eigCG(10,50), GMRESDR(55,34), and RecycledCG(50,30) methods, at 830, 1000, and 1100 iterations. For each method and case, smallest eigenvalues correspond to leftmost bars. []{data-label="fig:comparisons"}](SymEvalAccur_EigCG_GMRESDR_recycled){width="\textwidth"}

For comparison, we also report some results from running the GMRESDR directly on the nonsymmetric Dirac operator. This is sometimes reported to be twice as fast as CG on the normal equations because it avoids the squaring of the condition number. However, such results are often obtained from heavier quark masses, where the problem is not as difficult or interesting. In our $12\times12^4$ problem, and with the same accuracy and computational platform, GMRESDR(55,34) took 1000 iterations and 1120 seconds. GMRESDR(85,60) improved convergence to 585 iterations, but its execution time was still 1001 seconds. For low quark masses, as in our case, GMRESDR is not only slower than eigCG, but it has more difficulty computing accurate eigenpairs even with more memory as shown in Figure \[fig:GMRESDR\].

![Accuracy of 10 smallest magnitude eigenvalues computed by GMRESDR(55,34) and GMRESDR(85,60), at 585, 1000, and 2200 iterations on the nonsymmetric operator. GMRESDR(85,60) is not reported at 2200 MV since it reached machine precision much earlier. []{data-label="fig:GMRESDR"}](NSymEvalAccur_GMRESDR_55_85){width="80.00000%"}

Incrementally increasing eigenvector accuracy and number {#sec:incremental}
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When the CG iteration converges before equally accurate 