Solving tensor E-eigenvalue problem faster

Bozorgmanesh, Hassan; Hajarian, Masoud

doi:10.1016/j.aml.2019.106020

Cited by 7 publications

(3 citation statements)

References 35 publications

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“…Based on Definition 13, there are m − 2 distinct members of every equivalence class, and it is enough for each equivalence class to find one of them [Theorem 5 in [23]]. So the number of all the eigenpairs is given by (m−1) r −1 m−2 , which is determined by the order and rank of the symmetirc tensor.…”

Section: Resultsmentioning

confidence: 99%

The Z-eigenpairs of orthogonally diagonalizable symmetric tensors

Wang¹,

Geng²

2021

Preprint

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In this paper, we focus on a special class of symmetric tensors, which can be orthogonally diagonalizable, and investigate their Z-eigenpairs problem. We show that the eigenpairs can be uniformly expressed using several basic eigenpairs, and the number of all the eigenpairs is uniquely determined by the order and rank of the symmetric tensor. In addition, we exploit the local optimality of each eigenpair by checking the second-order necessary condition.

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Section: Resultsmentioning

confidence: 99%

The Z-eigenpairs of orthogonally diagonalizable symmetric tensors

Wang¹,

Geng²

2021

Preprint

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“…Tensors are becoming increasingly used to describe and solve several problems of documents analysis [7], psychometrics [27], chemometrics [38] and medical engineering [35], for more details see [10,21,34]. Research related to tensors has increased dramatically in recent years [3,4,8,9,11,13,14,20,42]. Solving linear systems in higher dimensions is one of the most important research topics in tensors [2,19,31].…”

Section: Introductionmentioning

confidence: 99%

Solving coupled tensor equations via higher order LSQR methods

Hajarian

2020

Filomat

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Tensors have a wide application in control theory, data mining, chemistry, information sciences, documents analysis and medical engineering. The material here is motivated by the development of the efficient numerical methods for solving the coupled tensor equations (A1*M X *N B1 + C1 *M Y *N D1 = E1, A2 *M X *N B2 + C2 *M Y *N D2 = E2, with Einstein product. We propose the tensor form of the LSQR methods to find the solutions X and Y of the coupled tensor equations. Finally we give some numerical examples to illustrate that our proposed methods are able to accurately and efficiently find the solutions of tensor equations with Einstein product.

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“…Then the M-spectral radius of A is denoted byRecently, tensors with special structures, such as nonnegative tensors, M-tensors and H-tensors, are becoming the focus in recent research [2,24,[28][29][30][31]. Some effective algorithms for finding eigenvalue and the corresponding eigenvector have been implemented [1,24,[32][33][34][35]. For example, Bozorgmanesh et al [32] propose an algorithm that can solve E-eigenvalue problem faster.…”

mentioning

confidence: 99%

Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

Zhang¹,

Sun

Wang

2020

Mathematics

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The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.where (A · xy 2 ) i = ∑ j,k,l∈ [n] a ijkl x j y k y l , (Ax 2 y·) l = ∑ i,j,k∈ [n] a ijkl x i x j y k , then the scalar λ is called an M-eigenvalue of the tensor A, and x and y are called left and right M-eigenvectors of A associated with the M-eigenvalue. Then the M-spectral radius of A is denoted byRecently, tensors with special structures, such as nonnegative tensors, M-tensors and H-tensors, are becoming the focus in recent research [2,24,[28][29][30][31]. Some effective algorithms for finding eigenvalue and the corresponding eigenvector have been implemented [1,24,[32][33][34][35]. For example, Bozorgmanesh et al. [32] propose an algorithm that can solve E-eigenvalue problem faster. However, it is very difficult for these algorithms to compute all M-eigenvalues or E-eigenvalues. Thus, some researchers turned to investigating eigenvalue inclusion sets [4,7,[36][37][38][39][40][41]. Particularly, some bounds for the minimum H-eigenvalue of nonsingular M-tensors have been proposed [2,28,30,42,43]. Ding et al. [24] introduced a structured partially symmetric tensor named elasticity M-tensors and established important properties of elasticity M-tensors and nonsingular elasticity M-tensors.

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Solving tensor E-eigenvalue problem faster

Cited by 7 publications

References 35 publications

The Z-eigenpairs of orthogonally diagonalizable symmetric tensors

The Z-eigenpairs of orthogonally diagonalizable symmetric tensors

Solving coupled tensor equations via higher order LSQR methods

Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

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