TY - JOUR
T1 - Tensor Bi-CR Methods for Solutions of High Order Tensor Equation Accompanied by Einstein Product
AU - Hajarian , Masoud
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 998
EP - 1016
PY - 2021
DA - 2021/09
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2021-0057
UR - https://global-sci.org/intro/article_detail/nmtma/19527.html
KW - Hestenes-Stiefel (HS) type of bi-conjugate residual (Bi-CR) algorithm, Lanczos type
of bi-conjugate residual (Bi-CR) algorithm, high order tensor equation, Einstein product.
AB - <p style="text-align: justify;">Tensors have a wide application in control systems, documents analysis,
medical engineering, formulating an $n$-person noncooperative game and so on. It is
the purpose of this paper to explore two efficient and novel algorithms for computing
the solutions $\mathcal{X}$ and $\mathcal{Y}$ of the high order tensor equation $\mathcal{A}*_P\mathcal{X}*_Q\mathcal{B}+\mathcal{C}*_P\mathcal{Y}*_Q\mathcal{D}=\mathcal{H}$ with Einstein product. The algorithms are, respectively, based on the Hestenes-Stiefel (HS) and the Lanczos types of bi-conjugate residual (Bi-CR) algorithm. The theoretical results indicate that the algorithms terminate after finitely many iterations
with any initial tensors. The resulting algorithms are easy to implement and simple
to use. Finally, we present two numerical examples that confirm our analysis and
illustrate the efficiency of the algorithms.</p>