Try using TaylorDiff.jl instead of TaylorIntegration.jl

[nathanaelbosch]

Not quite sure what to gain here, but at the very least the dependency could be more lightweight.

The part that would be changed is this one:

ProbNumDiffEq.jl/src/initialization/taylormode.jl

Lines 35 to 49 in 70b6d57

	"""
	Compute initial derivatives of an IIP ODEProblem with TaylorIntegration.jl
	"""
	function taylormode_get_derivatives(u, f::SciMLBase.AbstractODEFunction{true}, p, t, q)
	tT = Taylor1(typeof(t), q)
	tT[0] = t
	uT = similar(u, Taylor1{eltype(u)})
	@inbounds @simd ivdep for i in eachindex(u)
	uT[i] = Taylor1(u[i], q)
	end
	duT = zero(uT)
	uauxT = similar(uT)
	TaylorIntegration.jetcoeffs!(f, tT, uT, duT, uauxT, p)
	return [evaluate.(differentiate.(uT, i)) for i in 0:q]
	end

Currently I'm not sure how to best get exactly this functionality from TaylorDiff.jl. If you do, please comment!

[nathanaelbosch]

Here is a small example to get this forward, that uses the current implementation (linked above):

using ProbNumDiffEq

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]
    return nothing
end
u0 = [1.0; 0.0; 0.0]
tspan = (0.0, 1.0)
p = nothing
prob = ODEProblem(lorenz, u0, tspan)

order = 10
taylor_coeffs = ProbNumDiffEq.taylormode_get_derivatives(prob.u0, prob.f, prob.p, prob.tspan[1], order)

returns

11-element Vector{Vector{Float64}}:
 [1.0, 0.0, 0.0]
 [-10.0, 28.0, 0.0]
 [380.0, -308.0, 28.0]
 [-6880.0, 10920.0, -942.6666666666667]
 [178000.00000000003, -201777.33333333337, 54593.77777777778]
 [-3.797773333333334e6, 5.029636888888889e6, -2.256960740740741e6]
 [8.827410222222221e7, -1.0087210725925928e8, 1.0874346553086421e8]
 [-1.891462094814815e9, 1.8127303928395057e9, -4.805741615341564e9]
 [3.7041924876543205e10, -3.093252391818924e9, 2.2031089507078738e11]
 [-4.013517726836212e11, -2.5410226724751836e12, -9.768228414733674e12]
 [-2.1396708997915617e13, 2.351267406708858e14, 4.275183771080162e14]

The goal of this issue would be to find a way to have exactly this functionality with TaylorDiff.jl. Ideally even with less allocations, e.g. writing directly into some pre-allocated output vector.