### A Pluto.jl notebook ###
# v0.14.8

using Markdown
using InteractiveUtils

# ╔═╡ 6a62bdd8-7139-11eb-157e-8b433dd0177b
using DSP.FFTW, Interpolations, NCA, KadanoffBaym, Plots

# ╔═╡ 3b1f4142-8f9b-11eb-37ac-8b3898176679
md"""
This notebook is for testing how many k-states for the c-electrons should be taken to approximately match the continuum c-electron bath in equilibrium ``\sum_k G^c_k(\omega)``.

It's important to note that as `` t \gg 1`` some spurious features may arise which are probably not desirable. 
"""

# ╔═╡ ba3e2e22-713a-11eb-108c-edd5ce13de5b
function ft(xs, ys; mode = +1)
    # @assert ordered_xs
    dx = xs[2] - xs[1]

    x̂s = fftfreq(length(xs), -2pi / dx)
    is = sortperm(x̂s)

    ℯⁱᵠ = dx * exp.(mode * 1.0im * xs[1] .* x̂s) * (mode == 1 ? 1 : length(xs))

    return x̂s[is], (ℯⁱᵠ.*(mode == 1 ? fft(ys) : ifft(ys)))[is]
end

# ╔═╡ 8ba39052-713a-11eb-2993-8f2045a516eb
begin
    ω = range(-20.0, 20.0; length = 2^13)
    β = 10

    ρ(ω) = NCA.gaussian(ω, 0.0, 0.5)
    f(ω) = NCA.nf(ω, β)

    ts, G_ = ft(ω, im * ρ.(ω) .* f.(-ω); mode = -1)
    GG_exact = extrapolate(interpolate((ts,), -G_, Gridded(Linear())), 0)
end

# ╔═╡ a8a20078-7525-11eb-37c1-073d70915987
begin
    using AMR

    function integrate(x::AbstractVector, y::AbstractVector)
        if isone(length(x))
            return zero(first(y))
        end
        @inbounds retval = (x[2] - x[1]) * (y[1] + y[2])
        @inbounds @fastmath @simd for i = 2:(length(y)-1)
            retval += (x[i+1] - x[i]) * (y[i] + y[i+1])
        end
        return 1 // 2 * retval
    end

    (x, y) = sample_function(ρ, range(-2.0, 2.0; length = 2^4); tol = 1e-6)

    GG_amr(t, t′) = integrate(x, -1.0im * exp.(-1.0im * (t - t′) .* x) .* y .* f.(-x))
end

# ╔═╡ 9ede3c9a-7139-11eb-2db7-4bd9267ba6a0
begin
    ω_local = range(-2.0, 2.0; length = 2^9)

    GG_sum(t, t′) =
        (ω_local[2] - ω_local[1]) *
        sum(-1.0im * exp(-1.0im * (t - t′) * ω) * ρ(ω) * f(-ω) for ω in ω_local)
end

# ╔═╡ 5da544ac-713a-11eb-29b2-59d2d3d33146
begin
    t = range(0.0, 700.0; length = 10000)
    plot(t, real(GG_exact.(t)))
    plot!(t, real(GG_sum.(t, 0.0)))
    # plot!(t, GG_amr.(t, 0.0) |> real)
end

# ╔═╡ e450f60e-7527-11eb-30e6-e39f2c497a78
plot(t, real(GG_exact.(t) .- GG_sum.(t, 0.0)))

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