A QR Algorithm for Symmetric Tensors

TL;DR

This paper introduces the QRST algorithm, extending the matrix QR algorithm to symmetric tensors, enabling efficient eigenpair computation with improved convergence and the ability to find multiple eigenpairs.

Contribution

The paper presents a novel QR-based algorithm for symmetric tensors, including a shifted version and a heuristic for finding multiple eigenpairs, advancing tensor eigenvalue computation methods.

Findings

QRST effectively finds stable and unstable eigenpairs.

Shifted QRST converges faster than previous methods.

Permutation heuristic enables multiple eigenpair discovery.

Abstract

We extend the celebrated QR algorithm for matrices to symmetric tensors. The algorithm, named QR algorithm for symmetric tensors (QRST), exhibits similar properties to its matrix version, and allows the derivation of a shifted implementation with faster convergence. We further show that multiple tensor eigenpairs can be found from a local permutation heuristic which is effectively a tensor similarity transform, resulting in the permuted version of QRST called PQRST. Examples demonstrate the remarkable effectiveness of the proposed schemes for finding stable and unstable eigenpairs not found by previous tensor power methods.