Summary of the Numerical Results

The numerical results obtained show that the improvement in convergence speed obtained with the center series depends first and foremost on whether or not the Fourier series involved are absolutely and uniformly convergent. This is as one would expect on general grounds. For example, in the case of the parabolic wave, which is continuous and differentiable, with Fourier coefficients $a_{k}$ that behave as $1/k^{3}$ for large values of $k$, so that the Fourier series is absolutely and uniformly convergent, the improvement of the ratio of the average number of added terms varies from about $1.2$ to about $2.8$, depending on the precision lever required. In the case of the triangular wave, which is continuous but not differentiable, with Fourier coefficients that behave as $1/k^{2}$ for large values of $k$, so that the Fourier series is still absolutely and uniformly convergent, the improvement of the same ratio is larger, varying from about $1.9$ to about $11.6$.

In the cases of the discontinuous functions with Fourier coefficients that behave as $1/k$ for large values of $k$, so that the Fourier series is point-wise convergent almost everywhere but not absolutely or uniformly convergent, the ratio varies from about $21$ to about $6600$, being therefore very significant, specially for the higher levels of precision. In the case of the functions associated to the slow-converging series, with Fourier coefficients that behave as $1/\sqrt{k}$ for large values of $k$, so that the Fourier series is also point-wise convergent almost everywhere but not absolutely or uniformly convergent, the same ratio varies from about $1.0\times 10^{4}$ to about $4.8\times 10^{6}$, where we consider only the three lower levels of precision, which were the only ones for which we were able to run the Fourier programs within a feasible amount of time.

Table 1: Coefficients of the exponential fits to the ratios of the average numbers of added terms, as functions o the target precision.

Function	$a_{0}$	$a_{1}$	$a_{2}$
One-cycle sawtooth wave	$0.6287$	$1.1530$	$0.7668$
Its conjugate function	$0.6259$	$1.1533$	$0.9987$
Standard square wave	$0.6378$	$1.1553$	$0.6206$
Its conjugate function	$0.6388$	$1.1547$	$0.7636$
Two-cycle sawtooth wave	$0.6446$	$1.1526$	$0.8442$
Its conjugate function	$0.6399$	$1.1530$	$1.0229$
Triangular wave	$0.5126$	$0.3863$	$0.2718$
Its conjugate function	$0.4710$	$0.3958$	$0.3994$
Shifted square wave	$0.6380$	$1.1552$	$0.6105$
Its conjugate function	$0.6389$	$1.1547$	$0.7585$
Parabolic wave	$0.4836$	$0.2016$	$0.3594$
Its conjugate function	$0.5442$	$0.1941$	$0.2319$
First slow-converging wave	N/A	N/A	N/A
Its conjugate function	$0.9695$	$3.0811$	$223.53$
Second slow-converging wave	$1.0041$	$3.0761$	$241.09$
Its conjugate function	$0.9693$	$3.0812$	$223.75$

In short, if the original DP Fourier series is already absolutely and uniformly convergent, as in the first two cases above, then the improvement obtained is modest. Otherwise, since the center series is always absolutely and uniformly convergent, the improvement obtained is large. On a finer scale, there is always some improvement when the first-order center series is used, and further improvement with the second-order one. This is a consequence of the fact that every factorization of the dominant singularities adds a factor of $k$ to the denominator of the coefficients of the center series. We see therefore that the greater improvements come when the coefficients go from the $1/k$ or $1/k^{(1/2)}$ behavior in the Fourier case to the $1/k^{2}$ or $1/k^{(3/2)}$ behavior in the center series case, so that the original series is not absolutely or uniformly convergent, while the corresponding center series is.

In all cases except the very slow-converging ones the ratios of the average number of added terms can be fit very well by increasing exponentials, as functions of the logarithms of the target precision $\varepsilon$,

$$r = a_{0}\,{\rm e}^{a_{1}\xi}+a_{2},$$ (1)

where $r$ is the ratio for the average number of added terms and $\xi=-\log_{10}(\varepsilon)$. The coefficients $a_{0}$, $a_{1}$ and $a_{2}$ are always positive and of the order of one, and the correlation coefficients of the fits are very close to one, while the mean square error is of the order of about $1\%$ or less. The coefficients obtained for the parameters of these fits can be seen in Table 1. As one can see there, the fits were also worked out for some of the very slow-converging cases. In these cases, since we had only three data points available and three constants to fit, the fits were, of course, exact ones. The only possible justification for our producing these fits is, of course, that they work so well on the other cases. Using these fits one can estimate that, under the circumstances in which we executed the runs, the Fortran runs for the slow-converging cases, for a precision of $10^{-8}$, would take something like a few tens of thousands of years to complete. This is to be compared to the $30$ seconds or so that it took to run the center series for these functions, with the same precision.

We see therefore that with the utilization of the coefficients in Table 1 the formula in Equation 1 above can be used as a comparative performance predictor for higher levels of precision. This should work well for the average number of added terms, and may also give some idea of the running time, if one takes into consideration the hardware involved. However, it will probably produce an underestimation of the running time in the case of the higher precision cases, since it is observed that in these cases the standard Fourier runs tend to take a disproportionally large amount of time to complete. This is probably related to technical optimization issues and may depend strongly on the compiler used and on the details of the hardware.