  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.
