An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor

Liu, Yongjun; Zhou, Guanglu; Ibrahim, Nur Fadhilah

doi:10.1016/j.cam.2010.06.002

Cited by 103 publications

(98 citation statements)

References 19 publications

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“…where the third equality follows from (7). If x(J l ) = 0, then (ρ(T ), x(J l )) is a nonnegative eigenpair of tensor T J l ; and if x(J l ) = 0, then we have…”

Section: L}} Is Irreducible Andmentioning

confidence: 98%

Strictly nonnegative tensors and nonnegative tensor partition

Huang

2013

Sci. China Math.

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In this paper, we introduce a new class of nonnegative tensors -strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T , there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

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“…where the third equality follows from (7). If x(J l ) = 0, then (ρ(T ), x(J l )) is a nonnegative eigenpair of tensor T J l ; and if x(J l ) = 0, then we have…”

Section: L}} Is Irreducible Andmentioning

confidence: 98%

Strictly nonnegative tensors and nonnegative tensor partition

Huang

2013

Sci. China Math.

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“…We propose an algorithm for testing the positive definiteness of such a multivariate form. It should be pointed out that the class of multivariate forms studied in [18] is a special case of our model. We do not need the assumption that the diagonal entries are positive.…”

Section: A = Si − B Where S > 0 and B Is A Nonnegative Matrix For Wmentioning

confidence: 99%

“…Based on this observation, we next propose an iterative method for testing the the positive definiteness of f (x) with a Z-tensor. For this purpose, we first design an algorithm for computing the spectral radius of the nonnegative tensor C. This algorithm is a modified version of [18]. The substantial difference is that the modified version is always convergent for any nonnegative tensor, but the algorithm in [18] may be not convergent for some reducible nonnegative tensors.…”

Section: Applications Of M -Tensorsmentioning

confidence: 99%

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$M$-Tensors and Some Applications

Zhang

Qi²,

Zhou³

2014

SIAM J. Matrix Anal. & Appl.

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244

155

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Abstract. We introduce M -tensors. This concept extends the concept of M -matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M -tensors must be Ztensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M -tensor must be nonnegative. A symmetric M -tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an Mtensor is its smallest H + -eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M -tensor if and only if all its H + -eigenvalues are nonnegative. Some further spectral properties of M -tensors are given. We also introduce strong M -tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M -tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form. 1. Introduction. Tensors are increasingly ubiquitous in various areas of applied, computational, and industrial mathematics and have wide applications in data analysis and mining, information science, signal/image processing, and computational biology as well [5,9,15,17]. A tensor can be regarded as a higher-order generalization of a matrix, which takes the form Key words. M -tensors, H

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“…where i j D 1; 2; ; n for j 2 OEm [1][2][3][4][5]. It is obvious that a vector is an order 1 tensor and a matrix is an order 2 tensor.…”

Section: Introductionmentioning

confidence: 99%

Bounds for the Z-eigenpair of general nonnegative tensors

Liu

2016

Open Mathematics

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Abstract:In this paper, we consider the Z-eigenpair of a tensor. A lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Furthermore, upper bounds of Z-spectral radius of nonnegative tensors and general tensors are given. The proposed bounds improve some existing ones. Numerical examples are reported to show the effectiveness of the proposed bounds.

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An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor

Abstract: a b s t r a c tIn this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor. We prove this method converges for any irreducible nonnegative tensor. We also apply this method to study the positive definiteness of a multivariate form.

Cited by 103 publications

References 19 publications

Strictly nonnegative tensors and nonnegative tensor partition

Strictly nonnegative tensors and nonnegative tensor partition

$M$-Tensors and Some Applications

Bounds for the Z-eigenpair of general nonnegative tensors

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