in QCD [@deForcrand:Progress], but effective deflation methods have only appeared recently. In section \[sec:Background\] we review deflation methods that either precompute the required eigenpairs or use a restarted method to compute eigenvectors while solving linear systems. Both approaches, however, are unnecessarily expensive.

In this paper we present an algorithm that computes eigenvalue and eigenvector approximations of a Hermitian operator by reusing information from the [*unrestarted*]{} CG method. This is achieved by keeping a search space that includes current eigenvector approximations and only the last few CG iteration vectors. The crucial step is how we restart this search space to keep computations tractable. The CG iteration is completely unaffected. Our experiments show that eigenvector convergence is similar to unrestarted Lanczos; an impressive achievement yet to be understood theoretically.

Our motivating application is the computation of nucleon-nucleon scattering lengths with all to all propagators, where, for a time discretization between 64 and 128, 1500-3000 right hand sides must be solved. Our algorithms are equally applicable to the more classic problem of computing nucleon form factors using sequential propagators where 120 right hand sides are required. In a first phase, we solve 24 right hand sides with our method. Unconverged eigenvectors from one 