Universality for the Toda algorithm to compute the largest eigenvalue of a random matrix

TL;DR

This paper proves that the time it takes for the Toda algorithm to find the largest eigenvalue of large random matrices exhibits universal statistical behavior, relying on recent advances in random matrix theory.

Contribution

It establishes universality for the halting time fluctuations of the Toda algorithm in computing the largest eigenvalue of random matrices, connecting algorithmic performance to eigenvalue statistics.

Findings

Halting time fluctuations are universal across matrix ensembles.

Relies on eigenvalue delocalization, rigidity, and edge universality.

Provides rigorous proof linking random matrix theory to algorithmic complexity.

Abstract

We prove universality for the fluctuations of the halting time for the Toda algorithm to compute the largest eigenvalue of real symmetric and complex Hermitian matrices. The proof relies on recent results on the statistics of the eigenvalues and eigenvectors of random matrices (such as delocalization, rigidity and edge universality) in a crucial way.