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Numerical Algorithms

, Volume 80, Issue 2, pp 595–616 | Cite as

A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

  • Chun-Hua GuoEmail author
  • Wen-Wei Lin
  • Ching-Sung Liu
Original Paper
  • 37 Downloads

Abstract

We propose a modified Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. When the tensor is irreducible, all nonnegative eigenpairs are known to be positive. We prove local quadratic convergence of the new iteration to any positive eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration. A big advantage of the modified Newton iteration is that it seems capable of finding a nonnegative eigenpair starting with any positive unit vector. Special attention is paid to transition probability tensors.

Keywords

Nonnegative tensor Transition probability tensor Nonnegative Z-eigenpair Modified Newton iteration Quadratic convergence 

Mathematics Subject Classification (2010)

65F15 65F50 

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Notes

Acknowledgments

The authors thank the two referees for their helpful comments. This work was started when C.-H. Guo visited ST Yau Center at Chiao-Da in Taiwan in late 2015; he thanks the Center for its hospitality.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of ReginaReginaCanada
  2. 2.Department of Applied MathematicsNational Chiao Tung UniversityHsinchuTaiwan
  3. 3.Department of Applied MathematicsNational University of KaohsiungKaohsiungTaiwan

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