Universality for the Conjugate Gradient and MINRES Algorithms on Sample Covariance Matrices

We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. prove central limit theorem norms the residual vectors that are produced by conjugate gradient and MINRES algorithms when applied to wide class sample covariance matrices satisfying some standard moment conditions. The proof involves establishing four so-called spectral measure, implying, in particular, universality matrix Lanczos iteration. then implies an almost-deterministic iteration count iterative question.