Solving Multi-linear Systems with $$\mathcal {M}$$ M -Tensors

Ding, Weiyang; Wei, Yimin

doi:10.1007/s10915-015-0156-7

Cited by 182 publications

(150 citation statements)

References 33 publications

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“…Example Consider the homogenous system

\frac{normald {boldx}^{[m - 1]}}{normald t} = scriptA {boldx}^{m - 1}

with

- scriptA

being a nonsingular

scriptℳ

‐tensor . We construct a 3rd‐order nonsingular

scriptℳ

‐tensor

- scriptA = s scriptI - scriptℬ

{double-struckR}^{2 \times 2 \times 2}

as follows.…”

Section: Applicationsmentioning

confidence: 99%

Tensor logarithmic norm and its applications

Ding

Hou

Wei

2016

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Summary Matrix logarithmic norm is an important quantity, which characterize the stability of linear dynamical systems. We propose the logarithmic norms for tensors and tensor pairs, and extend some classical results from the matrix case. Moreover, the explicit forms of several tensor logarithmic norms and semi‐norms are also derived. Employing the tensor logarithmic norms, we bound the real parts of all the eigenvalues of a complex tensor and study the stability of a class of nonlinear dynamical systems. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Example Consider the homogenous system

\frac{normald {boldx}^{[m - 1]}}{normald t} = scriptA {boldx}^{m - 1}

with

- scriptA

being a nonsingular

scriptℳ

‐tensor . We construct a 3rd‐order nonsingular

scriptℳ

‐tensor

- scriptA = s scriptI - scriptℬ

{double-struckR}^{2 \times 2 \times 2}

as follows.…”

Section: Applicationsmentioning

confidence: 99%

Tensor logarithmic norm and its applications

Ding

Hou

Wei

2016

Numerical Linear Algebra App

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“…A tensor of order (also called mode) m , is an m −way array whose entries are addressed via m indices, for example, vectors are tensors of order one, matrices are tensors of order two. Tensors have wide applications in the practical fields such as medical engineering, documents analysis, higher‐order web link analysis, n ‐person noncooperative game, partial differential equation, and so on. The study in tensor has attracted people's much attention.…”

Section: Introductionmentioning

confidence: 99%

“…Li and Ng proposed some iterative methods for solving a set of sparse nonnegative tensor equations arising from data mining applications and obtained the linear convergence of the Jacobi and Gauss‐Seidel methods under suitable conditions. Ding and Wei extended some basic iterative methods including the Jacobi and Gauss‐Seidel methods for the solution of linear equations to tensor equations. Convergence of those methods were well‐established.…”

Section: Introductionmentioning

confidence: 99%

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Splitting methods for tensor equations

Xie

2017

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Summary The Jacobi, Gauss‐Seidel and successive over‐relaxation methods are well‐known basic iterative methods for solving system of linear equations. In this paper, we extend those basic methods to solve the tensor equation scriptAxm−1−b=0, where scriptA is an mth‐order n−dimensional symmetric tensor and b is an n‐dimensional vector. Under appropriate conditions, we show that the proposed methods are globally convergent and locally r‐linearly convergent. Taking into account the special structure of the Newton method for the problem, we propose a Newton‐Gauss‐Seidel method, which is expected to converge faster than the above methods. The proposed methods can be extended to solve a general symmetric tensor equations. Our preliminary numerical results show the effectiveness of the proposed methods.

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“…Aromatic polyimide (PI) is one class of high performance materials with excellent mechanical properties and thermal stabilities and it has been extensively investigated and applied in aviation, electronics, protection and filtrations, especially in high temperature areas [1][2][3][4][5][6]. In last decades, many researchers have prepared PI nanofibers (nonwovens, short nanofibers, aligned nanofiber belts and single nanofibers) by electrospinning and applied them in composites and battery separators [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%