Several tensor eigenpair definitions have been put forth in the past decade, but these can all be unified under generalized tensor eigenpair framework, introduced by Chang, Pearson, and Zhang [J. Math. Anal. Appl., 350 (2009), pp. 416–422]. Given mth-order, n-dimensional real-valued symmetric tensors $A$ and $B$, the goal is to find $\lambda \in R$ and $x \in R^n, x \neq 0$ such that $Ax^{m-1} = \lambda Bx^{m-1}$. Different choices for $B$ yield different versions of the tensor eigenvalue problem. We present our generalized eigenproblem adaptive power (GEAP) method for solving the problem, which is an extension of the shifted symmetric higher-order power method (SS-HOPM) for finding Z-eigenpairs. A major drawback of SS-HOPM is that its performance depended on choosing an appropriate shift, but our GEAP method also includes an adaptive method for choosing the shift automatically.
tensor eigenvalues, E-eigenpairs, Z-eigenpairs, l2-eigenpairs, generalized tensor eigenpairs, shifted symmetric higher-order power method (SS-HOPM), generalized eigenproblem adaptive power (GEAP) method
@article{KoMa14,
author = {Tamara G. Kolda and Jackson R. Mayo},
title = {An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs},
journal = {SIAM Journal on Matrix Analysis and Applications},
volume = {35},
number = {4},
pages = {1563--1581},
month = {December},
year = {2014},
doi = {10.1137/140951758},
}