  This paper introduces the Nystr\"om PCG algorithm for solving a symmetric
positive-definite linear system. The algorithm applies the randomized Nystr\"om
method to form a low-rank approximation of the matrix, which leads to an
efficient preconditioner that can be deployed with the conjugate gradient
algorithm. Theoretical analysis shows that preconditioned system has constant
condition number as soon as the rank of the approximation is comparable with
the number of effective degrees of freedom in the matrix. The paper also
develops adaptive methods that provably achieve similar performance without
knowledge of the effective dimension. Numerical tests show that Nystr\"om PCG
can rapidly solve large linear systems that arise in data analysis problems,
and it surpasses several competing methods from the literature.
