Programmable criteria for strong \(\mathcal {H}\)-tensors
A real tensor \(\mathcal A\) of order \(m\) and dimension \(n\) is called an \(\mathcal M\)-tensor if there exists a nonnegative tensor \(\mathcal B\) and a positive real number \(\eta \geq \rho({\mathcal B})\) such that \({\mathcal A}=\eta I - {\mathcal B}\). Here \(\rho(.)\) denotes the spectral radius. An \(\mathcal M\)-tensor \(\mathcal A=\eta I - {\mathcal B}\) is called a strong \(\mathcal M\)-tensor if \(\eta > \rho({\mathcal B})\). A tensor \(\mathcal A\) is called a strong \(\mathcal H\)-tensor if its comparison tensor is a strong \(\mathcal M\)-tensor. The definition of a comparison tensor is too complicated to be included here. However, since tensors generalize the notion of matrices, one could imagine a comparison tensor as an extension of a comparison matrix, quite well researched in the literature. The main contribution of the work reported, is in obtaining several sufficient conditions on the entries of a tensor \(\mathcal A\) which guarantee that \(\mathcal A\) is a strong \(\mathcal H\)-tensor. Consequently, an algorithm to identify strong \(\mathcal H\)-tensors is proposed. Numerical examples are used as illustrations.
scientific article; zbMATH DE number 6674017
scientific article