Genericfft

[daanhb]

This pull request makes the generic FFT and DCT routines in FastTransforms available for all AbstractFloats. Some changes to the codeflow are described in #62. An earlier issue was #34.

The code is written such that the more efficient plans of FFTW are still returned when T is one of the fftwNumber types. There is currently no check for this, which is dangerous. I suppose we could check on the type of the plan returned by plan_fft, but I'm not sure how to check which method of fft ends up being invoked. I would not be surprised if this pull request breaks things. The implementation remains a lot slower than FFTW of course, and it is also less accurate.

My goal was to make it work for DoubleFloats.jl types, such as Double64. This works now:

julia> using FFTW, FastTransforms, DoubleFloats

julia> N = 10000; z = rand(N); z2 = Double64.(z); z3 = BigFloat.(z);

julia> v = fft(z); v2 = fft(z2); v3 = fft(z3);

julia> maximum(abs.(v-v2))
2.0438760310276306e-13

julia> maximum(abs.(v2-v3))
4.673372870301638733235131886202073426336931382556029192698665209605575831732632e-26

julia> @time fft(z);
  0.000491 seconds (58 allocations: 315.766 KiB)

julia> @time fft(z2);
  0.097739 seconds (548.98 k allocations: 33.332 MiB, 3.70% gc time)

julia> @time fft(z3);
  1.978109 seconds (17.87 M allocations: 952.290 MiB, 42.06% gc time)

So, for this example, Double64 is faster than BigFloat by a factor of about 20. It is slower than FFTW by a factor of about 200.

[MikaelSlevinsky]

Awesome! It looks like this might work with Intervals from ValidatedNumerics.jl as well.

@dlfivefifty, do these changes work with the ecosystem?

[MikaelSlevinsky]

I think the build failures mainly point to too broad of support. We can drop 0.6 now. The Appveyor builds also should have 0.7/1.0 builds instead of 0.6 and keep latest on allowable failure.

[dlfivefifty]

It looks fine to me.

[daanhb]

I'm afraid I did not get this right the first time. The code accidentally hijacks some of the FFTW plans, resulting in significant slowdowns elsewhere. I'm looking at it and will add some tests to detect similar situations in the future.

[daanhb]

I've made a new pull request: #66.