# An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs

### Abstract

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.

Type
Publication
SIAM Journal on Matrix Analysis and Applications
Date
Tags
Citation
T. G. Kolda, J. R. Mayo. An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs. SIAM Journal on Matrix Analysis and Applications, Vol. 35, No. 4, pp. 1563-1581, 2014. https://doi.org/10.1137/140951758

### Keywords

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

### BibTeX

@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},
}