IFTDuals.jl

IFTDuals.jl is a lightweight Julia package for computing higher order derivatives of a function, $y=g(\theta)$, implicitly defined through the implicit function theorem (IFT) using dual numbers. IFT gives the relationship between $g$ and $\theta$ implicitly through the solution of the equation $f\left(g(\theta\right), \theta) = 0$. This package currently only supports derivatives obtained through ForwardDiff.jl

Installation

You can install IFTDuals.jl via Julia's package manager. In the Julia REPL, enter the package manager by pressing ] and then run:

pkg> add IFTDuals

Usage

Here is a simple example demonstrating how to use IFTDuals.jl to compute higher order derivatives using the implicit function theorem. For more detailed discussions, refer to the documentation.

using IFTDuals
using DifferentiationInterface

# Define the implicit function f(y, θ) = 0
f(y, θ) = ...

function get_y(θ)
    θ_primal = nested_pvalue(θ)
    y = root_solver(f, θ_primal)  # Any root solver that finds the primal value of y such that f(y, θ_primal) = 0
    return ift(y, f, θ, θ_primal) # compute derivatives if duals are present
end

grad = second_derivative(get_y, AutoForwardDiff(), θ) # use existing interfaces to compute derivatives 

Alternatives

ImplicitDifferentiation.jl is a different Julia package which provides the functionality to compute first order derivatives implicitly using any AD backend.

Contributing

Contributions to IFTDuals.jl are welcome! If you find a bug or have a feature request, please open an issue on the GitHub repository. If you'd like to contribute code, feel free to fork the repository and submit a pull request.