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(
+API reference · ImplicitDifferentiation.jl API reference
Index
ImplicitDifferentiation.ImplicitDifferentiationImplicitDifferentiation.AbstractLinearSolverImplicitDifferentiation.DirectLinearSolverImplicitDifferentiation.ImplicitFunctionImplicitDifferentiation.ImplicitFunctionImplicitDifferentiation.ImplicitFunctionImplicitDifferentiation.IterativeLinearSolverChainRulesCore.rrule
Docstrings
ImplicitDifferentiation.ImplicitDifferentiation — ModuleImplicitDifferentiation
A Julia package for automatic differentiation of implicit functions.
Its main export is the type ImplicitFunction.
sourceImplicitDifferentiation.AbstractLinearSolver — TypeAbstractLinearSolver
All 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 object A for which it is cheaper to solve several linear systems with different vectors b (a typical example would be to perform LU factorization).solve(linear_solver, A, b): Returns a vector x satisfying Ax = b. If the linear system has not been solved to satisfaction, every element of x should be a NaN of the appropriate floating point type.
sourceImplicitDifferentiation.DirectLinearSolver — TypeDirectLinearSolver
An implementation of AbstractLinearSolver using the built-in backslash operator.
Fields
verbose::Bool: Whether to throw a warning when the solver fails (defaults to true)
sourceImplicitDifferentiation.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 of AbstractLinearSolver, defines how the linear system will be solvedconditions_backend::B: either nothing or a subtype of AbstractDifferentiation.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...) = y and conditions(x, y, args...; kwargs...) = c - Byproduct:
forward(x, args...; kwargs...) = (y, z) and conditions(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.
Warning The byproduct z and the other positional arguments args... beyond x are considered constant for differentiation purposes.
sourceImplicitDifferentiation.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.
sourceImplicitDifferentiation.ImplicitFunction — MethodImplicitFunction(
forward,
conditions;
linear_solver=IterativeLinearSolver(),
conditions_backend=nothing,
-)
Construct an ImplicitFunction with default parameters.
sourceImplicitDifferentiation.IterativeLinearSolver — TypeIterativeLinearSolver
An implementation of AbstractLinearSolver using Krylov.gmres.
Fields
verbose::Bool: Whether to throw a warning when the solver fails (defaults to true)
sourceChainRulesCore.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.
sourceSettings
This document was generated with Documenter.jl version 0.27.25 on Thursday 10 August 2023. Using Julia version 1.9.2.
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.
