Refactor integration and QuadraticInterpolation

[SouthEndMusic]

@sathvikbhagavan I split this out from #356 for ease of reviewing. I need the QuadraticInteprolation refactor for the integration refactor, and the integration refactor for #356.

[sathvikbhagavan]

Left one comment. Are we getting any speedup from this implementation of QuadraticInterpolation than the earlier version? Otherwise, Looks good!

[sathvikbhagavan]

Just to confirm, can't u_tangent for QuadraticInterpolation be reformulated using α and β parameters you added?

[SouthEndMusic]

I'ts possible, but I suspect this is more efficient. Also, that might break AD types that do not support mutation when those parameters are cached?

[SouthEndMusic]

And you still need to have the logic somewhere to see which mode (forward/backward looking interpolation) was used for the u gradient.

[sathvikbhagavan]

Alright, then we can keep this.

[SouthEndMusic]

using DataInterpolations
using Random
using BenchmarkTools

Random.seed!(2)

t = cumsum(rand(5))
t_eval = range(first(t), last(t), length = 100)

### No cached parameters

## Scalar u
u = rand(5)
A = QuadraticInterpolation(u, t)
@btime A.($t_eval)
# Before:
# 1.110 μs (4 allocations: 1.12 KiB)
# After:
# 1.090 μs (4 allocations: 1.11 KiB)

## Vector u
u = rand(2, 5)
A = QuadraticInterpolation(u, t)
@btime A.($t_eval)
# Before:
# 19.100 μs (2804 allocations: 110.50 KiB)
# After:
# 17.200 μs (2204 allocations: 87.05 KiB)

### Cached parameters

## Scalar u
u = rand(5)
A = QuadraticInterpolation(u, t; cache_parameters = true)
@btime A.($t_eval)
# Before:
# 1.030 μs (4 allocations: 1.12 KiB)
# After:
# 976.471 ns (4 allocations: 1.11 KiB)

## Vector u
u = rand(2, 5)
A = QuadraticInterpolation(u, t; cache_parameters = true)
@btime A.($t_eval)
# Before:
# 10.600 μs (1404 allocations: 55.81 KiB)
# After:
# 4.829 μs (604 allocations: 24.55 KiB)