How long does it take to compute the eigenvalues of a random symmetric matrix?

Summary

We present the results of an empirical study of the performance of the QR algorithm (with and without shifts) and the Toda algorithm on random symmetric matrices. The random matrices are chosen from six ensembles, four of which lie in the Wigner class. For all three algorithms, we observe a form of universality for the deflation time statistics for random matrices within the Wigner class. For these ensembles, the empirical distribution of a normalized deflation time is found to collapse onto a curve that depends only on the algorithm, but not on the matrix size or deflation tolerance provided the matrix size is large enough. For the QR algorithm with the Wilkinson shift, the observed universality is even stronger and includes certain non-Wigner ensembles. Our experiments also provide a quantitative statistical picture of the accelerated convergence with shifts.

1. Introduction

We present the results of a statistical study of the performance of the QR and

Toda eigenvalue algorithms on random symmetric matrices. Our work is mainly

inspired by progress in quantifying the “probability of difficulty” and “typical

behavior” for several numerical algorithms [Demmel 1988; Goldstine and von

Neumann 1951]. This approach has led to a deeper understanding of the efficacy

of fundamental numerical algorithms such as Gaussian elimination and the

simplex method [Rudelson and Vershynin 2008; Sankar et al. 2006; Smale

1983; Tao and Vu 2010]. It has also stimulated new ideas in random matrix

theory [Dumitriu and Edelman 2002; Edelman 1988; Edelman and Sutton 2007].

Testing eigenvalue algorithms with random input continues this effort. In related

work [Pfrang 2011], we have also studied the performance of a version of the

matrix sign algorithm. However, these results are of a different character, and apart from some theoretical observations, we do not present any experimental results for this algorithm (see [Pfrang 2011] for more information). Our study is empirical—a study of the eigenvalue problem from the viewpoint of complexity theory is presented in [Armentano 2014].

1.1. Algorithms and ensembles.It is natural to study the QR algorithm because of its elegance and fundamental practical importance. But in fact all the algorithms we study are linked by a common framework. In each case, an initial matrix L₀ is diagonalized via a sequence of isospectral iterates L_m. The gist of the framework is that the L_m correspond exactly to the flow of a completely integrable Hamiltonian system evaluated at integer times.