So after having spent a while fixing my implementation of the Fourier Transform, a couple of early results seem to be:
+Using either FFTW or my own Fourier Transform, the run time is remarkably unaffected by the pattern size. I guess the theory tells us this much, but it’s still surprising to see an average run-time of 0.36 across about 7 different pattern sizes!
+Using my own Fourier Transform, the O(n log m) version is always quicker than the O(n log m) version and the larger n/m is, the greater the difference
+Using FFTW, this doesn’t quite follow. A graph of the the O (n log m) version would show a curve, starting and finishing higher (as m varies) than the O(n log n) version, but it does appear to dip below in the middle.
Some more tests will run overnight which should firm up these initial findings. It would seem then that the overhead is with FFTW and not the Fourier Transform per se. Though FFTW does, unsurprisingly, out-performs my code by a very long way, except for when the pattern is very very short. I hope to get to the bottom of this bizarre time increase for very small patterns tomorrow.