73 lines
2.4 KiB
Julia
73 lines
2.4 KiB
Julia
using Roots, LinearAlgebra, Plots
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include("../EC.jl")
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include("../common.jl")
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include("../p_space.jl")
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μ = 0.5
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V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q)) # ResonanceEC: Eq. (20)
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# determining c0 with EC
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temp_c = range(1.1, 0.9, 3)
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p, w = get_mesh([0, 8 - 3im], [512])
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H0 = get_T_matrix(p, μ)
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dim = size(H0, 1)
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V = get_V_matrix(V_system(1), p, w)
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EC = affine_EC(H0, V, w)
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train!(EC, temp_c; ref_eval=-0.2, CAEC=false, verbose=false)
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quick_extrapolate(c) = minimum(abs2, get_extrapolated_evals(EC.H0_EC, EC.H1_EC, EC.N_EC, c, 0))
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c0 = find_zero(quick_extrapolate, 0.85)
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# Calculation of training vectors
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training_c = range(1.2, 0.9, 9) # original: range(1.35, 0.9, 5)
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ref_E = -0.3
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order::Int = ceil((length(training_c) - 1) / 2) # order of the Pade approximant
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training_E = ComplexF64[]
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training_vecs = Vector{ComplexF64}[]
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for c in training_c
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println("Training for c = $c")
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H = H0 + c .* V
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evals, evecs = eigs(H, sigma=ref_E, maxiter=5000, ritzvec=true, check=1)
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val = nearest(evals, ref_E)
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push!(training_E, val)
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vec = evecs[:, nearestIndex(evals, val)]
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vec ./= sqrt(only(transpose(vec) * vec)) # normalize
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push!(training_vecs, vec)
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end
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training_vecs = hcat(training_vecs...)
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# Calculation of target E
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target_c = range(0.78, 0.45, 7) # original: range(0.75, 0.40, 8)
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target_E = [quick_pole_E(V_system(c)) for c in target_c]
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# Solve coefficients as a linear system
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M_left_element(c, i) = complex(c - c0)^(i/2)
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M_left = M_left_element.(training_c, (0:order)')
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a = zeros(ComplexF64, dim, order+1)
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b = zeros(ComplexF64, dim, order+1)
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for i in 1:dim
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println("Fitting coefficients $i/$dim")
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M_right = -training_vecs[i, :] .* M_left[:, 2:end] # remove the first column
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M = hcat(M_left, M_right) # M = [M_left | M_right]
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sol = M \ training_vecs[i, :]
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a[i, :] .= sol[1:order+1]
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b[i, :] .= [1; sol[order+2:end]]
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end
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# Pade approximant
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polynomial(a, c) = sum(i -> a[:, i+1] .* complex(c - c0)^(i/2), 0:order)
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pade_approx(c) = polynomial(a, c) ./ polynomial(b, c)
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# Extrapolate
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extrapolating_c = [training_c; target_c]
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extrapolated_vec = pade_approx.(extrapolating_c)
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extrapolated_E = [only(transpose(vec) * (H0 + c .* V) * vec) for (vec, c) in zip(extrapolated_vec, extrapolating_c)]
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# Plotting
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scatter(real.(training_E), imag.(training_E), label="training")
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scatter!(real.(target_E), imag.(target_E), label="target")
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scatter!(real.(extrapolated_E), imag.(extrapolated_E), label="extrapolated", m=:star5)
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