Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 109, 22 pages      arXiv:1605.06438
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

Smoothed Analysis for the Conjugate Gradient Algorithm

Govind Menon a and Thomas Trogdon b
a) Division of Applied Mathematics, Brown University, 182 George St., Providence, RI 02912, USA
b) Department of Mathematics, University of California, Irvine, Rowland Hall, Irvine, CA, 92697-3875, USA

Received May 23, 2016, in final form October 31, 2016; Published online November 06, 2016

The purpose of this paper is to establish bounds on the rate of convergence of the conjugate gradient algorithm when the underlying matrix is a random positive definite perturbation of a deterministic positive definite matrix. We estimate all finite moments of a natural halting time when the random perturbation is drawn from the Laguerre unitary ensemble in a critical scaling regime explored in Deift et al. (2016). These estimates are used to analyze the expected iteration count in the framework of smoothed analysis, introduced by Spielman and Teng (2001). The rigorous results are compared with numerical calculations in several cases of interest.

Key words: conjugate gradient algorithm; Wishart ensemble; Laguerre unitary ensemble; smoothed analysis.

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