Inexact Newton Method for M-Tensor Equations

Abstract

We first investigate properties of M-tensor equations. In particular, we show that if the constant term of the equation is nonnegative, then finding a nonnegative solution of the equation can be done by finding a positive solution of a lower dimensional M-tensor equation. We then propose an inexact Newton method to find a positive solution to the lower dimensional equation and establish its global convergence. We also show that the convergence rate of the method is quadratic. At last, we do numerical experiments to test the proposed Newton method. The results show that the proposed Newton method has a very good numerical performance.

Newton’s Method for M-Tensor Equations

“…We first test the performance of Algorithm 2.4 (Newton's Method denoted by 'NM'). In order to test the effectiveness of the proposed method, we compare the Newton method with Inexact Newton Method (denoted by 'INM') proposed in [14]. We take the parameter of NM be ǫ = 0.1, σ = 0.1 and ρ = 0.5.…”

“…Taking limits as k → ∞ with k ∈ K in both sides of the last inequality, we get Aū m−1 = 0. Since A is a strong M-tensor, from Theorem 2.3 in [14], we get a contradiction.…”

“…Our purpose is to find one nonnegative or positive solution of the equation. By Theorem 2.6 in [14], a nonnegative solution of (1.2) has zero elements if and only if A is reducible with respect to some I ⊆ I 0 , where…”

“…Recently, Bai, He, Ling and Zhou [2] proposed a nonnegativity preserving algorithm to solve M-Teq with b ≥ 0. Li, Guan and Wang [18] proposed a monotone iterative method to solve the M-Teq with arbitrary b. Li, Guan and Xu [14] proposed an inexact Newton method with b > 0 and extended the method to solving the M-Teq with b ≥ 0.…”

“…Our purpose is to find a nonnegative solution of the equation with b ≥ 0. As we know in [14], finding a nonnegative solution of the M-tensor equation can be done by finding a positive solution of a lower dimensional M-tensor equation with nonnegative constant term. It is noting that the constant term of that lower dimensional equation is still not guaranteed to be positive.…”