Therefore we do not expect any spurious eigenvalues to show up while solving one linear system.

The structure of $T_m$ after restart is reminiscent of the thick restarted Lanczos (TRLAN) [@TRLAN]. However, after the first restart, the residual of TRLAN (equivalently of ARPACK [@arpack] or of GMRESDR [@RBMorgan_2002]) is the residual produced by $2nev$ steps of [*some other Lanczos process with a different initial vector. Our residuals continue to be the Lanczos vectors of the original CG/Lanczos process.*]{} Therefore, the convergence of the CG is unaffected. Also, eigenvalue convergence in eigCG is expected to be different from TRLAN or GMRESDR.

Recently, the recycled MINRES (RMINRES) algorithm was developed independently in [@Wang_Sturler_Paulino]. It is based on the same technique of reusing some of the MINRES residuals in a basis $V$. The critical difference with eigCG is that RMINRES restarts as in TRLAN by keeping only the $nev$ harmonic Ritz vectors closest to zero and not the previous directions (i.e., $k=0$ in (\[eq:restartedV\])). As a result, [*eigenvalues converge only to a very low accuracy and then stagnate*]{}. This is acceptable in [@Wang_Sturler_Paulino] because their application involves systems of slightly varying matrices which cannot be deflated exactly by each other’s eigenvectors. Moreover, RMINRES 