Tensor Methods for Solving Symmetric $${\mathcal {M}}$$M-tensor Systems

Tensor systems involving tensor-vector products (or polynomial systems) are considered. We solve these tensor systems, especially focusing on symmetric $${\mathcal {M}}$$M-tensor systems, by some tensor methods. A new tensor method is proposed based on the rank-1 approximation of the coefficient tensor. Numerical examples show that the tensor methods could be more efficient than the Newton method for some $${\mathcal {M}}$$M-tensor systems.

[1]  Ngai Wong,et al.  Symmetric tensor decomposition by an iterative eigendecomposition algorithm , 2014, J. Comput. Appl. Math..

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

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

[4]  Kung-Ching Chang,et al.  On eigenvalue problems of real symmetric tensors , 2009 .

[5]  Bobby Schnabel,et al.  On the Performance of Tensor Methods for Solving Ill-Conditioned Problems , 2007, SIAM J. Sci. Comput..

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

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

[8]  Dan Feng,et al.  Tensor Methods for Equality Constrained Optimization , 1996, SIAM J. Optim..

[9]  Yi-min Wei,et al.  ℋ-tensors and nonsingular ℋ-tensors , 2016 .

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

[11]  R. Schnabel,et al.  Tensor Methods for Nonlinear Equations. , 1984 .

[12]  Naihua Xiu,et al.  The sparsest solutions to Z-tensor complementarity problems , 2015, Optim. Lett..

[13]  Ali Bouaricha,et al.  Tensor Methods for Large, Sparse Unconstrained Optimization , 1996, SIAM J. Optim..

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

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

[16]  Dan Feng,et al.  Tensor-GMRES Method for Large Systems of Nonlinear Equations , 1997, SIAM J. Optim..

[17]  Brett W. Bader.  Tensor-Krylov Methods for Solving Large-Scale Systems of Nonlinear Equations , 2005, SIAM J. Numer. Anal..

[18]  Yimin Wei,et al.  Theory and Computation of Tensors: Multi-Dimensional Arrays , 2016 .

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

[20]  Bobby Schnabel,et al.  Algorithm 768: TENSOLVE: a software package for solving systems of nonlinear equations and nonlinear least-squares problems using tensor methods , 1997, TOMS.

[21]  Trond Steihaug,et al.  Rate of convergence of higher order methods , 2013 .

[22]  Dan Feng,et al.  Local convergence analysis of tensor methods for nonlinear equations , 1993, Math. Program..

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

[24]  Gene H. Golub,et al.  Symmetric Tensors and Symmetric Tensor Rank , 2008, SIAM J. Matrix Anal. Appl..

[25]  Bobby Schnabel,et al.  Curvilinear Linesearch for Tensor Methods , 2003, SIAM J. Sci. Comput..