A power-like method for finding the spectral radius of a weakly irreducible nonnegative symmetric tensor

Abstract

The Perron–Frobenius theorem says that the spectral radius of a weakly irreducible nonnegative tensor is the unique positive eigenvalue corresponding to a positive eigenvector. With this fact in mind, the purpose of this paper is to find the spectral radius and its corresponding positive eigenvector of a weakly irreducible nonnegative symmetric tensor. By transforming the eigenvalue problem into an equivalent problem of minimizing a concave function on a closed convex set, we derive a simpler and cheaper iterative method called power-like method, which is well-defined. Furthermore, we show that both sequences of the eigenvalue estimates and the eigenvector evaluations generated by the power-like method Q-linearly converge to the spectral radius and its corresponding eigenvector, respectively. To accelerate the method, we introduce a line search technique. The improved method retains the same convergence property as the original version. Plentiful numerical results show that the improved method performs quite well.