TenEig swMATH ID: 14671 Software Authors: Chen, Liping; Han, Lixing; Zhou, Liangmin Description: Computing tensor eigenvalues via homotopy methods. We introduce the concept of mode-k generalized eigenvalues and eigenvectors of a tensor and prove some properties of such eigenpairs. In particular, we derive an upper bound for the number of equivalence classes of generalized tensor eigenpairs using mixed volume. Based on this bound and the structures of tensor eigenvalue problems, we propose two homotopy continuation type algorithms to solve tensor eigenproblems. With proper implementation, these methods can find all equivalence classes of isolated generalized eigenpairs and some generalized eigenpairs contained in the positive dimensional components (if there are any). We also introduce an algorithm that combines a heuristic approach and a Newton homotopy method to extract real generalized eigenpairs from the found complex generalized eigenpairs. A MATLAB software package TenEig has been developed to implement these methods. Numerical results are presented to illustrate the effectiveness and efficiency of TenEig for computing complex or real generalized eigenpairs. Homepage: http://users.math.msu.edu/users/chenlipi/teneig.html Dependencies: Matlab Related Software: TensorToolbox; PHCpack; Matlab; PHoM; HOM4PS; NACLab; Bertini; SPLATT; Algorithm 862; PHClab; MixedVol; libtropicana; Hom4PS-3; Gfan; PoSSo; DEMiCs; LAMG; Seigtool; Eigtool; ARPACK Cited in: 30 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Computing tensor eigenvalues via homotopy methods. Zbl 1376.15017Chen, Liping; Han, Lixing; Zhou, Liangmin 2016 all top 5 Cited by 51 Authors 6 Qi, Liqun 4 Chen, Yannan 4 Han, Lixing 3 Chang, Jingya 3 Chen, Liping 3 Liu, Jianzhou 3 Xiong, Liang 3 Zhou, Liangmin 2 Lin, Wen-Wei 2 Liu, Ching-Sung 2 Mo, Changxin 2 Sang, Caili 2 Wang, Yi 2 Wei, Yimin 2 Zhao, Jianxing 1 Bao, Yanhong 1 Bozorgmanesh, Hassan 1 Che, Maolin 1 Chen, Tianran 1 Ding, Weiyang 1 Fan, Yizheng 1 Guo, Chun-Hua 1 Hajarian, Masoud 1 Hu, Futao 1 Huang, Zheng-Hai 1 Jaffe, Ariel 1 Kuo, Yueh-Cheng 1 Li, Chaoqian 1 Li, Guoyin 1 Li, Wen 1 Liang, Maolin 1 Liu, Xifu 1 Nadler, Boaz 1 Ni, Guyan 1 Ni, Qin 1 Qin, Qi 1 Sheng, Zhou 1 Song, Yisheng 1 Wan, Jiang-Chao 1 Wang, Qun 1 Wang, Xuezhong 1 Weiss, Roi 1 Wen, Yaqiong 1 Xu, Yangyang 1 Yan, Hong 1 Yin, Hongxia 1 Zhang, Guofeng 1 Zhang, Mengshi 1 Zhao, Ruijuan 1 Zheng, Bing 1 Zhu, Ming all top 5 Cited in 21 Serials 3 Journal of Computational and Applied Mathematics 3 Computational and Applied Mathematics 2 Journal of Optimization Theory and Applications 2 Acta Applicandae Mathematicae 2 SIAM Journal on Matrix Analysis and Applications 2 Journal of Scientific Computing 2 SIAM Journal on Scientific Computing 1 Applied Mathematics and Computation 1 Calcolo 1 Discrete & Computational Geometry 1 Applied Mathematics Letters 1 Numerical Algorithms 1 Linear Algebra and its Applications 1 Journal of Algebraic Combinatorics 1 Computational Optimization and Applications 1 Numerical Linear Algebra with Applications 1 The Electronic Journal of Combinatorics 1 Quantum Information Processing 1 Frontiers of Mathematics in China 1 Analysis and Mathematical Physics 1 Numerical Algebra, Control and Optimization all top 5 Cited in 11 Fields 23 Linear and multilinear algebra; matrix theory (15-XX) 20 Numerical analysis (65-XX) 6 Operations research, mathematical programming (90-XX) 5 Combinatorics (05-XX) 3 Quantum theory (81-XX) 1 Commutative algebra (13-XX) 1 Convex and discrete geometry (52-XX) 1 Differential geometry (53-XX) 1 Mechanics of particles and systems (70-XX) 1 Mechanics of deformable solids (74-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) Citations by Year