Iterative Methods For Determining Strong ?-Tensors

Tensor theory plays an indispensable role in data mining and processing,neural networks,image processing,stoichiometry and psychometrics,elasticity analysis in physics and so on.As a part of the theory of tensors,identifying the strong ?-tensors play a crucial role in determining the positive definiteness of even-order multivariate homogeneous polynomials.However,it is difficult to determine whether a given tensor is a strong ?-tensor or not.Therefore,determining a tensor is strong ?-tensor or not has important theoretical significance and practical application background.In this paper,we mainly research the criteria for identifying strong ?-tensors.Firstly,we propose a new iterative algorithm for identifying strong ?-tensors.The properties of strong ?-tensors are used to prove that the algorithm is stopped in a finite step and its convergence rate is linear convergent.This algorithm can determine whether any given tensor is a strong ?-tensor or not.Secondly,based on the method of calculating the maximum eigenvalue of symmetric nonnegative tensors.We design an iterative algorithm for identifying symmetric strong ?-tensors.And it is proved that the algorithm converges when the input tensor is a symmetric tensor.Finally,based on the iterative algorithms of determining strong ?-tensors in this paper.We establish two iterative algorithms for identifying positive definiteness of even-order multivariate homogeneous polynomials.Some numerical examples are given to verify the effectiveness of the algorithms in this paper.