Some DFT issues

[timholy]

I'm working on a PR to fix these (which will also generalize FFT to non-1 indices), but I may get interrupted for a couple of days so I thought I'd at least document some problems I noticed just from browsing the current code.

First, at least rfft fails for anything other than StridedArrays:

julia> a = randn(10)
10-element Array{Float64,1}:
  0.0243352
 -1.70058  
  2.23913  
 -0.0130746
 -1.10423  
 -0.259103 
  0.871204 
 -0.799567 
 -0.0846184
 -0.676859 

julia> b = view(a, 1:8)
8-element SubArray{Float64,1,Array{Float64,1},Tuple{UnitRange{Int64}},true}:
  0.0243352
 -1.70058  
  2.23913  
 -0.0130746
 -1.10423  
 -0.259103 
  0.871204 
 -0.799567 

julia> rfft(b)
5-element Array{Complex{Float64},1}:
 -0.741877+0.0im     
 -0.446848-0.904787im
  -4.19023+1.14704im 
   2.70397+1.83107im 
   4.80277+0.0im     

julia> b = view(a, [1:8;])
8-element SubArray{Float64,1,Array{Float64,1},Tuple{Array{Int64,1}},false}:
  0.0243352
 -1.70058  
  2.23913  
 -0.0130746
 -1.10423  
 -0.259103 
  0.871204 
 -0.799567 

julia> rfft(b)
ERROR: MethodError: no method matching plan_rfft(::SubArray{Float64,1,Array{Float64,1},Tuple{Array{Int64,1}},false}, ::UnitRange{Int64})
Closest candidates are:
  plan_rfft{N}(::Union{Base.ReshapedArray{Float32,N,A<:DenseArray,MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N}}},DenseArray{Float32,N},SubArray{Float32,N,A<:Union{Base.ReshapedArray{T,N,A<:DenseArray,MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N}}},DenseArray},I<:Tuple{Vararg{Union{Base.AbstractCartesianIndex,Colon,Int64,Range{Int64}},N}},L}}, ::Any; flags, timelimit) at fft/FFTW.jl:636
  plan_rfft{N}(::Union{Base.ReshapedArray{Float64,N,A<:DenseArray,MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N}}},DenseArray{Float64,N},SubArray{Float64,N,A<:Union{Base.ReshapedArray{T,N,A<:DenseArray,MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N}}},DenseArray},I<:Tuple{Vararg{Union{Base.AbstractCartesianIndex,Colon,Int64,Range{Int64}},N}},L}}, ::Any; flags, timelimit) at fft/FFTW.jl:636
  plan_rfft{T<:Union{Integer,Rational{T<:Integer}}}(::AbstractArray{T<:Union{Integer,Rational},N}, ::Any; kws...) at dft.jl:191
  ...
 in #plan_rfft#7(::Array{Any,1}, ::Function, ::SubArray{Float64,1,Array{Float64,1},Tuple{Array{Int64,1}},false}) at ./dft.jl:38
 in rfft(::SubArray{Float64,1,Array{Float64,1},Tuple{Array{Int64,1}},false}) at ./dft.jl:36

Second, Float16 is an AbstractFloat but is not supported by FFTW, so:

julia> a = randn(Float16, 8)
8-element Array{Float16,1}:
 -1.7412  
  0.38818 
  0.066711
 -0.64893 
  1.3623  
 -0.48633 
 -1.1904  
 -0.023666

julia> fft(a)
ERROR: MethodError: no method matching plan_fft(::Array{Complex{Float16},1}, ::UnitRange{Int64})
Closest candidates are:
  plan_fft{T<:Union{Complex{Float32},Complex{Float64}},N}(::Union{Base.ReshapedArray{T<:Union{Complex{Float32},Complex{Float64}},N,A<:DenseArray,MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N}}},DenseArray{T<:Union{Complex{Float32},Complex{Float64}},N},SubArray{T<:Union{Complex{Float32},Complex{Float64}},N,A<:Union{Base.ReshapedArray{T,N,A<:DenseArray,MI<:Tuple{Vararg{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64},N}}},DenseArray},I<:Tuple{Vararg{Union{Base.AbstractCartesianIndex,Colon,Int64,Range{Int64}},N}},L}}, ::Any; flags, timelimit) at fft/FFTW.jl:585
  plan_fft{T<:Real}(::AbstractArray{T<:Real,N}, ::Any; kws...) at dft.jl:185
  plan_fft{T<:Union{Integer,Rational{T<:Integer}}}(::AbstractArray{Complex{T<:Union{Integer,Rational}},N}, ::Any; kws...) at dft.jl:187
  ...
 in fft(::Array{Float16,1}, ::UnitRange{Int64}) at ./dft.jl:184 (repeats 2 times)

even though

julia> a = round(Int, 10*randn(8))
8-element Array{Int64,1}:
  -8
  -3
   1
 -20
  -2
  15
  -5
   2

julia> fft(a)
8-element Array{Complex{Float64},1}:
    -20.0+0.0im    
 -3.17157+22.2843im
     -6.0-30.0im   
 -8.82843+34.2843im
     -8.0+0.0im    
 -8.82843-34.2843im
     -6.0+30.0im   
 -3.17157-22.2843im

works fine.