Solving Multi-linear Systems with $$\mathcal {M}$$ M -Tensors

Abstract

This paper is concerned with solving some structured multi-linear systems, especially focusing on the equations whose coefficient tensors are $$\mathcal {M}$$M-tensors, or called $$\mathcal {M}$$M-equations for short. We prove that a nonsingular $$\mathcal {M}$$M-equation with a positive right-hand side always has a unique positive solution. Several iterative algorithms are proposed for solving multi-linear nonsingular $$\mathcal {M}$$M-equations, generalizing the classical iterative methods and the Newton method for linear systems. Furthermore, we apply the $$\mathcal {M}$$M-equations to some nonlinear differential equations and the inverse iteration for spectral radii of nonnegative tensors.