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Forward Mode Automatic Differentiation for Julia

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In a recent Slack discussion, @mohamed82008 posted this useful code snippet that shouldn't go to waste. With a little bit of work, this could be turned into a macro that...

This changes the meaning of `==` for dual numbers, to demand that both real and ~~imaginary~~ dual parts match. Fixes #197, fixes #407. Edit -- also closes #490, closes #506,...

This follows LinearAlgebra in explicitly forbidding offset arrays. They don't work anyway, but this replaces a BoundsError with something deliberate: ```julia julia> ForwardDiff.gradient(sum, OffsetArray([1,2,3], 3)) # before ERROR: BoundsError: attempt...

if a package needs to do AD through a `StaticArray`, then they need to load `StaticArrays` anyway

It would be very nice to have support for AD in `Base.expm`. I would like to contribute to it, but since I am unfamiliar with the AD architecture in Julia...

```julia julia> ForwardDiff.Dual{Nothing, Float64}(1.0) ERROR: StackOverflowError: Stacktrace: [1] convert(#unused#::Type{ForwardDiff.Dual{Nothing, Float64}}, x::Float64) @ Base .\number.jl:7 [2] (ForwardDiff.Dual{Nothing, Float64})(x::Float64) @ ForwardDiff C:\Users\Hossein Pourbozorg\.julia\packages\ForwardDiff\wAaVJ\src\dual.jl:79 --- the last 2 lines are repeated 39990 more...

The calculation of the gradient from a multivariate normal prior involves left-division by a triangular matrix. Currently, when the covariance matrix is fixed, and the random vector depends on model...

This is picking up where #495 left off. Some simple examples of derivatives involving `fft` are now working. I wasn't able to get `dct` or `r2r` working, yet, as they...

I'm trying to calculate complex derivatives of analytic functions, using [a trick](https://discourse.julialang.org/t/automatic-differentiation-of-complex-valued-functions/30263/3) by @tpapp: ```julia using ForwardDiff ∂(f, x::Real) = ForwardDiff.derivative(f, x) function ∂(f, z::Complex) # https://discourse.julialang.org/t/automatic-differentiation-of-complex-valued-functions/30263/3 ff = ((x,y),)...

Hi, I am interested in calculating the hessian of a function that requires calls to multiple other built-in and local functions. What is otherwise a run-able function quickly becomes incompatible...