Customize backend for conditions

[gdalle]

Add a conditions_backend kwarg to the ImplicitFunction constructor to decide which backend will be used for condition differentiation.
If conditions_backend is nothing, the same backend will be used as for the implicit function.

[gdalle]

This doesn't work with conditions_backend = AD.ZygoteBackend() or AD.ReverseDiffBackend(). Here's the error:

ArgumentError: Tuple contains 2 elements, must contain exactly 1 element
  Stacktrace:
    [1] only(x::Tuple{Float64, Float64})
      @ Base.Iterators ./iterators.jl:1531
    [2] (::AbstractDifferentiation.var"#14#16"{Vector{Float64}, ImplicitDifferentiation.var"#6#8"{Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, ImplicitFunction{var"#forward#8", var"#conditions#9", IterativeLinearSolver, AbstractDifferentiation.ReverseRuleConfigBackend{ZygoteRuleConfig{Zygote.Context{false}}}}, Vector{Float64}}, Tuple{Vector{Float64}}})(::Float64, ::Vararg{Float64})
      @ AbstractDifferentiation ~/.julia/packages/AbstractDifferentiation/eEkWP/src/AbstractDifferentiation.jl:172

I thought it might be linked to JuliaDiff/AbstractDifferentiation.jl#63 for Zygote but that may not explain the ReverseDiff aspect

[gdalle]

Also doesn't work with AD.FiniteDifferencesBackend(), for a different reason

MethodError: no method matching (::ImplicitDifferentiation.var"#6#8"{Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, ImplicitFunction{var"#forward#8", var"#conditions#9", IterativeLinearSolver, AbstractDifferentiation.FiniteDifferencesBackend{FiniteDifferences.AdaptedFiniteDifferenceMethod{5, 1, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}}}}, Vector{Float64}})(::Vector{Float64}, ::Vector{Float64})
  
  Closest candidates are:
    (::ImplicitDifferentiation.var"#6#8")(::Any)
     @ ImplicitDifferentiation ~/Work/GitHub/Julia/ImplicitDifferentiation.jl/src/operators.jl:11
  
  Stacktrace:
    [1] (::FiniteDifferences.var"#88#89"{ImplicitDifferentiation.var"#6#8"{Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, ImplicitFunction{var"#forward#8", var"#conditions#9", IterativeLinearSolver, AbstractDifferentiation.FiniteDifferencesBackend{FiniteDifferences.AdaptedFiniteDifferenceMethod{5, 1, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}}}}, Vector{Float64}}})(xs::Tuple{Vector{Float64}, Vector{Float64}})
      @ FiniteDifferences ~/.julia/packages/FiniteDifferences/SDe8k/src/grad.jl:64
    [2] jvp(fdm::FiniteDifferences.AdaptedFiniteDifferenceMethod{5, 1, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}}, f::FiniteDifferences.var"#88#89"{ImplicitDifferentiation.var"#6#8"{Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, ImplicitFunction{var"#forward#8", var"#conditions#9", IterativeLinearSolver, AbstractDifferentiation.FiniteDifferencesBackend{FiniteDifferences.AdaptedFiniteDifferenceMethod{5, 1, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}}}}, Vector{Float64}}}, ::Tuple{Tuple{Vector{Float64}, Vector{Float64}}, Tuple{Float64, Float64}})
      @ FiniteDifferences ~/.julia/packages/FiniteDifferences/SDe8k/src/grad.jl:59

[codecov]

Codecov Report

Patch coverage: 84.61% and project coverage change: +0.17% 🎉

Comparison is base (9d34d8b) 96.27% compared to head (de5876e) 96.44%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #87      +/-   ##
==========================================
+ Coverage   96.27%   96.44%   +0.17%     
==========================================
  Files           9        9              
  Lines         161      169       +8     
==========================================
+ Hits          155      163       +8     
  Misses          6        6              

Files Changed	Coverage Δ
src/implicit_function.jl	70.00% <33.33%> (ø)
ext/ImplicitDifferentiationChainRulesExt.jl	100.00% <100.00%> (ø)
ext/ImplicitDifferentiationForwardDiffExt.jl	100.00% <100.00%> (ø)

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