Mismatch between returned value type and `eltype` for `Float16` data

[lcontento]

When interpolating Float16 data, the interpolator reports its element type as Float64 even if it correctly returns Float16 data when called.

julia> using Interpolations

julia> itp = interpolate(rand(Float16, 3, 3), BSpline(Linear()));

julia> isa(itp, Interpolations.BSplineInterpolation{Float16})
false

julia> eltype(itp)
Float64

julia> itp(Float16(1.1), Float16(1.1))
Float16(0.9077)

It looks to be due to the tweight function defaulting to Float64 and I can apparently solve the problem by specializing tweight for Float16 or changing its default value to be the same type as the input array. Since I am not very familiar with the codebase I am not sure if that's all it should be done.

[tomasaschan]

After we decided to have a functional interface rather than an array one (see e.g. #198, #206, #242), I'm not sure it makes sense to expect eltype to indicate the type returned by interpolating.

If #242 results in actually dropping AbstractArray inheritance, we should probably remove it altogether (this is my preference, but I'm no longer an active user of the project, so that's based solely on design principles and not on actual usage), while if it results in re-defining what the array interface of an interpolation object refers to, it might still, in at least some cases, make sense to have eltype(itp) != typeof(itp(x)).

In other words, I'm not sure this should be changed at all, and if it should, the question of how is probably still not settled.

[lcontento]

I see, I was mistaken in believing that eltype was referring to the output type. However, I still believe that the current behaviour for Float16 should be changed for consistency's sake. We have that

let A = rand(T, 3, 3), itp = interpolate(A, BSpline(Linear()))
    isa(itp, Interpolations.BSplineInterpolation{T}) && eltype(itp) == T
end

is true for all the basic floating point types T (e.g., T=Float32), so as a user I would expect it to hold true also for T=Float16. By the way, a similar situation also happens with Rationals since tweight is only specialized for Rational{Int} but not for smaller Integer types.

[timholy]

Until #242 happens, eltype should be the type of the return value when it is "indexed" with Int. It may or may not be true for other index types, and indeed it's often useful that it is not: e.g., passing dual numbers as "indexes" to compute gradients.

Since itp(2, 2) does return a Float16, this is indeed a bug. A fix would be welcome.