基于强H-张量的齐次多项式正定性的判定算法研究

发布时间：2018-06-04 20:29

本文选题：齐次多项式正定性 + 强-张量　；参考：《曲阜师范大学》2017年硕士论文

【摘要】：张量是近年来新发展起来的大数据分析中的新工具,是矩阵的推广.作为-矩阵的推广、-张量拥有着特殊的结构并在张量分析及运算上扮演着重要的角色,张量及强-张量的结构性质、判定准则以及迭代算法近年来广受专家学者的关注.对于张量本身的既约情况直接影响算法的设计与实现,因此对其可约性的研究也备受关注.此外,由于偶数阶齐次多项式正定性在医学成像、非线性自制系统的稳定性分析、多元网络可行性分析等方面的广泛应用使其判定算法成为一个重要的研究课题.由于强-张量与张量正定性具有一致性,因此可基于此解决多项式正定性的问题.本文中,我们主要探究-张量的性质,提出一些新的判定强-张量的迭代准则,并根据所得准则设计不含参数的判定强-张量的算法.基于多项式与其相应张量正定性的一致性及强-张量与张量正定性之间的关系,我们所提出的算法亦是齐次多项式正定性的判定算法.本文的文章结构安排如下:第一章,主要介绍张量、齐次多项式的正定性、二者间关系的研究背景和发展现状,以及本文的主要研究成果.第二章,首先给出了强-张量的定义及相关性质.然后给出张量不可约及弱不可约的定义及相关结论并基于此提出新的判定强-张量的迭代准则.最后,基于一类广义对角产物占优我们提出了判定强-张量的新的迭代准则.并且本章所得结论是-矩阵相应结论在高阶上的推广亦是现存强-张量结论的进一步优化.第三章,首先将上一章中基于弱可约概念提出的一些新的迭代准则用算法实现,从而得到判定强-张量的一个新的不含参数的算法.其次基于一类广义对角产物占优得出了判定强-张量的新的迭代准则,给出新的判定强-张量的无参算法.算法的准确性都将给出证明,并由数值算例验证其有效性.第四章,对所研究的内容作简要总结,并就日后将要开展的工作做一些规划.
[Abstract]:Tensor is a new tool in the new development of large data analysis in recent years. It is a generalization of matrix. As a generalization of the matrix, the tensor has a special structure and plays an important role in the tensor analysis and operation, the structural properties of tensor and tensor, and the decision criteria and iterative algorithms are widely concerned by experts and scholars in recent years. The research of the reducibility of the tensor itself directly affects the design and implementation of the algorithm, so the research on its reducibility is also paid much attention. In addition, the extensive application of the even order homogeneous polynomial positive determinability in medical imaging, the stability analysis of the nonlinear self-made system and the feasibility analysis of multiple networks make its decision algorithm a In this paper, we mainly explore the properties of the tensor and propose some new iterative criteria for determining the strong tensor, and design a strong tensor algorithm without parameter based on the obtained criteria. In the first chapter, we mainly introduce the tensor, the positive definite property of the homogeneous polynomial, the research background and the development status of the relationship between the two. In the second chapter, the definition and the related properties of the strong tensor are given. Then the definition of the tensor irreducibility and the weak irreducibility and the related conclusions are given. Based on this, a new iterative criterion for determining the strong tensor is proposed. Finally, based on a class of generalized diagonal products, we propose a new superposition of strong tensor. The conclusion of this chapter is that the generalization of the corresponding conclusion of the matrix in the higher order is also the further optimization of the existing strong tensor conclusion. In the third chapter, first, some new iterative criteria based on the weakly reducible concept in the last chapter are implemented, and a new algorithm for determining the strong tensioned quantity is obtained. A new iterative criterion for determining strong tensor is obtained from a class of generalized diagonal products, and a new non parametric algorithm for determining strong tensor is given. The accuracy of the algorithm will be proved and the validity of the algorithm is verified by numerical examples. The fourth chapter gives a brief summary of the contents of the study, and makes some plans for the work to be carried out in the future.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O183

【参考文献】