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.

References in zbMATH (referenced in 19 articles , 1 standard article )

Showing results 1 to 19 of 19.
Sorted by year (citations)

  1. Bao, Yan-Hong; Fan, Yi-Zheng; Wang, Yi; Zhu, Ming: A combinatorial method for computing characteristic polynomials of starlike hypergraphs (2020)
  2. Bozorgmanesh, Hassan; Hajarian, Masoud: Solving tensor E-eigenvalue problem faster (2020)
  3. Xiong, Liang; Liu, Jianzhou: (Z)-eigenvalue inclusion theorem of tensors and the geometric measure of entanglement of multipartite pure states (2020)
  4. Zhang, Mengshi; Ni, Guyan; Zhang, Guofeng: Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement (2020)
  5. Chen, Liping; Han, Lixing; Yin, Hongxia; Zhou, Liangmin: A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor (2019)
  6. Chen, Tianran: Unmixing the mixed volume computation (2019)
  7. Guo, Chun-Hua; Lin, Wen-Wei; Liu, Ching-Sung: A modified Newton iteration for finding nonnegative (Z)-eigenpairs of a nonnegative tensor (2019)
  8. Han, Lixing: A continuation method for tensor complementarity problems (2019)
  9. Mo, Changxin; Li, Chaoqian; Wang, Xuezhong; Wei, Yimin: (Z)-eigenvalues based structured tensors: (\mathcalM_Z)-tensors and strong (\mathcalM_Z)-tensors (2019)
  10. Qi, Liqun; Huang, Zheng-Hai: Tensor complementarity problems. II: Solution methods (2019)
  11. Chang, Jingya; Ding, Weiyang; Qi, Liqun; Yan, Hong: Computing the (p)-spectral radii of uniform hypergraphs with applications (2018)
  12. Jaffe, Ariel; Weiss, Roi; Nadler, Boaz: Newton correction methods for computing real eigenpairs of symmetric tensors (2018)
  13. Kuo, Yueh-Cheng; Lin, Wen-Wei; Liu, Ching-Sung: Continuation methods for computing Z-/H-eigenpairs of nonnegative tensors (2018)
  14. Che, Maolin; Li, Guoyin; Qi, Liqun; Wei, Yimin: Pseudo-spectra theory of tensors and tensor polynomial eigenvalue problems (2017)
  15. Chen, Liping; Han, Lixing; Zhou, Liangmin: Linear homotopy method for computing generalized tensor eigenpairs (2017)
  16. Chen, Yannan; Qi, Liqun; Zhang, Xiaoyan: The Fiedler vector of a Laplacian tensor for hypergraph partitioning (2017)
  17. Chang, Jingya; Chen, Yannan; Qi, Liqun: Computing eigenvalues of large scale sparse tensors arising from a hypergraph (2016)
  18. Chen, Liping; Han, Lixing; Zhou, Liangmin: Computing tensor eigenvalues via homotopy methods (2016)
  19. Chen, Yannan; Qi, Liqun; Wang, Qun: Computing extreme eigenvalues of large scale Hankel tensors (2016)