Add an option to use plan_r2r of FFTW

[ptiede]

For FFTW it provides a built in Hartley transform with, its r1r function which may save time and memory. Did we want to use this instead?

[ptiede]

https://juliamath.github.io/FFTW.jl/dev/fft/#FFTW.r2r

[kazuakiyama]

@ptiede Yes this was the first thing I checked. I tested FFTW's r2r before pushing this to the public repository.

I would note that FFTW's FFTW_DHT mode for r2r transform does NOT provide the exact multi-dimensional transform for the two or larger dimensions. See the note below in FFTW's documentation:

If FFTW_DHT is specified for multiple dimensions of a multi-dimensional transform, FFTW computes the separable product of 1d DHTs along each dimension. Unfortunately, this is not quite the same thing as a true multi-dimensional DHT; you can compute the latter, if necessary, with at most rank-1 post-processing passes [see e.g. H. Hao and R. N. Bracewell, Proc. IEEE 75, 264–266 (1987)].
https://www.fftw.org/doc/The-Discrete-Hartley-Transform.html

It turns out that FFTW's implementation of multidimensional transform is not only inaccurate but also slow. I tested with 2 and 3 dimensional cases, and found that pure Julia implementation (basically using real(ffted array) .+ imag(ffted array) seems to be at least a factor of two times faster. This is simply because FFTW does a separable transform.

Having said that it is possible that we can force to use FFTW's r2r for one dimensional transform --- indeed for 1 dimensional case FFTW is slightly faster. I will think about that. At least I confirmed that my implementation is consistent with FFTW for 1 dimensional transform.

[ptiede]

Ahh makes sense! This looks good then!

[kazuakiyama]

OK I have added an option to use r2r (ndim(A) > 1 will show a warn message). For 1d array, it will use FFTW's r2r transform as it is faster.

[kazuakiyama]

I will close this issue as I have officially released the first version (which is now being registered to the Julia package repo). Feel free to open the issue again if needed.