0065
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