Restarted Arnoldi [@Sorensen_92], so while GMRESDR solves the linear system, the $nev$ Ritz pairs also converge to the smallest magnitude eigenpairs. This elegant solution transfers also to the Hermitian case, where GMRESDR becomes equivalent to thick restarted Lanczos [@TRLAN; @AS_YS_KW_98].

For Hermitian systems, however, the GMRESDR approach presents three disadvantages. First, it is based on GMRES which is much more expensive per step than CG. Second, restarting GMRES every $m$ steps impairs the optimal convergence of CG. Third, restarting also impairs the convergence of unrestarted Lanczos to the $nev$ required eigenpairs. In the context of this paper, the latter is an important disadvantage, because the eigenspaces may not be obtained at the accuracy required for deflation of other systems. In that case, we would like to incrementally improve on the approximate eigenspace during the solution of subsequent linear systems. This is performed in [@Giraud_incremental], but only for GMRESDR. In Section \[sec:ourmethod\] we present a way to compute eigenpairs from the [*unrestarted*]{} CG. One of the components of our method is similar to a method developed independently by Wang et al. [@Wang_Sturler_Paulino]. As we show later, our method is not only cheaper but by making the appropriate restarting choices it yields 