tries to identify and maintain directions that tend to repeat across Krylov subspaces of different linear systems. To be effective, the basis $V$ must be kept orthogonal to these vectors, thus increasing the expense of its MINRES iteration. Our eigCG focuses on getting the eigenvectors accurately which can later be deflated inexpensively with init-CG.

The use of Rayleigh Ritz and thick restart guarantee monotonic convergence of the $nev$ Ritz values from $V$. Beyond that, it is not obvious why eigCG should converge to any eigenpairs, let alone with a convergence rate identical to that of unrestarted Lanczos! With thick restarting alone (as in RMINRES) the important information that is discarded cannot be reintroduced in $V$ as future residuals of the unrestarted CG are orthogonal to it. By using the restarting of eq. (\[eq:restartedV\]), with modest $nev$ values, almost all Lanczos information regarding the $nev$ eigenpairs is kept in condensed form in $V$. Then, the new CG residuals do in fact represent the steepest descent for our Ritz vectors, and eigCG behaves like unrestarted Lanczos.

The left graph of Figure \[fig:eigCG\_conv\] shows residual convergence for the smallest eigenpair under various eigCG($nev,m$) runs on a matrix of size $12\times 8^4 = 49152$ 