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We consider the conjugate gradient algorithm applied to a general class ofspiked sample covariance matrices. The main result of the paper is that thenorms of the error and residual vectors at any finite step concentrate ondeterministic values determined by orthogonal polynomials with respect to adeformed Marchenko--Pastur law. The first-order limits and fluctuations areshown to be universal. Additionally, for the case where the bulk eigenvalueslie in a single interval we show a stronger universality result in that theasymptotic rate of convergence of the conjugate gradient algorithm only dependson the support of the bulk, provided the spikes are well-separated from thebulk. In particular, this shows that the classical condition number bound forthe conjugate gradient algorithm is pessimistic for spiked matrices.

The conjugate gradient algorithm on a general class of spiked covariance matrices