Abstract
We prove a universal limit theorem for the halting time, or iteration count, of the power/inverse power methods and the QR eigenvalue algorithm. Specifically, we analyze the required number of iterations to compute extreme eigenvalues of random, positive definite sample covariance matrices to within a prescribed tolerance. The universality theorem provides a complexity estimate for the algorithms which, in this random setting, holds with high probability. The method of proof relies on recent results on the statistics of the eigenvalues and eigenvectors of random sample covariance matrices (i.e., delocalization, rigidity, and edge universality).
Original language | English (US) |
---|---|
Pages (from-to) | 2835-2862 |
Number of pages | 28 |
Journal | SIAM Journal on Numerical Analysis |
Volume | 55 |
Issue number | 6 |
DOIs | |
State | Published - 2017 |
Keywords
- Eigenvalue computation
- Random matrix theory
- Universality
ASJC Scopus subject areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics