Choice of $\epsilon$ for truncation

The criterion for choosing $\epsilon$ is that it be close to, but larger than, the machine precision[161] $\epsilon_{mach} \approx 10^{-16}$. I have found that pushing $\epsilon$ too close to $\epsilon_{mach}$ (or smaller) does affect the quality of the returned eigenvector ${\mathbf x}^{(1)}$ from the recipe in the previous section. One simple measure of this is the difference $\delta f \equiv [{\mathbf x}^{(1)}]^{{\mbox{\tiny T}}} F {\mathbf x}^{(1)} - \lambda_1^{-1}$, which should be zero. For $\epsilon = 10^{-16}$ I found $\vert\delta f\vert < 10^{-6}$ while for $\epsilon = 10^{-14}$ it drops to $\vert\delta f\vert < 10^{-9}$. It is unknown how the contamination by other (error) eigenvectors varies with $\epsilon$. However I believe that since the errors mainly occur in the evaluation of small eigenvalue/eigenvector pairs of $F$, any contamination is mainly by harmless (null-space) vectors.

On the other hand, $\epsilon$ also provides an approximate lower bound for the tension that can be reported. Therefore, in order to preserve the full depth of the tension minima, as small an $\epsilon$ as possible should be used. In practice, tension minima rarely are smaller than $10^{-8}$ using RPWs, but with better basis sets they can reach typical values of $10^{-11}$ in systems under investigation.

A choice of $\epsilon = 10^{-14}$ has proved optimal for calculations in this and the next chapter.