Computing Extreme Eigenvalues of Large Scale Hankel Tensors

Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of m th order n dimensional Hankel tensors, where n is large. Owing to the fast Fo...

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Bibliographic Details
Published in:Journal of scientific computing 2016-08, Vol.68 (2), p.716-738
Main Authors: Chen, Yannan, Qi, Liqun, Wang, Qun
Format: Article
Language:English
Subjects:
Algorithms
Computational Mathematics and Numerical Analysis
Eigenvalues
Eigenvectors
Fast Fourier transformations
Iterative methods
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Optimization
Search methods
Spectrum analysis
Tensors
Theoretical
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Summary:Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of m th order n dimensional Hankel tensors, where n is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about O ( m n log ( m n ) ) . Using the Cayley transform, we obtain an effective curvilinear search scheme. Then, we show that every limiting point of iterates generated by the new algorithm is an eigen-pair of Hankel tensors. Without the assumption of a second-order sufficient condition, we analyze the linear convergence rate of iterate sequence by the Kurdyka–Łojasiewicz property. Finally, numerical experiments for Hankel tensors, whose dimension may up to one million, are reported to show the efficiency of the proposed curvilinear search method.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-015-0155-8