几类非负张量有向超图性质的研究

The numerical analysis for nonnegative tensors is one of hot topic in the field of computational mathematics. The nonnegative tensors have good spectral properties. The spectral radius is an eigenvalue of nonnegative tensor. And the spectral radius has an unique positive eigenvector if the tensor is irreducible. Although the weakly irreducible tensor also has this spectral property, the eigenvalue which has an nonnegative eigenvector may not be the spectral radius. It is notice that the irreducibility is just only a sufficient condition for an nonnegative tensor has a positive spectral eigenvector. In this project, we will give a necessary and sufficient condition for a nonnegative has a positive spectral eigenvector. The properties of irreducibility and having positive spectral eigenvector are the structure properties of nonnegative tensors. The directed hypergraph is one of useful tool for studying the structure properties of non- negative tensors. This project will study the relationship between edges, path of directed hypergraph and the nonnegative entries of nonnegative tensor and give the equivalent condition of irreducibility in directed hypergraph theory. And we will design new algorithm for testing the irreducibility of an nonnegative tensor and give the computational complexity of the new algorithm. The project will also study the necessary and sufficient condition for an nonnegative tensor has a positive spectral eigenvector by studying the relationship between the block expression of eigen-equation and the classification of directed hypergraph.

非负张量的数值分析是近年来国内外计算数学领域的热点课题之一。非负张量有良好的谱性质，谱半径是特征值，并且非负不可约张量的谱半径有唯一的正特征向量。弱不可约张量虽然也有这个谱性质，但是有非负特征向量的特征值不一定是谱半径。所以给出不可约性的等价描述是有意义的。另外，不可约性只是非负张量有正的谱特征向量的充分条件。因此，本项目将研究非负张量有正的谱特征向量的充要条件。不可约性及有正的谱特征向量是非负张量的结构性质。而非负张量的有向超图是研究非负张量结构性质的有力工具之一。因此，本项目将通过研究有向超图中的边、路径等与非负张量元素之间的关系给出不可约张量在有向超图理论中的等价条件；并且将结合不可约张量的有向超图性质设计验证不可约性的新算法并给出复杂度分析。另外，本项目将研究非负张量的特征方程的分块表达式与有向超图中分类的关系，给出非负张量有正的谱特征向量的充要条件。