Finding the nearest minimum

The abrupt transition from one parabola to the next therefore happens very close to the midpoints between the $k_\mu$. Therefore it is possible to use the quadratic form to make an efficient search routine for the nearest state to a given $k$: $f(k_1)$ and $f(k_2)$ can be used to solve for the current $k_\mu$ and $\epsilon_\mu$, given that $k_1$ and $k_2$ are nearest to the same $k_\mu$. This can be iterated until $k_\mu$ settles to some accuracy. I will not present detailed results, however the search seems to require a few (less than 10) tension evaluations per state found. Essentially, this is the Newton-Rapheson [161] scheme applied to the derivative $f'(k)$. This method is similar to the `improved PWDM' of Li [135], who claims 2-3 evaluations per state found (if high accuracy is needed). In the Boundary Integral Method (BIM or BEM) literature, a recent innovation named the `Multiple Reciprocity BEM' has arisen [101,100], which seems very similar (it seems to be application of `power iterations' [81]).

The accuracy of the norm matrix $G$ need not be very high for location of tension minima (a fractional error of $O(1)$ is even acceptable). Therefore estimation by a few interior points is fine (see Section 5.4). But the following question arises: Would an inaccurate norm create false fluctuations in the tension which would be mistaken for minima? The answer turns out to be no. This is because the wavefunction associated with the maximum (generalized) eigenvalue $\lambda_1$, being a scaling eigenfunction, is actually very stable (this was noted in [91]) as $k$ is changed through $k_\mu$, until the abrupt transition to the next state happens. Therefore the inaccuracy makes a jump only at these transitions, not around $k_\mu$, so no spurious minima are created.

In the next chapter we will see that special properties of a certain weighting function $w{({\mathbf s})}$ allow a vastly superior method, which bypasses the need to `hunt' for states at all.