A tensor Alternating Anderson–Richardson method for solving multilinear systems with M-tensors

Abstract

It is well-known that a multilinear system with a nonsingular $M$-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson–Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson–Richardson method, finally, we periodically employ the tensor Anderson–Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.