[WIP] feat: initial implementation of SCCNonlinearProblem codegen

[AayushSabharwal]

I've checked this on the trivial case in SCCNonlinearSolve's tests. However, there is one major caveat: There can be multiple orderings of the SCCs. As a result, solving the same system with NonlinearProblem and SCCNonlinearProblem leads to different ordering of the solution vector, because one uses the order of unknowns in the system and the other uses the order of SCCs. I'm figuring out how to use the same sorted order of SCCs in the simplified system.

[AayushSabharwal]

Example:

julia> @variables u(t)[1:8] [irreducible = true]
julia> @mtkbuild sys = NonlinearSystem([
                 0 ~ u[5]^2 + u[4]
                 0 ~ u[3]^2 + u[5]
                 0 ~ cos(u[2]) - u[1]
                 0 ~ 2u[4] + u[3] + 1.0
                 0 ~ u[1] + u[2] + u[3] + u[4] + u[5]    + 2.0u[6] + 2.5u[7] + 1.5u[8]
                 0 ~ u[1] + u[2] + u[3] + 2.0u[4] + u[5] + 4.0u[6] - 1.5u[7] + 1.5u[8]
                 0 ~ u[1] + 2.0u[2] + 3.0u[3] + 5.0u[4] + 6.0u[5] + u[6] - u[7] - u[8]
                 0 ~ sin(u[1] + u[2]) + u[2]
             ])
julia> prob =  SCCNonlinearProblem{true}(sys, unknowns(sys) .=> zeros(8));
julia> sol = solve(prob, NewtonRaphson())
julia> sol
retcode: Success
u: 8-element Vector{Float64}:
 -0.443600343333633
  0.9032122743166665
 -0.41964337760708054
 -0.17610056436947866
 -0.6477988712610427
 -4.179650698308442
  1.0580471331304233
  2.204144548445586
julia> nprob = NonlinearProblem(sys, unknowns(sys) .=> zeros(8))
julia> nsol = solve(nprob, NewtonRaphson())
retcode: Success
u: 8-element Vector{Float64}:
 -0.41964337760708054
 -0.17610056436947866
 -0.6477988712610426
 -0.44360034333363063
  0.903212274316663
 -4.179650698308441
  1.0580471331304235
  2.204144548445586

Note how both solutions have the same answer, but in different permutations.

[ChrisRackauckas]

But SII should take care of that?

[ChrisRackauckas]

1. Is there a way to toggle the initialization problem to be SCC or not?