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  本文選題：齊次多項式正定性 + 強-張量�。� 參考：《曲阜師范大學》2017年碩士論文

【摘要】：張量是近年來新發(fā)展起來的大數(shù)據(jù)分析中的新工具,是矩陣的推廣.作為-矩陣的推廣、-張量擁有著特殊的結(jié)構(gòu)并在張量分析及運算上扮演著重要的角色,張量及強-張量的結(jié)構(gòu)性質(zhì)、判定準則以及迭代算法近年來廣受專家學者的關注.對于張量本身的既約情況直接影響算法的設計與實現(xiàn),因此對其可約性的研究也備受關注.此外,由于偶數(shù)階齊次多項式正定性在醫(yī)學成像、非線性自制系統(tǒng)的穩(wěn)定性分析、多元網(wǎng)絡可行性分析等方面的廣泛應用使其判定算法成為一個重要的研究課題.由于強-張量與張量正定性具有一致性,因此可基于此解決多項式正定性的問題.本文中,我們主要探究-張量的性質(zhì),提出一些新的判定強-張量的迭代準則,并根據(jù)所得準則設計不含參數(shù)的判定強-張量的算法.基于多項式與其相應張量正定性的一致性及強-張量與張量正定性之間的關系,我們所提出的算法亦是齊次多項式正定性的判定算法.本文的文章結(jié)構(gòu)安排如下:第一章,主要介紹張量、齊次多項式的正定性、二者間關系的研究背景和發(fā)展現(xiàn)狀,以及本文的主要研究成果.第二章,首先給出了強-張量的定義及相關性質(zhì).然后給出張量不可約及弱不可約的定義及相關結(jié)論并基于此提出新的判定強-張量的迭代準則.最后,基于一類廣義對角產(chǎn)物占優(yōu)我們提出了判定強-張量的新的迭代準則.并且本章所得結(jié)論是-矩陣相應結(jié)論在高階上的推廣亦是現(xiàn)存強-張量結(jié)論的進一步優(yōu)化.第三章,首先將上一章中基于弱可約概念提出的一些新的迭代準則用算法實現(xiàn),從而得到判定強-張量的一個新的不含參數(shù)的算法.其次基于一類廣義對角產(chǎn)物占優(yōu)得出了判定強-張量的新的迭代準則,給出新的判定強-張量的無參算法.算法的準確性都將給出證明,并由數(shù)值算例驗證其有效性.第四章,對所研究的內(nèi)容作簡要總結(jié),并就日后將要開展的工作做一些規(guī)劃.
[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.
【學位授予單位】：曲阜師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O183

【參考文獻】

相關期刊論文 前1條

1 HU ShengLong;HUANG ZhengHai;QI LiQun;;Strictly nonnegative tensors and nonnegative tensor partition[J];Science China(Mathematics);2014年01期

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本文編號：1978726