Some Superlinear Convergence Results for the Conjugate Gradient Method

The conjugate gradient method is applied to a linear equation, with a uniformly positive definite coefficient operator, on a real, separable Hilbert space. We observe that, if the coefficient operator is a compact perturbation of the identity, then the method is R-superlinearly convergent. Furthermore, if the perturbation belongs to a certain trace class, then a rate of superconvergence can be given. The results can be considered as infinite-dimensional analogs of the fact that the method has finite termination property on a finite-dimensional space.