that represents the spectrum of a typical Wilson fermion matrix with light quark mass. The eigCG(1,3) holds only three vectors in $V$, which cannot capture the information well, and stagnates. Keeping as few as three Ritz vectors (eigCG(3,9)) improves the stagnation point dramatically, while with eight Ritz vectors we were able to reach accuracy of 1e-12. The figure only shows convergence until step 1540 which is where the linear system converged. It is remarkable, however, that until it reaches the stagnation point eigCG converges at the rate of unrestarted, fully reorthogonalized Lanczos. We have observed the same property for all $8$ smallest eigenvalues, whose convergence is depicted in the right graph in Figure \[fig:eigCG\_conv\]. A theoretical analysis of this surprising behavior will be the focus of our future research.

![Left graph: eigCG convergence with various values of ($nev,m$), matching unrestarted Lanczos. Right graph: convergence of 8 smallest eigenpairs of eigCG(8,20).[]{data-label="fig:eigCG_conv"}](history_plot "fig:"){width=".5\textwidth"} ![Left graph: eigCG convergence with various values of ($nev,m$), matching unrestarted Lanczos. Right graph: convergence of 8 smallest eigenpairs of eigCG(8,20).[]{data-label="fig:eigCG_conv"}](history_nevs "fig:"){width=".5\textwidth"}

Figure \[fig:comparisons\] compares the effectiveness of eigCG in approximating many eigenpairs to those of the competing GMRESDR and RecycledCG methods (the latter implementing the RMINRES ideas on 