A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

TL;DR

This paper introduces a modified Newton iteration method for efficiently computing nonnegative Z-eigenpairs of nonnegative tensors, with proven local quadratic convergence and ability to find eigenpairs from arbitrary positive vectors, especially for transition probability tensors.

Contribution

The paper presents a novel modified Newton iteration that guarantees convergence to nonnegative eigenpairs of nonnegative tensors, improving robustness and applicability over existing methods.

Findings

Proven local quadratic convergence for positive eigenpairs.

Method can find eigenpairs from any positive starting vector.

Effective for transition probability tensors.

Abstract

We propose a modified Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. When the tensor is irreducible, all nonnegative eigenpairs are known to be positive. We prove local quadratic convergence of the new iteration to any positive eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration. A big advantage of the modified Newton iteration is that it seems capable of finding a nonnegative eigenpair starting with any positive unit vector. Special attention is paid to transition probability tensors.