A positivity preserving inverse iteration for finding the Perron pair of an irreducible nonnegative third order tensor

Ching-Sung Liu National University of Kaohsiung Chun-Hua Guo University of Regina Wen-Wei Lin National Chiao Tung University

Numerical Linear Algebra mathscidoc:1605.31001

We propose an inverse iterative method for computing the Perron pair of an irreducible nonnegative third order tensor. The method involves the selection of a parameter $\theta_k$ in the $k$th iteration. For every positive starting vector, the method converges quadratically and is positivity preserving in the sense that the vectors approximating the Perron vector are strictly positive in each iteration. It is also shown that $\theta_k=1$ near convergence. The computational work for each iteration of the proposed method is less than four times (three times if the tensor is symmetric in modes two and three, and twice if we also take the parameter to be $1$ directly) that for each iteration of the Ng--Qi--Zhou algorithm, which is linearly convergent for essentially positive tensors.
inverse iteration, nonnegative tensor, $M$-matrix, nonnegative matrix, positivity preserving, quadratic convergence, Perron vector, Perron root
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@inproceedings{ching-sunga,
  title={A positivity preserving inverse iteration for finding the Perron pair of an irreducible nonnegative third order tensor},
  author={Ching-Sung Liu, Chun-Hua Guo, and Wen-Wei Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160526115626172445039},
}
Ching-Sung Liu, Chun-Hua Guo, and Wen-Wei Lin. A positivity preserving inverse iteration for finding the Perron pair of an irreducible nonnegative third order tensor. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160526115626172445039.
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