type stability

[stevengj]

Makes the quadgk function type-stable. Closes #21. Closes #19. Closes #15. Closes #20.

Also fixes handling of some other other number types, and in particular fixes dimensionful types (#17). Mostly supersedes https://github.com/PainterQubits/UnitfulIntegration.jl, cc @ajkeller34:

julia> using Unitful, QuadGK, Test

julia> @inferred quadgk(x -> x^2, 0u"m", 1u"m")
(0.3333333333333333 m^3, 5.551115123125783e-17 m^3)

It still doesn't completely address #13, because it only works if real(one(T)) returns the underlying real numeric type for computing the quadrature weights etc. This doesn't work for dual numbers (ForwardDiff.jl) or measurements (JuliaPhysics/Measurements.jl#31), and in general I haven't been able to think of a way to do this that works across packages.

[stevengj]

Note that the code will now fail if the integrand returns one (or more) types when it is evaluated at the initial quadrature points, and then suddenly returns another type (that is incompatible under convert) when it is evaluated at refined points. In principle, I know how to handle this (by reallocating the segs array in adapt … see the todo). In practice, I'm doubtful that it is worth the trouble. Fixed.

[ararslan]

That to me seems like something that is unlikely to occur in most user code, so I'd agree that it's not likely worth the trouble. Though perhaps a warning about it in the documentation would be warranted, in the off chance someone runs into it?

[MasonProtter]

It still doesn't completely address #13, because it only works if real(one(T)) returns the underlying real numeric type for computing the quadrature weights etc.

That's frustrating. Why does the quadrature weights calculation need real numbers?

Perhaps one thing that could be done is put in a function value and a base definition value(x::Number) = real(x) and then put a note in the documentation that if someone is trying to make their custom number type work with QuadGK, they need to define value on that number type.

ie.

QuadGK.value(x::ForwardDiff.Dual) = x.value
QuadGK.value(x::Unitful.Quantity) = x.val

[stevengj]

I pushed a fix for type-unstable functions; it turned out to be not too hard.

[stevengj]

Perhaps one thing that could be done is put in a function value and a base definition value(x::Number) …

Then any package that wanted this to work would have to depend on QuadGK. This really needs to go into Base.

[stevengj]

Maybe packages that define new number types should define AbstractFloat(x::MyNumber) to give the raw-floating-point value? (They should define a separate float(x::MyNumber) method to convert to a floating-point version of the same type.)

[giordano]

What is the "raw-floating-point value"?