Some DifferentialEquations.jl solvers fail when using `ArrayPartition`

[Sleort]

Note the use of ArrayPartition below

julia> using DifferentialEquations,RecursiveArrayTools

julia> function lorenz(du,u,p,t)
           du[1] = 10.0*(u[2]-u[1])
           du[2] = u[1]*(28.0-u[3]) - u[2]
           du[3] = u[1]*u[2] - (8/3)*u[3]
       end
lorenz (generic function with 1 method)

julia> u0 = ArrayPartition([1.0,0.0],[0.0])
([1.0, 0.0], [0.0])

julia> tspan = (0.0,100.0)
(0.0, 100.0)

julia> prob = ODEProblem(lorenz,u0,tspan)
ODEProblem with uType ArrayPartition{Float64,Tuple{Array{Float64,1},Array{Float64,1}}} and tType Float64. In-place: true
timespan: (0.0, 100.0)
u0: [1.0, 0.0][0.0]

julia> sol = solve(prob,Tsit5());

julia> sol = solve(prob,AutoTsit5(Rosenbrock23()));
ERROR: MethodError: no method matching ldiv!(::LinearAlgebra.LU{Float64,Array{Float64,2}}, ::ArrayPartition{Float64,Tuple{Array{Float64,1},Array{Float64,1}}})
Closest candidates are:
  ldiv!(::LinearAlgebra.Transpose{#s561,#s560} where #s560<:LinearAlgebra.LowerTriangular where #s561, ::AbstractArray{T,1} where T) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/triangular.jl:1217
  ldiv!(::LinearAlgebra.Transpose{#s561,#s560} where #s560<:LinearAlgebra.LowerTriangular where #s561, ::AbstractArray{T,1} where T, ::AbstractArray{T,1} where T) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/triangular.jl:1201
  ldiv!(::LinearAlgebra.Transpose{#s561,#s560} where #s560<:LinearAlgebra.UnitLowerTriangular where #s561, ::AbstractArray{T,1} where T) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/LinearAlgebra/src/triangular.jl:1235
  ...

(Everything works fine without ArrayPartition, i.e. u0 = [1.0, 0.0, 0.0])

I'm using DifferentialEquations v5.3.1, RecursiveArrayTools v0.18.3, and Julia 1.0.1 on Ubuntu 16.04.

[YingboMa]

There is no linear algebra defined on ArrayPartition. We should fix that. Thanks for reporting!

[vnvasquez]

Adding another MWE to this discussion showing the current incompatibility of ArrayPartitions with NLsolve:

using NLsolve
using RecursiveArrayTools
function mymodel(F, vars)
    x = vars.x[1]    
    F.x[1][1] = (x[1]+3)*(x[2]^3-7)+18
    F.x[1][2] = sin(x[2]*exp(x[1])-1)
    y=vars.x[2]
    F.x[2][1] = (y[1]+3)*(y[2]^3-7)+18
    F.x[2][2] = sin(y[2]*exp(y[1])-1)
end

# To show that the function works
F = ArrayPartition([0.0 0.0],[0.0, 0.0])
u0= ArrayPartition([0.1; 1.2], [0.1; 1.2])
result = mymodel(F, u0) 

# To show the NLsolve error that results with ArrayPartitions: 
nlsolve(f!, ArrayPartition([0.1; 1.2], [0.1; 1.2]))

[ChrisRackauckas]

@YingboMa what do you think about just defining a slow fallback for now? I don't think it would actually be a major performance hit since the factorization is just a normal array factorization, we just need to define ldiv! to convert the ArrayPartition to an Array and then do the quick step?

[ChrisRackauckas]

These examples a lot more are all fixed. Go ham.