The Conjugate Gradient Algorithm on Well-Conditioned Wishart Matrices is Almost Deterministic

Percy Deift, Thomas Trogdon

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that the number of iterations required to solve a random positive definite linear system with the conjugate gradient algorithm is almost deterministic for large matrices. We treat the case of Wishart matrices W = XX* where X is n x m and n/m ̰ d for0 < d < 1. Precisely, we prove that for most choices of error tolerance, as the matrix increases in size, the probability that the iteration count deviates from an explicit deterministic value tends to zero. In addition, for a fixed iteration count, we show that the norm of the error vector and the norm of the residual converge exponentially fast in probability, converge in mean, and converge almost surely.

Original languageEnglish (US)
JournalQuarterly of Applied Mathematics
Volume79
Issue number1
DOIs
StatePublished - Jul 9 2020

ASJC Scopus subject areas

  • Applied Mathematics

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