First pass

[ChrisRackauckas]

• Convert sparse matrix to graph
• Perform coloring on graph
• spit out color vector
• make finite differencing choose directions based on color vector
• add the Jacobian decompression
• make dual numbers in the directions of the color vector
• decompress the partials into the Jacobian

[pkj-m]

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

[ChrisRackauckas]

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.

[ChrisRackauckas]

@pkj-m nicely done!

[pkj-m]

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