IA.jl

using Plots; gr(); 
ENV["PLOTS_TEST"] = "true"
using IntervalArithmetic

N    = 27

x0 = @interval(0.11)
x0_sol = 0.11

f(x) = 4*x*(1-x)
sol(n, x0) = ( 1 - cos(2^n * acos(1-2x0) ) ) / 2

x_sol = sol.(1:N, x0_sol)

xhi = zeros(N)
xlo = zeros(N)
x = x0
for n in 1:N
    x = f(x)
    xhi[n] = x.hi
    xlo[n] = x.lo
end

plot(1:N, xlo, label="lo", ls=:dash)
plot!(1:N, xhi, label="hi", ls=:dash)
plot!(1:N, x_sol, label="exact")
savefig("IA.png")

IA500.jl

using Plots; gr(); 
ENV["PLOTS_TEST"] = "true"
using IntervalArithmetic

setprecision(Interval,500)
N    = 251

x0 = @interval(0.11)
x0_sol     = big"0.11"

f(x) = 4*x*(1-x)

sol(n, x0) = (big"1" - cos(big"2"^n*acos(big"1"-big"2"*x0)))/big"2"

x_sol = sol.(1:N, x0_sol)

xhi = zeros(N)
xlo = zeros(N)
x = x0
for n in 1:N
    x = f(x)
    xhi[n] = x.hi
    xlo[n] = x.lo
end

plot(1:N, xlo, label="lo", ls=:dash)
plot!(1:N, xhi, label="hi", ls=:dash)
plot!(1:N, x_sol, label="exact")
savefig("IA_500.png")