Bounds And Calculation Methods Of The Spectral Radius Of Nonnegative Tensors

Nonnegative tensors are an important extension of nonnegative matrices.The estimation and calculation of their spectral radius(largest eigenvalue)have very important significance in numerical algebra,and it is also a research topic that has attracted much attention.This paper mainly studies the bounds and calculation methods of the spectral radius of nonnegative tensors.The content is as follows:Firstly,we obtain the improved results of the upper and lower bounds for the spectral radius of a nonnegative tensor by its majorization matrix's digraph.Through tensor analysis,the estimation inequality of the upper and lower bounds for the spectral radius of a nonnegative tensor with general form is given.The results improve the estimation results of the upper and lower bounds of the spectral radius of nonnegative tensors in related literatures.Secondly,the problem of calculating the spectral radius of a nonnegative tensor that satisfies the following conditions is solved by applying a diagonal similarity transformation similar to a matrix:(1)a nonnegative tensor whose majorization matrix is irreducible;(2)constructs a matrix by selecting elements in different positions of the tensor,and the matrix is an irreducible nonnegative matrix.Therefore,algorithms for calculating the spectral radius of irreducible nonnegative tensors are extended to weakly irreducible nonnegative tensors.In process of constructing algorithms,by introducing appropriate parameters,algorithms significantly reduce the number of iterations and iteration time under the condition that the calculation amount is basically unchanged,and improves computational efficiency.Thirdly,as an application of algorithm for spectral radius of nonnegative tensors,the iterative discriminant method of M-tensors is given which solves M-tensors iterative discrimination problem of tensors.