A nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors

Abstract

This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular $\mathcal{M}$-tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative decreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method. Particularly, when some elements of the right-hand side vector are zeros, the proposed algorithm works well while existing state-of-the-art solvers may not produce a nonnegative solution.