# A homotopy method for solving multilinear systems with M-tensors

## TL;DR

This paper introduces a homotopy method designed to efficiently find the unique positive solution of multilinear systems involving nonsingular M-tensors, with proven convergence and demonstrated numerical effectiveness.

## Contribution

The paper presents a novel homotopy approach specifically tailored for solving multilinear systems with M-tensors, including convergence analysis and numerical validation.

## Key findings

- The method converges to the unique positive solution.
- Numerical experiments confirm the effectiveness of the approach.
- The approach is applicable to systems arising in various applications.

## Abstract

Multilinear systems of equations arise in various applications, such as numerical partial differential equations, data mining, and tensor complementarity problems. In this paper, we propose a homotopy method for finding the unique positive solution to a multilinear system with a nonsingular M-tensor and a positive right side vector. We analyze the method and prove its convergence to the desired solution. We report some numerical results based on an implementation of the proposed method using a prediction-correction approach for path following.