On the infinite-dimensional QR algorithm

TL;DR

This paper introduces an infinite-dimensional QR algorithm that can compute spectra and eigenvectors of certain operators with guaranteed convergence rates and error control, advancing spectral analysis in infinite-dimensional spaces.

Contribution

It generalizes the QR algorithm to infinite-dimensional operators and proves its effectiveness with convergence guarantees and practical demonstrations.

Findings

Algorithm recovers extremal spectra and eigenvectors with convergence guarantees.

Demonstrates the algorithm's effectiveness on various examples.

In some cases, the algorithm outperforms theoretical predictions.

Abstract

Spectral computations of infinite-dimensional operators are notoriously difficult, yet ubiquitous in the sciences. Indeed, despite more than half a century of research, it is still unknown which classes of operators allow for computation of spectra and eigenvectors with convergence rates and error control. Recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation, and no convergence rates nor error control is possible. This begs the question: which classes of operators allow for computations with convergence rates and error control? In this paper we address this basic question, and the algorithm used is an infinite-dimensional version of the QR algorithm. Indeed, we generalise the QR algorithm to infinite-dimensional operators. We prove that not only is the algorithm executable on a finite machine, but one can also recover the extremal parts of the spectrum and corresponding eigenvectors, with convergence rates and error control. This allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture. The algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches (that are notorious for providing false solutions).We also find that in some cases the IQR algorithm performs better than predicted by theory and make conjectures for future study.