Support Complex Numbers in Stiff Routines

[ChrisRackauckas]

After a long journey with @AlexanderKoshkarov, we found that the OrdinaryDiffEq.jl non-stiff (Runge-Kutta) routines are compatible with complex numbers (with adaptivity), but the implicit and Rosenbrock methods are not. This is because the Calculus.jl (now FiniteDiff.jl) and ForwardDiff.jl dependencies cannot take derivatives of complex numbers:

johnmyleswhite/FiniteDiff.jl#14
JuliaDiff/ForwardDiff.jl#157

Also, NLsolve.jl is incompatible with complex numbers. On the other hand, the ODEInterface and Sundials algorithms do not seem to be compatible with complex numbers, and so all of this is a dead-end.

Ways around this are:

• Implement Jacobian-free nonlinear solvers (Jacobian-Free Newton Krylov)
• Implement a simple package just for Jacobians and derivatives with complex numbers
• Get a PR into FiniteDiff / ForwardDiff

I don't think I'll have the time for this right now, but am documenting this in case anyone else has an idea / wants to help.

(@mauro3 do you have routines for Jacobians?)

[mauro3]

Sorry, I'm no help here. What about @pwl's DASSL?

[ChrisRackauckas]

Looks like Calculus.jl might be easy to solve: JuliaMath/Calculus.jl#115

[ChrisRackauckas]

Solved by DiffEqDiffTools.jl replacing Calculus.jl (and gets like a 50x speedup!). ForwardDiff.jl will still fail though, but that should be fixed by it changing to a Cassette.jl backend.

[jebej]

I am still getting errors with stiff routines, for example with Rosenbrock23:

ERROR: ArgumentError: Cannot create a dual over scalar type Complex{Float64}. If the type behaves as a scalar, define FowardDiff.can_dual.
Stacktrace:
 [1] throw_cannot_dual(::Type) at C:\Users\Jeremy\.julia\packages\ForwardDiff\cXTw0\src\dual.jl:36
 [2] ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.TimeGradientWrapper{Liouvillian{SparseMatrixCSC{Complex{Float64},Int64},2,Tuple{typeof(drivem),typeof(drivep)},Schrodinger.var"#46#47"{SparseMatrixCSC{Complex{Float64},Int64},Tuple{SparseMatrixCSC{Complex{Float64},Int64},Array{Complex{Float64},2}},Tuple{typeof(drivem),typeof(drivep)},Tuple{Array{Float64,1},Array{Float64,1}}},2},Array{Complex{Float64},1},DiffEqBase.NullParameters},Complex{Float64}},Complex{Float64},1}(::Complex{Float64}, ::ForwardDiff.Partials{1,Complex{Float64}}) at C:\Users\Jeremy\.julia\packages\ForwardDiff\cXTw0\src\dual.jl:18
 [3] Dual at C:\Users\Jeremy\.julia\packages\ForwardDiff\cXTw0\src\dual.jl:55 [inlined]
 [4] Dual at C:\Users\Jeremy\.julia\packages\ForwardDiff\cXTw0\src\dual.jl:62 [inlined]
 [5] Dual at C:\Users\Jeremy\.julia\packages\ForwardDiff\cXTw0\src\dual.jl:66 [inlined]
 [6] _broadcast_getindex_evalf at .\broadcast.jl:631 [inlined]
 [7] _broadcast_getindex at .\broadcast.jl:604 [inlined]
 [8] getindex at .\broadcast.jl:564 [inlined]
 [9] copy at .\broadcast.jl:854 [inlined]
 [10] materialize(::Base.Broadcast.Broadcasted{Base.Broadcast.DefaultArrayStyle{1},Nothing,Type{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.TimeGradientWrapper{Liouvillian{SparseMatrixCSC{Complex{Float64},Int64},2,Tuple{typeof(drivem),typeof(drivep)},Schrodinger.var"#46#47"{SparseMatrixCSC{Complex{Float64},Int64},Tuple{SparseMatrixCSC{Complex{Float64},Int64},Array{Complex{Float64},2}},Tuple{typeof(drivem),typeof(drivep)},Tuple{Array{Float64,1},Array{Float64,1}}},2},Array{Complex{Float64},1},DiffEqBase.NullParameters},Complex{Float64}},V,N} where N where V},Tuple{Array{Complex{Float64},1},Array{Complex{Float64},1}}}) at .\broadcast.jl:820
 [11] build_grad_config(::Rosenbrock23{0,true,DefaultLinSolve,DataType}, ::Function, ::Function, ::Array{Complex{Float64},1}, ::Float64) at C:\Users\Jeremy\.julia\packages\OrdinaryDiffEq\RYCSN\src\derivative_wrappers.jl:175
[...]

[jebej]

Note that now that I see the stacktrace more clearly this might somehow be due to the custom functor type.

[jebej]

No, this is indeed due to ForwardDiff not supporting complex numbers. Maybe I am doing something wrong, any advice here?

[ChrisRackauckas]

Use autodiff=false until JuliaDiff/ForwardDiff.jl#455