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A power-like method for finding the spectral radius of a weakly irreducible nonnegative symmetric tensor

Author

Listed:
  • Xueli Bai

    (Guangdong University of Foreign Studies)

  • Dong-Hui Li

    (South China Normal University)

  • Lei Wu

    (Jiangxi Normal University)

  • Jiefeng Xu

    (South China Normal University)

Abstract

The Perron–Frobenius theorem says that the spectral radius of a weakly irreducible nonnegative tensor is the unique positive eigenvalue corresponding to a positive eigenvector. With this fact in mind, the purpose of this paper is to find the spectral radius and its corresponding positive eigenvector of a weakly irreducible nonnegative symmetric tensor. By transforming the eigenvalue problem into an equivalent problem of minimizing a concave function on a closed convex set, we derive a simpler and cheaper iterative method called power-like method, which is well-defined. Furthermore, we show that both sequences of the eigenvalue estimates and the eigenvector evaluations generated by the power-like method Q-linearly converge to the spectral radius and its corresponding eigenvector, respectively. To accelerate the method, we introduce a line search technique. The improved method retains the same convergence property as the original version. Plentiful numerical results show that the improved method performs quite well.

Suggested Citation

  • Xueli Bai & Dong-Hui Li & Lei Wu & Jiefeng Xu, 2024. "A power-like method for finding the spectral radius of a weakly irreducible nonnegative symmetric tensor," Computational Optimization and Applications, Springer, vol. 89(3), pages 895-926, December.
    • Handle: RePEc:spr:coopap:v:89:y:2024:i:3:d:10.1007_s10589-024-00601-8
    DOI: 10.1007/s10589-024-00601-8
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    References listed on IDEAS

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    2. Gaohang Yu & Zefeng Yu & Yi Xu & Yisheng Song & Yi Zhou, 2016. "An adaptive gradient method for computing generalized tensor eigenpairs," Computational Optimization and Applications, Springer, vol. 65(3), pages 781-797, December.
    3. Qin Ni & Liqun Qi, 2015. "A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map," Journal of Global Optimization, Springer, vol. 61(4), pages 627-641, April.
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