per step decreases with $m$ and for large enough $m > 10nev$ it approaches $O(4 N nev)$ flops per step. This is the same as the computation of $2nev$ Ritz vectors from the full basis of unrestarted Lanczos. Finally, the extra computations are fully parallelizable: First, the computation $V (QZ)$ incurs no synchronization and can be performed in a cache efficient way with level 3 BLAS routines. Second, the accumulation of the dot products $V^HAz$ can be delayed until the first dot product during the next CG iteration to avoid an extra synchronization.

Convergence and comparison with other methods {#sec:convergence}
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In exact arithmetic, the vectors in $V$ remain orthonormal (or $P$-orthonormal in case of preconditioning) as linear combinations of the Lanczos vectors. In floating point arithmetic, the Lanczos property guarantees $z_j \perp V$ and the numerical accuracy of the $T_m$ coefficients until some eigenpairs converge to square root of machine precision ($\sqrt{\epsilon_{mach}}\|A\|$). After that, spurious eigenvalues start to appear in $T_{m}$ but without compromising the accuracy of the correct ones. Converging but unwanted eigenpairs, lying in the high end of the spectrum, pose little stability threat because they are purged at every restart (step 11.7). Nevertheless, we should choose 