GLOBALLY CONVERGENT INVERSE ITERATION ALGORITHM FOR FINDING THE LARGEST EIGENVALUE OF A NONNEGATIVE WEAKLY IRREDUCIBLE TENSOR

In this paper, we propose an inverse iteration algorithm for finding the largest eigenvalue of a nonnegative weakly irreducible tensor. The positive property of approximate eigenvector is preserved at each iteration for any initial positive vector, as we all know, this is crucial during the computation. The proposed algorithm involves a multilinear equation at each iteration, which can be solved by the Newton method. An important part of the paper consists of proving that the algorithm is globally convergent. Numerical examples are reported to illustrate the proposed algorithm is efficient and promising. We show an application of this algorithm to determine the positive definiteness of a weakly irreducible 2-tensor, which is done on this 2-tensor directly. The numerical results indicated that it is capable of testing the positive definiteness of weakly irreducible 2-tensors.