Universality for eigenvalue algorithms on sample covariance matrices

Percy Deift, Thomas Trogdon

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)2835-2862
Number of pages28
JournalSIAM Journal on Numerical Analysis
Issue number6
StatePublished - 2017


  • Eigenvalue computation
  • Random matrix theory
  • Universality

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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