Finding a Nonnegative Solution to an M-Tensor Equation

Abstract

We are concerned with the tensor equation with an M-tensor, which we call the M-tensor equation. We first derive a necessary and sufficient condition for an M-tensor equation to have nonnegative solutions. We then develop a monotone iterative method to find a nonnegative solution to an Mtensor equation. The method can be regarded as an approximation to Newton's method for solving the equation. At each iteration, we solve a system of linear equations. An advantage of the proposed method is that the coefficient matrices of the linear systems are independent of the iteration. We show that if the initial point is appropriately chosen, then the sequence of iterates generated by the method converges to a nonnegative solution of the M-tensor equation monotonically and linearly. At last, we do numerical experiments to test the proposed methods. The results show the efficiency of the proposed methods.

Inexact Newton Method for M-Tensor Equations

“…There are also a few methods that can solve M-Teq (1.1) without restriction b ∈ R n ++ or that A is an M tensor. Those methods include the splitting method by Li, Guan and Wang [18], and Li, Xie and Xu [15], the alternating projection method by Li, Dai and Gao [17], the alternating iterative methods by Liang, Zheng and Zhao [23] etc.. Related works can also be found in [4,5,16,24,29,30,31,32,33,34].…”

“…However, in many cases, the standard Newton method may fail to work or loss its quadratic convergence property when applied to solve tensor equation (1.1). We refer to [18] for details.…”

“…[18] We solve tensor equation(1.1) where A is a non-symmetric strong M-tensor of order m (m = 3, 4, 5) in the form A = sI − B, and tensor B is nonnegative tensor whose entries are uniformly distributed in (0, 1). The parameter s is set tos = (1 + 0.01) • max i=1,2,...,n(Be m−1 ) i .…”

Fast and flexible preconditioners for solving multilinear systems

“…As an application, a tensor splitting algorithm for solving the multi-linear model of higher-order Markov chains was proposed. Li et al, in [16] firstly derived a necessary and sufficient condition for an M-tensor equation to have nonnegative solutions. Secondly, developed a monotone iterative method to find a nonnegative solution to an Mtensor equation.…”

A nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors

“…However, the state-of-the-art solvers, e.g., HM and QCA, tailored for (1.3) with a positive vector b may not produce a desired nonnegative solution in some situations. In addition, just after this article has been completed, Li, Guan and Wang [12] proposed an algorithm for finding a nonnegative solution of the multilinear system (1.3). It is shown that an increasing sequence is generated by the algorithm in [12] and it converges to a nonnegative solution of the multilinear system (1.3), while our proposed algorithm produces a decreasing sequence.…”