Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form Ax^{m-1} = λx subject to ||x||=1, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs.
Computation of extreme eigenvalues in higher dimensions using block tensor train format
Max-Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进会数学与科学研究所）；Institute of Numerical Mathematics of Russian Academy of Sciences（俄罗斯科学院计算数学研究所）；University of Southampton（南安普顿大学）
Multi-subspace power method for decomposing partially symmetric tensors
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Department of Mathematics, University of Georgia（佐治亚大学数学系）；Department of Mathematics, University of Texas at Austin（德克萨斯大学奥斯汀分校数学系）
