# Newton correction methods for computing real eigenpairs of symmetric tensors

## TL;DR

This paper introduces a Newton-based iterative method for efficiently computing real eigenpairs of symmetric tensors, demonstrating quadratic convergence and superior performance over existing methods.

## Contribution

It proposes a novel Newton correction method with proven convergence properties and empirical advantages for finding real eigenpairs of symmetric tensors.

## Key findings

- Method converges quadratically near eigenpairs.
- Finds more eigenpairs than previous methods.
- Typically finds all real eigenpairs with multiple initializations.

## Abstract

Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based method to compute real eigenpairs of symmetric tensors. We derive sufficient conditions for a real eigenpair to be a stable fixed point for our method, and prove that given a sufficiently close initial guess, the convergence rate is quadratic. Empirically, our method converges to a significantly larger number of eigenpairs compared to previously proposed iterative methods, and with enough random initializations typically finds all real eigenpairs. In particular, for a generic symmetric tensor, the sufficient conditions for local convergence of our Newton-based method hold simultaneously for all its real eigenpairs.