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\iteman{io-port 05810129}
\itemau{Choulakian, Vartan}
\itemti{Some numerical results on the rank of generic three-way arrays over $\Bbb{R}$.}
\itemso{SIAM J. Matrix Anal. Appl. 31, No. 4, 1541-1551 (2010).}
\itemab
The author introduces a new method for the numerical computation of the rank of a three-way array $X\in\Bbb R^{I\times J\times K}$ over ${\Bbb R}$ showing that the rank is intimately related to the real solution set of a system of polynomial equations.  The first part is introductory in nature.  The second part presents the main Lemma which provides a necessary and sufficient condition for a three-way array $X\in\Bbb R^{I\times J\times K},$ $2\leq K\leq J\leq I$, to be expressed as a sum of a fixed number of decomposed tensors $$X=\sum_{\alpha=1}^I a_\alpha\otimes b_\alpha\otimes c_\alpha,$$ where $\{a_\alpha\mid\alpha=1,\dots,I\}$ is a basis for $\Bbb {R}^I,$ $c_\alpha\in\Bbb R^K$ and $b_\alpha\in{\Bbb R}^J.$ The third part concerns the rank computation showing how the Lemma can be applied to compute the rank of a generic tensor over ${\Bbb R}$ via Gr\"obner basis numerically for some cases.  The main conclusions and perspectives are presented in the last part.
\itemrv{R. Militaru (Craiova)}
\itemcc{}
\itemut{tensors; three-way arrays; candecomp/Parafac; generic rank; typicl rank; degree of Segre variety; Gr\"obner bases; numerical examples; system of polynomial equations}
\itemli{doi:10.1137/08073531X}
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