CG). The experiments are run on a Wilson Fermion lattice of size $12\times 12^4$ with periodic boundary conditions and quark mass equal to the critical mass. We use odd-even preconditioning (which yields a matrix of half the size) and focus on the symmetric normal equations. We run the experiments in Matlab 7.5, on a Mac Pro, dual processor, dual-core, 3 GHz Intel Xeon. We test eigCG(10,50), GMRESDR(55,34), and RecycledCG(50,30) so that all methods have similar memory requirements. We plot the residual norms of the 10 smallest eigenvalues computed by the three methods at 830, 1000, and 1100 matrix-vector products. GMRESDR is effective at computing the lowest three eigenvalues but the accuracy of the rest does not seem to improve with further iterations. RecycledCG cannot improve the low accuracy obtained early during the iteration. Our method not only computes more accurate and more eigenvalues than both other methods, it also continues to improve them with more iterations. Moreover, GMRESDR took 1168 iterations and 1505 seconds to solve the linear system to accuracy $10^{-8}$, while eigCG took 1010 iterations and 410 seconds. Thus, eigCG was 3.7 times faster, while also computing a much better eigenspace.

![Accuracy of 10 smallest eigenvalues computed by 