Hello, i calculate equilibriums with `IntervalRootFinding.Newton`

The calculation time takes too long

Could it be some kind of error or a problem in the method itself?

I call the method of Newton using `fixedpoints`

from `ChaosTools.jl`

Code:

```
using StaticArrays, DifferentialEquations, DynamicalSystems, LinearAlgebra, IntervalRootFinding
@inbounds function TM(u, p, t)
U(y, p) = p[8] + p[9] / ( 1.0 + exp( -50.0 * (y - p[7]) ) )
σ(x, p) = 1.0 / ( 1.0 + exp( -20.0 * (x-p[6]) ) )
g(E, x, y, p, U_) = log( 1.0 + exp( (p[5] * U_ * x * E + p[11] ) / (p[1]) ) )
U_ = U(u[3], p)
du1 = (-u[1] + p[1] * g(u[1], u[2], u[3], p, U_) ) / p[2]
du2 = (1.0 - u[2]) / p[3] - U_*u[2]*u[1]
du3 = (-u[3])/p[4] + p[10] * σ(u[2], p)
return SVector(du1, du2, du3)
end
@inbounds function jacob_TM_(u, p, t)
U(y, p, exp50) = p[8] + p[9] / ( 1.0 + exp50 )
U_y(y, p, exp50) = (50.0 * p[9] * exp50) / (1.0 + exp50)^2
g(E, x, y, p, U_) = exp((p[5] * U_ * x * E + p[11]) / p[1])
σ_der(x, p) = exp( (-20.0) * (x - p[6]) )
exp50 = exp(-50.0 * (u[3] - p[7]))
U_ = U(u[3], p, exp50)
Uy = U_y(u[3], p, exp50)
g_ = g(u[1], u[2], u[3], p, U_)
σ_deri = σ_der(u[2], p)
g_plus = 1.0 + g_
g_mult = g_ * U_
g_plus_mult = p[2] * (g_plus)
u1p5 = p[5] * u[1]
Uyu2 = Uy * u[2]
E_E = (-1.0 + ((J * u[2] * g_mult)) / (g_plus) ) / p[2]
E_x = (u1p5 * g_mult) / (g_plus_mult)
E_y = (u1p5 * Uyu2 * g_) / (g_plus_mult)
x_E = -U_ * u[2]
x_x = -1.0 / p[3] - U_ * u[1]
x_y = -Uyu2 * u[1]
y_x = 20.0 * p[10] * σ_deri / (1.0 + σ_deri)^2
y_y = -1.0/p[4]
SMatrix{3,3}(E_E, x_E, 0.0,
E_x, x_x, y_x,
E_y, x_y, y_y)
end
t = 500.0; tt = 600.0; tstep = 0.001
trange = range(0.0, t, step = tstep)
integ_set = (alg = RK4(), adaptive = false, dt = tstep)
E = interval(-40, 40); x = interval(0, 1); y = interval(0, 1)
box = IntervalBox(E, x, y);
const τ = 0.013; const τD = 0.07993; const τy = 3.3; const J = 3.07; const β = 0.300
const xthr = 0.75; const ythr = 0.4
const α = 1.58; const ΔU0 = 0.305;
U0, I0 = 0.182490142150041, -1.33122064887577
p = SA[α, τ, τD, τy, J, xthr, ythr, U0, ΔU0, β, I0];
u0 = [7.485044731286254, 0.771932479798953, 0.5235934863993938]
ds = CoupledODEs(TM, u0, p, diffeq = integ_set)
fp, _, _ = fixedpoints(ds, box, jacob_TM_, method = IntervalRootFinding.Newton)
```