Avoid non-fftwNumber Chebyshev U transform, which breaks auto-diff (#44)

* Restrict usage of Chebyshev U transform to fftwNumber

* Update chebyshev.jl

* Update test_chebyshev.jl

* Update test_chebyshev.jl

Author: dlfivefifty

src/classical/chebyshev.jl

@@ -100,7 +100,8 @@ Jacobi(C::ChebyshevU{T}) where T = Jacobi(one(T)/2,one(T)/2)
 factorize(L::SubQuasiArray{T,2,<:ChebyshevT,<:Tuple{<:Inclusion,<:OneTo}}) where T =
     TransformFactorization(grid(L), plan_chebyshevtransform(Array{T}(undef, size(L,2))))
 
-factorize(L::SubQuasiArray{T,2,<:ChebyshevU,<:Tuple{<:Inclusion,<:OneTo}}) where T =
+# TODO: extend plan_chebyshevutransform
+factorize(L::SubQuasiArray{T,2,<:ChebyshevU,<:Tuple{<:Inclusion,<:OneTo}}) where T<:FastTransforms.fftwNumber =
     TransformFactorization(grid(L), plan_chebyshevutransform(Array{T}(undef, size(L,2))))
 
 
@@ -171,6 +172,7 @@ function \(w_A::WeightedChebyshevT, w_B::WeightedChebyshevU)
     _BandedMatrix(Vcat(Fill(one(T)/2, 1, ∞), Zeros{T}(1, ∞), Fill(-one(T)/2, 1, ∞)), ℵ₀, 2, 0)
 end
 
+\(w_A::WeightedChebyshevU, w_B::WeightedChebyshevT) = inv(w_B \ w_A)
 \(T::ChebyshevT, U::ChebyshevU) = inv(U \ T)
 
 ####

test/test_chebyshev.jl

@@ -150,6 +150,10 @@ import ContinuumArrays: MappedWeightedBasisLayout, Map
             @test sum(WT; dims=1)[:,1:10] ≈ [π zeros(1,9)]
             @test sum(WT[:,1]) ≈ π
             @test iszero(sum(WT[:,2]))
+
+            WU = Weighted(ChebyshevU())
+            @test (WT \ WU)[1:10,1:10] ≈ inv(WU \ WT)[1:10,1:10]
+            @test_skip (WU \ WT)[1,1] == 2
         end
 
         @testset "mapped" begin