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

## TL;DR

This paper demonstrates that for large well-conditioned Wishart matrices, the conjugate gradient algorithm's iteration count becomes nearly deterministic, with error and residual norms converging rapidly in probability and almost surely.

## Contribution

It establishes that the iteration count for conjugate gradient on large Wishart matrices is almost deterministic, providing explicit convergence results.

## Key findings

- Iteration count concentrates around a deterministic value
- Error and residual norms decay exponentially fast
- Convergence occurs in probability, mean, and almost surely

## 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 $n \times m$ and $n/m \sim d$ for $0 < d < 1$. 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.