Finding a Nonnegative Solution to an M-Tensor Equation

TL;DR

This paper establishes conditions for nonnegative solutions to M-tensor equations and introduces a monotone iterative method that converges linearly, with numerical experiments demonstrating its efficiency.

Contribution

It provides a necessary and sufficient condition for nonnegative solutions and develops a new iterative method with proven convergence for M-tensor equations.

Findings

The method converges monotonically and linearly.

Linear systems are solved at each iteration with fixed coefficient matrices.

Numerical experiments confirm the method's efficiency.

Abstract

We are concerned with the tensor equation with an M-tensor or Z-tensor, which we call the M- tensor equation or Z-tensor equation respectively. We derive a necessary and sufficient condition for a Z (or M)-tensor equation to have nonnegative solutions. We then develop a monotone iterative method to find a nonnegative solution to an M-tensor 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.