First pass about sparsedifftools.jl HOT 4 CLOSED

@pkj-m nicely done!

from sparsedifftools.jl.

JuliaDiff/FiniteDiff.jl#58 handles finite differencing based on color vector and Jacobian decompression.

from sparsedifftools.jl.

Now this is how you do forward-mode automatic differentiation in the direction of v: https://github.com/JuliaDiffEq/DiffEqOperators.jl/blob/master/src/jacvec_operators.jl#L7-L13 . Let me know if you'd like some starter code for this, or want to discuss how that's working. Essentially what needs to be done is a multi-dimensional partial in the direction of each color vector, so color .== i for each i < maximum(color), making one dual cache1 .= ForwardDiff.Dual{JacVecTag}.(x, v1, v2, v3, .., vn), solving with that dual, and extracting each vector.

For a first pass let it be allocating, and then we'll optimize it in the next step.

from sparsedifftools.jl.

For record, #17 implements dual numbers in the direction determined by the coloring vector.

from sparsedifftools.jl.