Newton's Method for M-Tensor Equations

TL;DR

This paper develops and analyzes a Newton method for solving M-tensor equations, demonstrating its convergence and efficiency through theoretical proofs and numerical experiments.

Contribution

The paper introduces a Newton method tailored for M-tensor equations with positive and nonnegative constant terms, proving convergence and demonstrating practical efficiency.

Findings

The proposed Newton method converges globally and quadratically for equations with positive constant terms.

The method is extended to handle nonnegative constant terms with proven convergence.

Numerical experiments confirm the method's efficiency in solving M-tensor equations.

Abstract

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.