API reference
Index
ImplicitDifferentiation.ImplicitDifferentiationImplicitDifferentiation.AbstractLinearSolverImplicitDifferentiation.DirectLinearSolverImplicitDifferentiation.ImplicitFunctionImplicitDifferentiation.ImplicitFunctionImplicitDifferentiation.ImplicitFunctionImplicitDifferentiation.IterativeLinearSolverChainRulesCore.rrule
Docstrings
ImplicitDifferentiation.ImplicitDifferentiation — ModuleImplicitDifferentiationA Julia package for automatic differentiation of implicit functions.
Its main export is the type ImplicitFunction.
ImplicitDifferentiation.AbstractLinearSolver — TypeAbstractLinearSolverAll linear solvers used within an ImplicitFunction must satisfy this interface.
It can be useful to roll out your own solver if you need more fine-grained control on convergence / speed / behavior in case of singularity. Check out the source code of IterativeLinearSolver and DirectLinearSolver for implementation examples.
Required methods
presolve(linear_solver, A, y): Returns a matrix-like objectAfor which it is cheaper to solve several linear systems with different vectorsb(a typical example would be to perform LU factorization).solve(linear_solver, A, b): Returns a vectorxsatisfyingAx = b. If the linear system has not been solved to satisfaction, every element ofxshould be aNaNof the appropriate floating point type.
ImplicitDifferentiation.DirectLinearSolver — TypeDirectLinearSolverAn implementation of AbstractLinearSolver using the built-in backslash operator.
Fields
verbose::Bool: Whether to throw a warning when the solver fails (defaults totrue)
ImplicitDifferentiation.ImplicitFunction — TypeImplicitFunction{F,C,L,B}Wrapper for an implicit function defined by a forward mapping y and a set of conditions c.
An ImplicitFunction object behaves like a function, and every call is differentiable with respect to the first argument x. When a derivative is queried, the Jacobian of y is computed using the implicit function theorem:
∂/∂y c(x, y(x)) * ∂/∂x y(x) = -∂/∂x c(x, y(x))This requires solving a linear system A * J = -B where A = ∂c/∂y, B = ∂c/∂x and J = ∂y/∂x.
Fields
forward::F: a callable, does not need to be compatible with automatic differentiationconditions::C: a callable, must be compatible with automatic differentiationlinear_solver::L: a subtype ofAbstractLinearSolver, defines how the linear system will be solvedconditions_backend::B: eithernothingor a subtype ofAbstractDifferentiation.AbstractBackend, defines how the conditions will be differentiated within the implicit function theorem
There are two possible signatures for forward and conditions, which must be consistent with one another:
- Standard:
forward(x, args...; kwargs...) = yandconditions(x, y, args...; kwargs...) = c - Byproduct:
forward(x, args...; kwargs...) = (y, z)andconditions(x, y, z, args...; kwargs...) = c.
In both cases, x, y and c must be arrays, with size(y) = size(c). In the second case, the byproduct z can be an arbitrary object generated by forward. The positional arguments args... and keyword arguments kwargs... must be the same for both forward and conditions.
The byproduct z and the other positional arguments args... beyond x are considered constant for differentiation purposes.
ImplicitDifferentiation.ImplicitFunction — Method(implicit::ImplicitFunction)(x::AbstractArray, args...; kwargs...)Return implicit.forward(x, args...; kwargs...), which can be either an array y or a tuple (y, z).
This call is differentiable.
ImplicitDifferentiation.ImplicitFunction — MethodImplicitFunction(
+ forward,
+ conditions;
+ linear_solver=IterativeLinearSolver(),
+ conditions_backend=nothing,
+)Construct an ImplicitFunction with default parameters.
ImplicitDifferentiation.IterativeLinearSolver — TypeIterativeLinearSolverAn implementation of AbstractLinearSolver using Krylov.gmres.
Fields
verbose::Bool: Whether to throw a warning when the solver fails (defaults totrue)
ChainRulesCore.rrule — Functionrrule(rc, implicit, x, args...; kwargs...)Custom reverse rule for an ImplicitFunction, to ensure compatibility with reverse mode autodiff.
This is only available if ChainRulesCore.jl is loaded (extension), except on Julia < 1.9 where it is always available.
We compute the vector-Jacobian product Jᵀv by solving Aᵀu = v and setting Jᵀv = -Bᵀu. Positional and keyword arguments are passed to both implicit.forward and implicit.conditions.
