A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor
- PDF / 529,900 Bytes
- 23 Pages / 439.37 x 666.142 pts Page_size
- 103 Downloads / 189 Views
A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor Yi-Yong Li1 · Qing-Zhi Yang1 · Xi He1
Received: 17 November 2015 / Revised: 20 March 2016 / Accepted: 14 June 2016 / Published online: 19 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported. Keywords Nonnegative tensor · Indirectly positive tensors · Linear convergence · Perturbation · Complexity Mathematics Subject Classification 74B99 · 15A18 · 15A69
Yi-Yong Li’s work was supported by the Ph.D. Candidate Research Innovation Fund of Nankai University. Qing-Zhi Yang’s work was supported by the National Natural Science Foundation of China (No. 11271206) and Doctoral Fund of Chinese Ministry of Education (No. 20120031110024).
B
Qing-Zhi Yang [email protected] Yi-Yong Li [email protected] Xi He [email protected]
1
School of Mathematical Sciences and Lab of Pure Mathematics and Combinatic, Nankai University, Tianjin 300071, China
123
4
Y.-Y. Li et al.
1 Introduction Eigenvalue problems of higher order tensors have become a more and more important topic. In theory, Chang et al. generalized the Perron–Frobenius Theorem from nonnegative matrices to nonnegative tensors in [1]. Y. Yang and Q. Yang extended their results in [2,3]. The latest result on the Perron–Frobenius Theorem is that the eigenvalues with modulus ρ(A) have the same geometric multiplicity in [4]. Some other results of nonnegative tensors were established in [5–12]. What is more, Ng, Qi, and Zhou proposed the NQZ method for finding spectral radius of a nonnegative irreducible tensor in [13]. Pearson obtained that the NQZ method would converge if the tensor with even order is essentially positive in [14]. In [15] Chang, Pearson and Zhang proved the convergence of the NQZ method for primitive tensors with any nonzero nonnegative initial vector. Zhang and Qi gave the linear convergence of the NQZ method for essentially positive tensors in [16]. Hu, Huang, and Qi [17] established the global R-linear convergence of the modified version of the NQZ method for nonnegative weakly irreducible tensors which were introduced by Friedland, Gaubert, and Han in [18]. Chen, Qi, Yang, et al showed an inexact power-type algorithm for finding spectral radius of nonnegative tensors in [19]. In this pa
Data Loading...
