eigenvectors can be obtained, these eigenvectors cannot deflate effectively the CG for the next right hand side. Deflating with the eigenvectors obtained in the example of Figure \[fig:eigCG\_conv\] yielded 10% faster convergence in the next linear system. As we mentioned earlier, using the resulting $V$ as a spectral preconditioner is much more effective, resulting in 50% faster convergence in the same example. Still, we would like to avoid this expense on every step of CG, and more importantly to obtain more eigenvectors of $A$ for more effective deflation.

A simple outer scheme can be designed that calls eigCG for $s_1 < s$ right hand sides and accumulates the resulting $nev$ approximate Ritz vectors from each run into a larger set, $U$.

If we assume that $U$ contains converged eigenvectors, the Ritz vectors produced in $V$ during the solution of the next linear system will be in the orthogonal complement of span$(U)$, as $x_0$ is deflated of $U$. If some of the vectors in $U$ are not converged enough, the eigCG produces $V$ with directions not only in new eigenvectors but also in directions that complement the unconverged Ritz vectors in $U$. Thus, the accuracy of $U$ incrementally improves up to 