Add batch/vectorized finite difference Jacobian evaluation

[ChrisRackauckas-Claude]

Summary

Closes #210.

• Adds batch=true keyword argument to finite_difference_jacobian and finite_difference_jacobian!
• When batch=true, the function f receives a matrix where each column is an input point, and should return a matrix where each column is the corresponding output
• For the in-place variant, f(FX, X) receives output matrix FX and input matrix X
• All perturbations are evaluated in a single call to f, enabling GPU-parallelized or otherwise vectorized evaluation
• Supports forward, central, and complex step methods

Implementation details

Out-of-place (finite_difference_jacobian):

• Forward: builds n×(n+1) matrix (base + n perturbations) or n×n if f_in provided → 1 call to f
• Central: builds n×2n matrix (forward + backward perturbations) → 1 call to f
• Complex: builds complex n×n matrix → 1 call to f

In-place (finite_difference_jacobian!):

• Same matrix construction, but f(FX, X) fills the pre-allocated output matrix

Example usage

# Out-of-place: f accepts matrix, returns matrix (columns = eval points)
function f_batch(X::AbstractMatrix)
    hcat([f_scalar(X[:, j]) for j in 1:size(X, 2)]...)
end

J = finite_difference_jacobian(f_batch, x; batch=true)

# In-place: f(FX, X) fills FX
function f_batch!(FX::AbstractMatrix, X::AbstractMatrix)
    for j in 1:size(X, 2)
        FX[:, j] .= f_scalar(X[:, j])
    end
end

finite_difference_jacobian!(J, f_batch!, x; batch=true)

Test plan

• Out-of-place batch Jacobian matches reference for forward, central, complex step
• In-place batch Jacobian matches reference for forward, central, complex step
• Batch results match non-batch results on a 3→3 function
• All existing tests still pass (core + coloring + out-of-place)
• CI passes

🤖 Generated with Claude Code

[gdalle]

Adding a note to draw inspiration from BatchSolve.jl by @mattsignorelli. In particular, the file batch-ad/finitediff.jl seems relevant