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.
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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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C.-H. Guo was supported in part by an NSERC Discovery Grant, W.-W. Lin was supported in part by the Ministry of Science and Technology, the National Center for Theoretical Sciences, and ST Yau Center at Chiao-Da in Taiwan, and C.-S. Liu was supported in part by the Ministry of Science and Technology in Taiwan.
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Guo, CH., Lin, WW. & Liu, CS. A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor. Numer Algor 80, 595–616 (2019). https://doi.org/10.1007/s11075-018-0498-y
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DOI: https://doi.org/10.1007/s11075-018-0498-y
Keywords
- Nonnegative tensor
- Transition probability tensor
- Nonnegative Z-eigenpair
- Modified Newton iteration
- Quadratic convergence