The scaling method in action

In Figs. 6.5 and 6.6 I show that the $k_\mu$ (and tensions $t_\mu$) found coincide with tension minima found by the sweep method of the previous chapter. The $k$ agreement is well within the sweep method error due to the finite rounding of the minima. Because the scaling method does not just compute the tension of a single state, it is able to distinguish close eigenvalues that cannot be distinguished using the sweep method (using the same basis set). How far this can be pushed has not yet been investigated. The finite tension minima are highly correlated between the methods, and are therefore a function of the limitation of the RPW basis set chosen.

The remarkable improvement in efficiency over the sweep methods of the previous chapter (or any other known method!) comes from the fact that $\sim (k{\mathsf{L}})^{d-1}$ billiard eigenstates are computed using a single diagonalization. The effort per state therefore scales like $\sim k^{2(d-1)}$ compared to $\sim k^{3(d-1)}$ for sweep methods. In $d=2$ this gives an improvement in efficiency of $\sim 10^3$ at $k\sim10^3$ if only moderate accuracy is required. More specifically, the basis set size is $N \approx N_{sc}$, and about $N/10$ usable states are returned per diagonalization (moderate accuracy, $\vert t_\mu\vert<10^{-5}$), or about $N/50$ (very high accuracy, $\vert t_\mu\vert<10^{-10}$). This assumes that the basis set is adequate for the particular billiard shape. Errors will be further analysed in Section 6.3.

Now I give a practical example. A very high energy state of the stadium billiard (odd-odd symmetry class) is shown in Fig. 6.7, with $k=1000.00275996$, and tension $8 \times 10^{-12}$. The error in $k$ is about $\pm 1$ in the last digit. At this $k$, corresponding to about the 142000'th odd-odd eigenstate, the basis size is $N = 950$. It takes about 10 seconds to fill $F$ and $dF/dk$, another 50 seconds for diagonalization, returning about 16-100 usable states (depending on required tension accuracy), and about 2 seconds per state to evaluate the wavefunction and its normal gradient on the perimeter (2000 sample points around the quarter-perimeter of the stadium). It takes a further 300 seconds to generate spatial plots of the type of Fig. 6.7 with resolution 1000*500 samples. Because the plotting algorithm generates plots of the scaling eigenfunctions simultaneously, it does not take much longer to evaluate more than one such plot (in each case $\sim 10^9$ sin or cos evaluations are required). This was performed on a Compaq XP1000 workstation, with alpha processor 21264, 667MHz, using C++ and LAPACK [5]. It is clear that obtaining spatial plots is the limiting factor above (due to the number of basis function evaluations required), rather than solving for the eigenstate $k_\mu$ and coefficients themselves!