Fast verified computation for positive solutions to M-tensor multi-linear systems and Perron vectors of a kind of weakly irreducible nonnegative tensors

Two fast numerical algorithms are proposed for computing interval vectors containing positive solutions to M-tensor multi-linear systems. The first algorithm involves only two tensor–vector multiplications. The second algorithm is iterative one, and generally gives interval vectors narrower than those by the first algorithm. We also develop two verification algorithms for Perron vectors of a kind of weakly irreducible nonnegative tensors, which we call slightly positive tensors. The first and second algorithms have properties similar to those of the two algorithms for the solutions to the M-tensor systems. We clarify relations between slightly positive tensors and other tensor classes. Numerical results show efficiency of the algorithms.