An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs

Journal Article · Thu Dec 11 04:00:00 EST 2014 · SIAM Journal on Matrix Analysis and Applications

DOI:https://doi.org/10.1137/140951758· OSTI ID:1126939

Kolda, Tamara G.; Mayo, Jackson R.

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 $${\mathscr{A}}$$ and $$\boldsymbol{\mathscr{B}}$$, the goal is to find $$\lambda \in \mathbb{R}$$ and $$\mathbf{x} \in \mathbb{R}^{n}, \mathbf{x} \neq 0$$ such that $${\mathscr{A}}\mathbf{x}^{m-1} = \lambda {\mathscr{B}}\mathbf{x}^{m-1}$$. Different choices for $${\mathscr{B}}$$ 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.

• Research Organization: Sandia National Laboratories (SNL-CA), Livermore, CA (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• DOE Contract Number: AC04-94AL85000
• OSTI ID: 1126939
• Report Number(s): SAND2014--0027J; 493218
• Journal Information: SIAM Journal on Matrix Analysis and Applications, Journal Name: SIAM Journal on Matrix Analysis and Applications Journal Issue: 4 Vol. 35; ISSN 0895-4798
• Publisher: SIAM
• Country of Publication: United States
• Language: English

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97 MATHEMATICS AND COMPUTING