Exact and Inexact Iterative Methods for Finding the Largest Eigenpair of a Weakly Irreducible Nonnegative Tensor

Abstract

In tensor computations, tensor–vector multiplication is one of the main computational costs. We recently studied algorithms with wider applicability and more computational potential for computing the largest eigenpair of a weakly irreducible nonnegative mth-order tensor ${\mathscr {A}}$, called higher-order Noda iteration (HONI). This method is an eigenvalue solver which uses an inner-outer scheme. The outer iteration is the update of the approximate eigenpair(s), while in the inner iteration a multilinear system has to be solved, often iteratively. For the inner iteration, we also provide a Newton-type method to solve multilinear systems, and prove that the algorithm converges to the unique solution of multilinear systems and the convergence rate is quadratic. HONI has superior performance in terms of fast convergence and positivity preserving property, and its main advantage is to use simple recursive relations to compute the approximate eigenvalue, which means that no additional tensor–vector multiplication is required. Moreover, we devise a practical relaxation criterion based on our theoretical results to improve the efficiency and practicality of HONI, called inexact HONI, and further explain the relationship between HONI and Newton–Noda iteration. Numerical experiments are provided to support the theoretical results.