An algorithm for computing the spectral radius of nonnegative tensors
- PDF / 370,265 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 78 Downloads / 190 Views
(2019) 38:90
An algorithm for computing the spectral radius of nonnegative tensors Qilong Liu1 · Zhen Chen1 Received: 27 November 2018 / Revised: 20 January 2019 / Accepted: 2 February 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we present an algorithm to find the weakly irreducible normal form of tensors. Based on the weakly irreducible normal form of nonnegative tensors, we present a convergent algorithm for computing the spectral radius of any nonnegative tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radius of any nonnegative tensors. Keywords Irreducible · Weakly irreducible · Weakly irreducible normal form of tensors · Spectral radius Mathematics Subject Classification 15A18 · 15A69 · 65F15
1 Introduction An mth-order n-dimensional real tensor is a multidimensional array of n m elements of the form A = (ai1 i2 ···im ), ai1 i2 ···im ∈ R, i j ∈ [n], j ∈ [m], where [n] = {1, 2, . . . , n} (Qi and Luo 2017). When m = 2, A is an n-by-n matrix. A tensor A is called nonnegative if each entry is nonnegative. Tensors have applications in many areas such as data analysis and mining, signal and image processing, and computational biology (Chang et al. 2013; De Lathauwer et al. 2000; Kolda and Bader 2009; Kolda and Mayo 2010; Ng et al. 2009). We shall denote the set of all real mth-order n-dimensional tensors by R[m,n] , and the set of all mth-order n-dimensional nonnegative tensors by R[m,n] . The mth-order + n-dimensional identity tensor (Yang and Yang 2010), denoted by I = (δi1 i2 ···im ) ∈ R[m,n] , is the tensor with entries,
Communicated by Jinyun Yuan.
B
Zhen Chen [email protected] Qilong Liu [email protected]
1
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, People’s Republic of China
123
90
Page 2 of 14
Q. Liu, Z. Chen
δi1 i2 ···im =
1, 0,
if i 1 = i 2 = · · · = i m , otherwise.
For a tensor A = (ai1 ···im ) ∈ R[m,n] , let |A| ∈ R[m,n] denotes the tensor whose + (i 1 , . . . , i m )-th entry is defined as |ai1 ···im |. For a set Λ, |Λ| denotes the cardinality of Λ. For an n-dimensional vector x = (x1 , x2 , . . . , xn )T , real or complex, we define n-dimensional vectors Axm−1 and x[m−1] , whose i-th component is (Axm−1 )i = aii2 ···im xi2 · · · xim , i 2 ,...,i m ∈[n]
and (x[m−1] )i = xim−1 , respectively. Let A ∈ R[m,n] and λ ∈ C. If there is a nonzero vector x ∈ Cn such that Axm−1 = λx[m−1] ,
then λ is said to be an eigenvalue of A and x is said to be an eigenvector of A corresponding to the eigenvalue λ. If the eigenvector x is real, then the eigenvalue is also real. In this case, λ and x are called an H-eigenvalue and an H-eigenvector of A, respectively. This definition was first introduced and studied by Qi (2005) and Lim (2005), respectively. The largest modulus of the eigenvalues of A, denoted by ρ(A), is called the spectral radius of A, and the set of all eigenvalues of A is said to be the spectrum of A, denoted as σ
Data Loading...
