Newton’s Method for M-Tensor Equations

We are concerned with the tensor equations whose coefficient tensor is an M-tensor. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend the method to solve the equation with a nonnegative constant term and establish its convergence. At last, we do numerical experiments to test the proposed methods. The results show that the proposed method is quite efficient.

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

[2]  Changfeng Ma,et al.  A Levenberg-Marquardt method for solving semi-symmetric tensor equations , 2018, J. Comput. Appl. Math..

[3]  Guanglu Zhou,et al.  A nonnegativity preserving algorithm for multilinear systems with nonsingular ℳ ${\mathcal M}$ -tensors , 2020, Numer. Algorithms.

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

[5]  Yimin Wei,et al.  Numerical solution to a linear equation with tensor product structure , 2017, Numer. Linear Algebra Appl..

[6]  Lixing Han,et al.  A homotopy method for solving multilinear systems with M-tensors , 2017, Appl. Math. Lett..

[7]  Huan Gao,et al.  Alternating projection method for a class of tensor equations , 2019, J. Comput. Appl. Math..

[8]  Chen Ling,et al.  Generalized tensor equations with leading structured tensors , 2018, Appl. Math. Comput..

[9]  Tan Zhang,et al.  Primitivity, the Convergence of the NQZ Method, and the Largest Eigenvalue for Nonnegative Tensors , 2011, SIAM Journal on Matrix Analysis and Applications.

[10]  Wen Li,et al.  The tensor splitting with application to solve multi-linear systems , 2018, J. Comput. Appl. Math..

[11]  Dong-Hui Li,et al.  Finding a Nonnegative Solution to an M-Tensor Equation , 2018, 1811.11343.

[12]  Jie Xu,et al.  Inexact Newton Method for M-Tensor Equations , 2020, 2007.13324.

[13]  L. Qi,et al.  Tensor Analysis: Spectral Theory and Special Tensors , 2017 .

[14]  Dong-Hui Li,et al.  Splitting methods for tensor equations , 2017, Numer. Linear Algebra Appl..

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

[16]  Yi-min Wei,et al.  Solving Multilinear Systems with M-Tensors , 2016 .

[17]  Yi-min Wei,et al.  A fast algorithm for solving circulant tensor systems , 2017 .

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

[19]  Chen Ling,et al.  A Globally and Quadratically Convergent Algorithm for Solving Multilinear Systems with $${{\mathcal {M}}}$$M-tensors , 2018, J. Sci. Comput..

[20]  Chen Ling,et al.  An efficient nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors , 2018, 1811.09917.

[21]  Liqun Qi,et al.  Tensor Eigenvalues and Their Applications , 2018 .

[22]  F. Facchinei,et al.  Beyond Monotonicity in Regularization Methods for Nonlinear Complementarity Problems , 1999 .

[23]  Chen Ling,et al.  A Globally and Quadratically Convergent Algorithm for Solving Multilinear Systems with M -tensors , 2017 .

[24]  M. Ng,et al.  Solving sparse non-negative tensor equations: algorithms and applications , 2015 .

[25]  Xiao-Qing Jin,et al.  Tensor Methods for Solving Symmetric $${\mathcal {M}}$$M-tensor Systems , 2017, J. Sci. Comput..

[26]  Na Li,et al.  Solving Multilinear Systems via Tensor Inversion , 2013, SIAM J. Matrix Anal. Appl..

[27]  Yimin Wei,et al.  Neural networks based approach solving multi-linear systems with M-tensors , 2019, Neurocomputing.

[28]  Daniel Kressner,et al.  Krylov Subspace Methods for Linear Systems with Tensor Product Structure , 2010, SIAM J. Matrix Anal. Appl..

[29]  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..

[30]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.