Invert interpolation integrals

[SouthEndMusic]

Fixes https://github.com/SciML/DataInterpolations.jl/issues/277

Checklist

• [x] Appropriate tests were added
• [x] Any code changes were done in a way that does not break public API
• [ ] All documentation related to code changes were updated
• [x] The new code follows the contributor guidelines, in particular the SciML Style Guide and COLPRAC.
• [ ] Any new documentation only uses public API

[SouthEndMusic]

Integral inversion POC:

using DataInterpolations
using Plots

t = collect(1:5)
u = [1.0, 1.0, 2.0, 4.0, 3.0]
A = LinearInterpolation(u, t)

ts = range(first(t), last(t), length = 100)
Is = DataInterpolations.integral.(Ref(A), ts)

A_intinv = DataInterpolations.invert_integral(A)
ts_ = A_intinv.(Is)

plot(ts, Is, label = "Original integral")
plot!(ts_, Is, label = "Integral inverse")

It handles the edge case of the interpolation being constant perfectly 🥳

[SouthEndMusic]

Speed boost when using cached integral values:

using DataInterpolations
using BenchmarkTools
using Random
Random.seed!(42)

N = 100
t = cumsum(rand(N))
ddu = rand(N)
du = rand(N)
u = rand(N)
A = QuinticHermiteSpline(ddu, du, u, t)

@btime DataInterpolations.integral(A, last(t))

# old: 1.320 μs (2 allocations: 32 bytes)
# new: 66.395 ns (2 allocations: 32 bytes)

[SouthEndMusic]

@sathvikbhagavan how shall I document this new feature?

[ChrisRackauckas]

Make a new docs page on it with a short tutorial and then the docstrings. And make a list of the methods which are compatible with it on there.

[SouthEndMusic]

I'm wondering what to do with the docstrings of the LinearInterpolationIntInv and ConstantInterpolationIntInv structs. For the other interpolation methods this describes the public constructor, but I've set up my code so that users will probably always create these objects via DataInterpolations.invert_integral.