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We prove that the number of iterations required to solve a random positivedefinite linear system with the conjugate gradient algorithm is almostdeterministic for large matrices. We treat the case of Wishart matrices $W =XX^*$ where $X$ is $n \times m$ and $n/m \sim d$ for $0 < d < 1$. Precisely, weprove that for most choices of error tolerance, as the matrix increases insize, the probability that the iteration count deviates from an explicitdeterministic value tends to zero. In addition, for a fixed iteration count, weshow that the norm of the error vector and the norm of the residual convergeexponentially fast in probability, converge in mean and converge almost surely.

The conjugate gradient algorithm on well-conditioned Wishart matrices is almost deterministic