The tensor splitting with application to solve multi-linear systems

Abstract In this paper, firstly, we introduce the variant tensor splittings, and present some equivalent conditions for a strong M -tensor based on the tensor splitting. Secondly, the existence and uniqueness conditions of the solution for multi-linear systems are given. Thirdly, we propose some tensor splitting algorithms for solving multi-linear systems with coefficient tensor being a strong M -tensor. As an application, a tensor splitting algorithm for solving the multi-linear model of higher order Markov chains is proposed. Numerical examples are given to demonstrate the efficiency of the proposed algorithms.

[1]  Changjiang Bu,et al.  The inverse, rank and product of tensors , 2014 .

[2]  B. Sturmfels,et al.  The number of eigenvalues of a tensor , 2010, 1004.4953.

[3]  Michael K. Ng,et al.  The uniqueness of multilinear PageRank vectors , 2017, Numer. Linear Algebra Appl..

[4]  Michael K. Ng,et al.  Some bounds for the spectral radius of nonnegative tensors , 2015, Numerische Mathematik.

[5]  Qingzhi Yang,et al.  Further Results for Perron-Frobenius Theorem for Nonnegative Tensors , 2010, SIAM J. Matrix Anal. Appl..

[6]  Kung-Ching Chang,et al.  Perron-Frobenius theorem for nonnegative tensors , 2008 .

[7]  Kung-Ching Chang,et al.  Some variational principles for Z-eigenvalues of nonnegative tensors , 2013 .

[8]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[9]  Yisheng Song,et al.  Spectral Properties of Positively Homogeneous Operators Induced by Higher Order Tensors , 2013, SIAM J. Matrix Anal. Appl..

[10]  Yimin Wei,et al.  Solving Multi-linear Systems with $$\mathcal {M}$$M-Tensors , 2016, J. Sci. Comput..

[11]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[12]  M. Ng,et al.  On the limiting probability distribution of a transition probability tensor , 2014 .

[13]  Michael K. Ng,et al.  Numerical ranges of tensors , 2016 .

[14]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[15]  K. Pearson.  Essentially Positive Tensors , 2010 .

[16]  Liqun Qi,et al.  M-Tensors and Some Applications , 2014, SIAM J. Matrix Anal. Appl..

[17]  A. Raftery.  A model for high-order Markov chains , 1985 .

[18]  L. Qi,et al.  M-tensors and nonsingular M-tensors , 2013, 1307.7333.

[19]  Michael K. Ng,et al.  Finding the Largest Eigenvalue of a Nonnegative Tensor , 2009, SIAM J. Matrix Anal. Appl..

[20]  J. Shao,et al.  On some properties of three different types of triangular blocked tensors , 2016, 1604.08442.

[21]  Wen Li,et al.  On the inverse of a tensor , 2016 .

[22]  J. Shao.  A general product of tensors with applications , 2012, 1212.1535.