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No commits in common. "main" and "ho_basis_proper" have entirely different histories.
main
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ho_basis_p
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@ -1,3 +1,6 @@
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# probably not recommended
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Project.toml
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# Compiled FORTRAN libraries
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# Compiled FORTRAN libraries
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*.so
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*.so
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@ -8,7 +11,7 @@ hpc/*
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# VS Code files for those working on multiple tools
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# VS Code files for those working on multiple tools
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.vscode/*
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.vscode/*
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# .vscode/settings.json
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!.vscode/settings.json
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!.vscode/tasks.json
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!.vscode/tasks.json
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!.vscode/launch.json
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!.vscode/launch.json
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!.vscode/extensions.json
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!.vscode/extensions.json
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@ -0,0 +1,51 @@
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using Plots, LinearAlgebra
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include("p_space.jl")
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berggren_mesh = get_mesh((0, 0.4 - 0.15im, 0.8, 6), 128)
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csm_mesh = get_mesh((0, 8 - 3im), 512)
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for mesh in (berggren_mesh, csm_mesh)
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p, w = mesh
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mesh_E = p.*p ./ (2*0.5)
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# ResonanceEC: Eq. (20)
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V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q))
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training_points = range(1.1, 0.9, 5) # original: range(1.35, 0.9, 5)
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training_E = Vector{ComplexF64}(undef, length(training_points))
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EC_basis = Matrix{ComplexF64}(undef, length(p), length(training_points))
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for (j, c) in enumerate(training_points)
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evals, evecs = eigen(get_H_matrix(V_system(c), p, w))
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i = identify_pole_i(p, evals)
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training_E[j] = evals[i]
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EC_basis[:, j] = evecs[:, i]
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end
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EC_basis = hcat(EC_basis, conj.(EC_basis)) # CA-EC
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EC_basis_w = EC_basis .* w
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N_EC = transpose(EC_basis_w) * EC_basis
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extrapolate_points = range(0.78, 0.45, 7) # original: range(0.75, 0.40, 8)
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exact_E = Vector{ComplexF64}(undef, length(extrapolate_points))
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extrapolate_E = Vector{ComplexF64}(undef, length(extrapolate_points))
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for (j, c) in enumerate(extrapolate_points)
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exact_E[j] = quick_pole_E(V_system(c))
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H = get_H_matrix(V_system(c), p, w)
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H_EC = transpose(EC_basis_w) * H * EC_basis
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evals = eigvals(H_EC, N_EC)
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i = argmin(abs.(evals .- exact_E[j]))
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extrapolate_E[j] = evals[i]
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end
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scatter(real.(training_E), imag.(training_E), label="training")
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scatter!(real.(exact_E), imag.(exact_E), label="exact")
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scatter!(real.(extrapolate_E), imag.(extrapolate_E), label="extrapolated")
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plot!(real.(mesh_E), imag.(mesh_E), label="contour")
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xlims!(-0.3,0.3)
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ylims!(-0.120,0.020)
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savefig("temp/" * string(rand(UInt16)) * ".pdf")
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end
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173
EC.jl
173
EC.jl
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@ -1,173 +0,0 @@
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using Statistics, SparseArrays, LinearAlgebra, Arpack, Plots
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include("common.jl")
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"EC model for a Hamiltonian family H(c) = H0 + c * H1"
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mutable struct affine_EC
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H0::AbstractMatrix{ComplexF64}
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H1::AbstractMatrix{ComplexF64}
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weights::Vector{ComplexF64}
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trained::Bool
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H0_EC
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H1_EC
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N_EC
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ensemble_size::Int
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H0_EC_ensemble
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H1_EC_ensemble
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N_EC_ensemble
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training_E::Vector{ComplexF64}
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exact_E::Vector{ComplexF64}
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extrapolated_E::Vector{ComplexF64}
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extrapolated_CI::Vector{ComplexF64}
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affine_EC(H0::AbstractMatrix{ComplexF64}, H1::AbstractMatrix{ComplexF64}, weights::Vector{ComplexF64}=ones(ComplexF64, size(H0, 1)); ensemble_size=0) =
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new(H0, H1, weights, false, nothing, nothing, nothing, ensemble_size, Matrix[], Matrix[], Matrix[], ComplexF64[], ComplexF64[], ComplexF64[], ComplexF64[])
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end
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"Train an EC model for a given range of c values.
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If a list is provided for ref_eval, they are used as reference values for picking the closest eigenvalues at each sampling point.
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If a single number is provided for ref_eval, it is used as a reference for the first point, and the previous eigenvalue is used as the reference for each successive point.
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If orthonormalize_threshold > 0, Gram-Schmidt orthonormalization is performed, using this value as the threshold for dropping redundant vectors."
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function train!(EC::affine_EC, c_vals; ref_eval=-10.0, CAEC=false, gram_schmidt_threshold=0, verbose=true, tol=1e-5)
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training_vecs = Vector{ComplexF64}[]
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for c in c_vals
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verbose && println("Training for c = $c")
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global current_E
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if ref_eval isa Number
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current_E = ref_eval
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ref_eval = nothing
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elseif !isnothing(ref_eval)
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current_E = popfirst!(ref_eval)
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end
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H = EC.H0 + c .* EC.H1
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evals, evecs = eigs(H, sigma=current_E, maxiter=5000, tol=tol, ritzvec=true, check=1)
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current_E = nearest(evals, current_E)
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push!(EC.training_E, current_E)
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push!(training_vecs, evecs[:, nearestIndex(evals, current_E)])
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end
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CAEC && append!(training_vecs, conj.(training_vecs))
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(EC.H0_EC, EC.H1_EC, EC.N_EC) = get_reduced_matrices(EC, training_vecs, gram_schmidt_threshold; verbose=true)
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for _ in 1:EC.ensemble_size
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subsample = resample(length(training_vecs))
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if gram_schmidt_threshold > 0
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(H0_EC, H1_EC, N_EC) = get_reduced_matrices(EC, training_vecs, gram_schmidt_threshold, subsample; verbose=false)
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push!(EC.H0_EC_ensemble, H0_EC)
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push!(EC.H1_EC_ensemble, H1_EC)
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push!(EC.N_EC_ensemble, N_EC)
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else
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push!(EC.H0_EC_ensemble, EC.H0_EC[subsample, subsample])
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push!(EC.H1_EC_ensemble, EC.H1_EC[subsample, subsample])
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push!(EC.N_EC_ensemble, EC.N_EC[subsample, subsample])
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end
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end
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EC.trained = true
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end
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function get_reduced_matrices(EC::affine_EC, training_vecs, gram_schmidt_threshold, subsample=1:length(training_vecs); verbose=false)
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vecs = deepcopy(training_vecs[subsample])
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if gram_schmidt_threshold > 0; vecs = gram_schmidt!(vecs, EC.weights, gram_schmidt_threshold; verbose=verbose); end
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EC_basis = hcat(vecs...)
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weights_mat = spdiagm(EC.weights)
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H0_EC = transpose(EC_basis) * weights_mat * EC.H0 * EC_basis
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H1_EC = transpose(EC_basis) * weights_mat * EC.H1 * EC_basis
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N_EC = transpose(EC_basis) * weights_mat * EC_basis
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return (H0_EC, H1_EC, N_EC)
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end
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resample(n::Int) = rand(1:n, n) |> unique |> sort
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"Extrapolate using a trained EC model for a given range of c values
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If a list is provided for ref_eval, they are used as reference values for picking the closest eigenvalues at each point.
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If a single number is provided for ref_eval, it is used as a reference for the first point, and the previous eigenvalue is used as the reference for each successive point.
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If precalculated_exact_E is provided, ref_eval is ignored.
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If pseudo_inv_tol > 0, the GEVP is avoided using Moore-Penrose psuedoinverse, using this value as the relative tolerance for dropping redundant vectors."
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function extrapolate!(EC::affine_EC, c_vals; ref_eval=EC.training_E[end], pseudo_inv_tol=0, verbose=true, tol=1e-5, precalculated_exact_E=nothing)
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@assert EC.trained "EC model must be trained using train() before extrapolation"
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for c in c_vals
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global current_E
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if isnothing(precalculated_exact_E)
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if ref_eval isa Number
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current_E = ref_eval
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ref_eval = nothing
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elseif !isnothing(ref_eval)
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current_E = popfirst!(ref_eval)
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end
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verbose && println("Extact solution for c = $c")
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H = EC.H0 + c .* EC.H1
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evals, _ = eigs(H, sigma=current_E, maxiter=5000, tol=tol, ritzvec=false, check=1)
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current_E = nearest(evals, current_E)
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else
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current_E = popfirst!(precalculated_exact_E)
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end
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push!(EC.exact_E, current_E)
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verbose && println("Extrapolating for c = $c")
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evals = get_extrapolated_evals(EC.H0_EC, EC.H1_EC, EC.N_EC, c, pseudo_inv_tol)
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push!(EC.extrapolated_E, nearest(evals, current_E))
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if EC.ensemble_size > 0
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E_ensemble = zeros(ComplexF64, EC.ensemble_size)
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for i in 1:EC.ensemble_size
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evals = get_extrapolated_evals(EC.H0_EC_ensemble[i], EC.H1_EC_ensemble[i], EC.N_EC_ensemble[i], c, pseudo_inv_tol)
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E_ensemble[i] = nearest(evals, current_E)
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end
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re_CI = std(real.(E_ensemble))
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im_CI = std(imag.(E_ensemble))
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push!(EC.extrapolated_CI, complex(re_CI, im_CI))
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end
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end
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end
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"Solve the GEVP with or without Moore-Penrose psuedoinverse"
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function get_extrapolated_evals(H0_EC, H1_EC, N_EC, c, pseudo_inv_tol)
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H_EC = H0_EC + c .* H1_EC
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if pseudo_inv_tol > 0
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inv_N_EC = pinv(N_EC; atol=pseudo_inv_tol)
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H_EC = inv_N_EC * H_EC
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return eigvals(H_EC)
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else
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return eigvals(H_EC, N_EC)
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end
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end
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"Export EC data as CSV file"
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exportCSV(EC::affine_EC, filename) = exportCSV(filename, (EC.training_E, EC.exact_E, EC.extrapolated_E), ("training", "exact", "extrapolated"))
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"Plot EC data and optionally save figure to a file"
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function plot(EC::affine_EC, save_fig_filename=nothing; basis_points=nothing, basis_contour=nothing, xlims=nothing, ylims=nothing)
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scatter(real.(EC.training_E), imag.(EC.training_E), label="training")
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scatter!(real.(EC.exact_E), imag.(EC.exact_E), label="exact", markercolor=:white)
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if EC.ensemble_size > 0
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scatter!(real.(EC.extrapolated_E), imag.(EC.extrapolated_E), xerror=real.(EC.extrapolated_CI), yerror=imag.(EC.extrapolated_CI), label="extrapolated", m=:x)
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else
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scatter!(real.(EC.extrapolated_E), imag.(EC.extrapolated_E), label="extrapolated", m=:x)
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end
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isnothing(basis_points) || scatter!(real.(basis_points), imag.(basis_points), m=:x, label="basis")
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isnothing(basis_contour) || plot!(real.(basis_contour), imag.(basis_contour), label="contour")
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isnothing(xlims) || xlims!(xlims...)
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isnothing(ylims) || ylims!(ylims...)
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isnothing(save_fig_filename) || savefig(save_fig_filename)
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end
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@ -0,0 +1,100 @@
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"using Plots, LinearAlgebra\n",
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"include(\"p_space.jl\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"vertices = (0, 0.4 - 0.15im, 0.8, 6)\n",
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"subdivisions = 128\n",
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"p, w = get_mesh(vertices, subdivisions)\n",
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"mesh_E = p.*p ./ (2*0.5)\n",
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"\n",
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"# ResonanceEC: Eq. (20)\n",
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"V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q))"
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]
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},
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{
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||||||
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"training_points = range(0.75, 0.45, 5)\n",
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"training_E = Vector{ComplexF64}(undef, length(training_points))\n",
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"EC_basis = Matrix{ComplexF64}(undef, length(p), length(training_points))\n",
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"\n",
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"for (j, c) in enumerate(training_points)\n",
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" evals, evecs = eigen(get_H_matrix(V_system(c), p, w))\n",
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||||||
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" i = identify_pole_i(p, evals)\n",
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" training_E[j] = evals[i]\n",
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" EC_basis[:, j] = evecs[:, i]\n",
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"end\n",
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"\n",
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"scatter(real.(training_E), imag.(training_E), label=\"training\")\n",
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"plot!(real.(mesh_E), imag.(mesh_E), label=\"contour\")\n",
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"xlims!(0,1)"
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]
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},
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{
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"cell_type": "code",
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|
"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"extrapolate_points = range(0.40, 0.25, 5)\n",
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"ref_E = 0.2 - 0.1im\n",
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"\n",
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"exact_E = Vector{ComplexF64}(undef, length(extrapolate_points))\n",
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"extrapolate_E = Vector{ComplexF64}(undef, length(extrapolate_points))\n",
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"\n",
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"EC_basis_w = EC_basis .* w\n",
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"N_EC = transpose(EC_basis_w) * EC_basis\n",
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"\n",
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"for (j, c) in enumerate(extrapolate_points)\n",
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" exact_E[j] = quick_pole_E(V_system(c))\n",
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"\n",
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" EC_basis_w = EC_basis .* w\n",
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" H = get_H_matrix(V_system(c), p, w)\n",
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" H_EC = transpose(EC_basis_w) * H * EC_basis\n",
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" evals = eigvals(H_EC, N_EC)\n",
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" i = argmin(abs.(evals .- ref_E))\n",
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" ref_E = evals[i]\n",
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" extrapolate_E[j] = evals[i]\n",
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"end\n",
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"\n",
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"scatter(real.(training_E), imag.(training_E), label=\"training\")\n",
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"scatter!(real.(exact_E), imag.(exact_E), label=\"exact\")\n",
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||||||
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"scatter!(real.(extrapolate_E), imag.(extrapolate_E), label=\"extrapolated\")\n",
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||||||
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"plot!(real.(mesh_E), imag.(mesh_E), label=\"contour\")\n",
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"xlims!(0,1)"
|
||||||
|
]
|
||||||
|
}
|
||||||
|
],
|
||||||
|
"metadata": {
|
||||||
|
"kernelspec": {
|
||||||
|
"display_name": "Julia 1.9.0",
|
||||||
|
"language": "julia",
|
||||||
|
"name": "julia-1.9"
|
||||||
|
},
|
||||||
|
"language_info": {
|
||||||
|
"file_extension": ".jl",
|
||||||
|
"mimetype": "application/julia",
|
||||||
|
"name": "julia",
|
||||||
|
"version": "1.9.0"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"nbformat": 4,
|
||||||
|
"nbformat_minor": 2
|
||||||
|
}
|
||||||
15
Project.toml
15
Project.toml
|
|
@ -1,15 +0,0 @@
|
||||||
[deps]
|
|
||||||
Arpack = "7d9fca2a-8960-54d3-9f78-7d1dccf2cb97"
|
|
||||||
CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
|
|
||||||
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
|
|
||||||
DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
|
|
||||||
FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
|
|
||||||
HDF5 = "f67ccb44-e63f-5c2f-98bd-6dc0ccc4ba2f"
|
|
||||||
LRUCache = "8ac3fa9e-de4c-5943-b1dc-09c6b5f20637"
|
|
||||||
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
|
|
||||||
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
|
|
||||||
QuadGK = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
|
|
||||||
Roots = "f2b01f46-fcfa-551c-844a-d8ac1e96c665"
|
|
||||||
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
|
|
||||||
SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
|
|
||||||
WignerSymbols = "9f57e263-0b3d-5e2e-b1be-24f2bb48858b"
|
|
||||||
|
|
@ -0,0 +1,55 @@
|
||||||
|
using LinearAlgebra, Plots
|
||||||
|
include("ho_basis.jl")
|
||||||
|
include("p_space.jl")
|
||||||
|
|
||||||
|
μω_gen = 0.5 * exp(-1im * 0.47 * pi)
|
||||||
|
μ = 0.5
|
||||||
|
l = 0
|
||||||
|
V1 = -5
|
||||||
|
R1 = sqrt(3)
|
||||||
|
V2 = 2
|
||||||
|
R2 = sqrt(10)
|
||||||
|
n_max = 15
|
||||||
|
|
||||||
|
ns = collect(0:n_max)
|
||||||
|
ls = fill(l, n_max + 1)
|
||||||
|
|
||||||
|
T = sp_T_matrix(ns, ls; μω_gen=μω_gen, μ=μ)
|
||||||
|
V = V1 .* V_Gaussian.(R1, l, ns, transpose(ns); μω_gen=μω_gen) + V2 .* V_Gaussian.(R2, l, ns, transpose(ns); μω_gen=μω_gen)
|
||||||
|
|
||||||
|
n_EC = 8
|
||||||
|
train_cs = (0.7 .+ 0.05 * randn(n_EC)) - 1im * (0.2 .+ 0.05 * randn(n_EC))
|
||||||
|
target_cs = range(0.77, 0.22, 6)
|
||||||
|
|
||||||
|
train_E = zeros(ComplexF64, n_EC)
|
||||||
|
EC_basis = zeros(ComplexF64, (n_max + 1, length(train_cs)))
|
||||||
|
exact_E = zeros(ComplexF64, length(target_cs))
|
||||||
|
extrapolate_E = similar(exact_E)
|
||||||
|
|
||||||
|
near_E = 0.2 + 0.2im
|
||||||
|
|
||||||
|
for (j, c) in enumerate(train_cs)
|
||||||
|
H = T + c .* V
|
||||||
|
evals, evecs = eigen(H)
|
||||||
|
i = argmin(abs.(evals .- near_E))
|
||||||
|
train_E[j] = evals[i]
|
||||||
|
EC_basis[:, j] = evecs[:, i]
|
||||||
|
end
|
||||||
|
|
||||||
|
N_EC = transpose(EC_basis) * EC_basis
|
||||||
|
|
||||||
|
for (j, c) in enumerate(target_cs)
|
||||||
|
exact_E[j] = quick_pole_E((p, q) -> c*(V1*g0(R1, p, q) + V2*g0(R2, p, q)), μ; cs_angle=0.5)
|
||||||
|
|
||||||
|
H = T + c .* V
|
||||||
|
H_EC = transpose(EC_basis) * H * EC_basis
|
||||||
|
evals = eigvals(H_EC, N_EC)
|
||||||
|
i = argmin(abs.(evals .- exact_E[j]))
|
||||||
|
extrapolate_E[j] = evals[i]
|
||||||
|
end
|
||||||
|
|
||||||
|
scatter(real.(train_E), imag.(train_E), label="training")
|
||||||
|
scatter!(real.(exact_E), imag.(exact_E), label="exact")
|
||||||
|
scatter!(real.(extrapolate_E), imag.(extrapolate_E), label="extrapolated")
|
||||||
|
xlims!(-0.2,0.3)
|
||||||
|
ylims!(-0.3,0.3)
|
||||||
22
berggren.jl
22
berggren.jl
|
|
@ -1,22 +0,0 @@
|
||||||
using SparseArrays
|
|
||||||
include("math.jl")
|
|
||||||
|
|
||||||
"berg_bases1/2 are lists of (1+l_max) matrices containing the eigenbases corresponding to 1st and 2nd DOFs respectively,
|
|
||||||
js are a list of tuples (j1, j2) corresponding to 1st and 2nd DOFs respectively,
|
|
||||||
and ws are the weights needed to evaluate the inner products"
|
|
||||||
function get_2p_p1p2_matrix(mesh_size, js, Λ, berg_bases1::Vector{Matrix{ComplexF64}}, berg_bases2::Vector{Matrix{ComplexF64}}, ks::Vector{ComplexF64}, ws::Vector{ComplexF64})
|
|
||||||
# TODO: Cache / precalculate
|
|
||||||
integral1(np, lp, n, l) = sum(ks .* berg_bases1[1 + lp][:, np] .* ws .* berg_bases1[1 + l][:, n])
|
|
||||||
integral2(np, lp, n, l) = sum(ks .* berg_bases2[1 + lp][:, np] .* ws .* berg_bases2[1 + l][:, n])
|
|
||||||
|
|
||||||
basis = iter_prod(js, 1:mesh_size, 1:mesh_size)
|
|
||||||
mat = zeros(ComplexF64, length(basis), length(basis))
|
|
||||||
Threads.@threads for idx in CartesianIndices(mat)
|
|
||||||
(ip, i) = Tuple(idx)
|
|
||||||
((j1p, j2p), n1p, n2p) = basis[ip]
|
|
||||||
((j1, j2), n1, n2) = basis[i]
|
|
||||||
val = racahs_reduction_formula(n1p, j1p, n2p, j2p, n1, j1, n2, j2, Λ, integral1, integral2)
|
|
||||||
if !(val ≈ 0); mat[idx] = val; end
|
|
||||||
end
|
|
||||||
return sparse(mat)
|
|
||||||
end
|
|
||||||
|
|
@ -1,75 +0,0 @@
|
||||||
using LinearAlgebra, SparseArrays, Arpack
|
|
||||||
include("common.jl")
|
|
||||||
include("p_space.jl")
|
|
||||||
include("ho_basis.jl")
|
|
||||||
include("berggren.jl")
|
|
||||||
|
|
||||||
println("No of threads = ", Threads.nthreads())
|
|
||||||
atol = 10^-5
|
|
||||||
maxevals = 10^5
|
|
||||||
R_cutoff = 16
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
μ = m/2 # due to simple relative coordinates
|
|
||||||
|
|
||||||
target = 4.0766890719636875 - 0.012758927741074495im
|
|
||||||
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
V_l(j, k, kp) = Vl_mat_elem(V_of_r, j, k, kp; atol=atol, maxevals=maxevals, R_cutoff=R_cutoff)
|
|
||||||
|
|
||||||
vertices = [0, 2 - 0.2im, 3, 4]
|
|
||||||
subdivisions = [15, 10, 10]
|
|
||||||
ks, ws = get_mesh(vertices, subdivisions)
|
|
||||||
|
|
||||||
jmax = 4
|
|
||||||
tri((j1, j2)) = triangle_ineq(j1, j2, Λ)
|
|
||||||
js = collect(Iterators.filter(tri, iter_prod(0:jmax, 0:jmax)))
|
|
||||||
|
|
||||||
basis = iter_prod(js, zip(ks, ws), zip(ks, ws)) # basis = ((j1, j2), (k1, w1), (k2, w2))
|
|
||||||
basis_size = length(js) * length(ks)^2
|
|
||||||
@assert length(basis) == basis_size "Something wrong with the basis"
|
|
||||||
println("Basis size = $basis_size")
|
|
||||||
|
|
||||||
# generate Berggren bases
|
|
||||||
@time "berg_bases" begin
|
|
||||||
berg_bases = Vector{Matrix{ComplexF64}}(undef, jmax + 1)
|
|
||||||
for j in 0:jmax
|
|
||||||
_, berg_basis = eigen(get_H_matrix((k, kp) -> V_l(j, k, kp), ks, ws, μ); permute=false, scale=false)
|
|
||||||
N_berg = sum(berg_basis.^2 .* ws, dims=1)
|
|
||||||
berg_bases[1 + j] = berg_basis ./ transpose(sqrt.(N_berg))
|
|
||||||
end
|
|
||||||
to_berg_basis(mat, j) = transpose(berg_bases[1 + j] .* ws) * mat * berg_bases[1 + j]
|
|
||||||
end
|
|
||||||
|
|
||||||
@time "U_berggren" begin
|
|
||||||
U_blocks = [kron(berg_bases[1 + j1], berg_bases[1 + j2]) for (j1, j2) in js]
|
|
||||||
U = blockdiag(sparse.(U_blocks)...)
|
|
||||||
end
|
|
||||||
|
|
||||||
@time "Block diagonal part" begin
|
|
||||||
Hb_blocks = [kron_sum(to_berg_basis(get_H_matrix((k, kp) -> V_l(j1, k, kp), ks, ws, μ), j1), to_berg_basis(get_H_matrix((k, kp) -> V_l(j2, k, kp), ks, ws, μ), j2)) for (j1, j2) in js]
|
|
||||||
Hb = blockdiag(sparse.(Hb_blocks)...)
|
|
||||||
end
|
|
||||||
|
|
||||||
@time "T_cross" T_cross = get_2p_p1p2_matrix(length(ks), js, Λ, berg_bases, berg_bases, ks, ws) ./ (2*μ)
|
|
||||||
|
|
||||||
E_max = 30
|
|
||||||
μω_global = 0.5
|
|
||||||
# due to simple relative coordinates
|
|
||||||
μω = μω_global * 2
|
|
||||||
μ = m/2
|
|
||||||
|
|
||||||
basis_ho = ho_basis_2B(E_max, Λ)
|
|
||||||
|
|
||||||
@time "V12_HO" V12_HO = get_src_V12_matrix(V_of_r, basis_ho, μω_global; atol=10^-6, maxevals=10^5)
|
|
||||||
|
|
||||||
@time "W" W = get_W_matrix(basis, basis_ho, μω, μω; weights=true)
|
|
||||||
|
|
||||||
@time "V12_p" V12_p = W * V12_HO * transpose(W)
|
|
||||||
@time "V12" V12 = transpose(U) * V12_p * U
|
|
||||||
|
|
||||||
@time "H" H = Hb + T_cross + V12
|
|
||||||
|
|
||||||
@time "Eigenvalues" evals, _ = eigs(H, sigma=target, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
|
||||||
display(evals)
|
|
||||||
|
|
@ -1,33 +0,0 @@
|
||||||
using Roots, DelimitedFiles
|
|
||||||
|
|
||||||
include("../EC.jl")
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
μ = 0.5
|
|
||||||
V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q)) # ResonanceEC: Eq. (20)
|
|
||||||
|
|
||||||
# determining c0 with EC
|
|
||||||
temp_c = range(1.1, 0.9, 3)
|
|
||||||
p, w = get_mesh([0, 8], [256])
|
|
||||||
H0 = get_T_matrix(p, μ)
|
|
||||||
V = get_V_matrix(V_system(1), p, w)
|
|
||||||
EC = affine_EC(H0, V, w)
|
|
||||||
train!(EC, temp_c; ref_eval=-0.2, CAEC=false, verbose=false)
|
|
||||||
quick_extrapolate(c) = minimum(abs2, get_extrapolated_evals(EC.H0_EC, EC.H1_EC, EC.N_EC, c, 0))
|
|
||||||
c0 = find_zero(quick_extrapolate, 0.85)
|
|
||||||
|
|
||||||
training_c = range(1.2, 0.9, 9) # original: range(1.35, 0.9, 5)
|
|
||||||
extrapolating_c = range(0.78, 0.45, 7) # original: range(0.75, 0.40, 8)
|
|
||||||
|
|
||||||
data_c = vcat(training_c, extrapolating_c)
|
|
||||||
data_E = [quick_pole_E(V_system(c)) for c in data_c]
|
|
||||||
|
|
||||||
# export to CSV
|
|
||||||
file = "temp/2body_data.csv"
|
|
||||||
delim = ','
|
|
||||||
open(file, "w") do f
|
|
||||||
writedlm(f, ["c" "re_E" "im_E"], delim)
|
|
||||||
writedlm(f, [c0 0 0], delim) # first entry for the threshold
|
|
||||||
writedlm(f, hcat(data_c, real.(data_E), imag.(data_E)), delim)
|
|
||||||
end
|
|
||||||
|
|
@ -1,40 +0,0 @@
|
||||||
include("../p_space.jl")
|
|
||||||
include("../EC.jl")
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
|
|
||||||
ϕ = 0.1
|
|
||||||
vertices = [0, 4 * exp(-1im * ϕ), 5, 6]
|
|
||||||
subdivisions = [40, 12, 15]
|
|
||||||
jmax = 6
|
|
||||||
|
|
||||||
E_max = 40
|
|
||||||
μω_global = 0.5
|
|
||||||
|
|
||||||
H0, weights = get_3b_H_matrix(jacobi, V_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m)
|
|
||||||
|
|
||||||
# Vp = perturbation to make the state artificially bound
|
|
||||||
Vp_of_r(r) = -exp(-(r/3)^2)
|
|
||||||
Vp, _ = get_3b_H_matrix(jacobi, Vp_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
training_c = [2.6, 2.4, 2.2, 2.0, 1.8]
|
|
||||||
extrapolating_c = 0.0 : 0.2 : 1.2
|
|
||||||
|
|
||||||
training_ref = -2.22 # complete list not needed because identification is simple
|
|
||||||
|
|
||||||
extrapolating_ref = [4.076662025307587-0.012709842443350328im,
|
|
||||||
3.613318119833891-0.007335804709990623im,
|
|
||||||
3.1453431847006783-0.004030580410326795im,
|
|
||||||
2.672967129943755-0.00211498327461944im,
|
|
||||||
2.196542557810288-0.0010719835443437104im,
|
|
||||||
1.7164583929199813-0.0005455212208182736im,
|
|
||||||
1.233088227541505-0.0003070320106485624im]
|
|
||||||
|
|
||||||
EC = affine_EC(H0, Vp, weights)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=true)
|
|
||||||
extrapolate!(EC, extrapolating_c; ref_eval=extrapolating_ref)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/Berggren_B2R.csv")
|
|
||||||
plot(EC, "temp/Berggren_B2R.pdf")
|
|
||||||
|
|
@ -1,42 +0,0 @@
|
||||||
include("../p_space.jl")
|
|
||||||
include("../EC.jl")
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
|
|
||||||
vertices = [0, 2 - 0.2im, 3, 4]
|
|
||||||
subdivisions = [15, 10, 10]
|
|
||||||
jmax = 4
|
|
||||||
|
|
||||||
E_max = 40
|
|
||||||
μω_global = 0.5
|
|
||||||
|
|
||||||
H0, weights = get_3b_H_matrix(jacobi, V_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m)
|
|
||||||
|
|
||||||
# Vp = perturbation to make the state artificially bound
|
|
||||||
Vp_of_r(r) = -exp(-(r/3)^2)
|
|
||||||
Vp, _ = get_3b_H_matrix(jacobi, Vp_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
training_c = [1.1, 0.9, 0.7, 0.5]
|
|
||||||
extrapolating_c = 0.0 : 0.2 : 1.2
|
|
||||||
|
|
||||||
training_ref = [1.4750633616275919 - 0.0003021770706749637im
|
|
||||||
1.9567078295375822 - 0.0007646829108872369im
|
|
||||||
2.4351117758403076 - 0.001281037843108658im
|
|
||||||
2.9096543462392357 - 0.002962488527470604im]
|
|
||||||
|
|
||||||
extrapolating_ref = [4.076662025307587-0.012709842443350328im,
|
|
||||||
3.613318119833891-0.007335804709990623im,
|
|
||||||
3.1453431847006783-0.004030580410326795im,
|
|
||||||
2.672967129943755-0.00211498327461944im,
|
|
||||||
2.196542557810288-0.0010719835443437104im,
|
|
||||||
1.7164583929199813-0.0005455212208182736im,
|
|
||||||
1.233088227541505-0.0003070320106485624im]
|
|
||||||
|
|
||||||
EC = affine_EC(H0, Vp, weights)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=false)
|
|
||||||
extrapolate!(EC, extrapolating_c; ref_eval=extrapolating_ref)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/Berggren_R2R.csv")
|
|
||||||
plot(EC, "temp/Berggren_R2R.pdf")
|
|
||||||
|
|
@ -1,40 +0,0 @@
|
||||||
include("../p_space.jl")
|
|
||||||
include("../EC.jl")
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
|
|
||||||
ϕ = 0.1
|
|
||||||
vertices = [0, 6 * exp(-1im * ϕ)]
|
|
||||||
subdivisions = [50]
|
|
||||||
jmax = 4
|
|
||||||
|
|
||||||
E_max = 40
|
|
||||||
μω_global = 0.5
|
|
||||||
|
|
||||||
H0, weights = get_3b_H_matrix(jacobi, V_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m)
|
|
||||||
|
|
||||||
# Vp = perturbation to make the state artificially bound
|
|
||||||
Vp_of_r(r) = -exp(-(r/3)^2)
|
|
||||||
Vp, _ = get_3b_H_matrix(jacobi, Vp_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
training_c = [2.6, 2.4, 2.2, 2.0, 1.8]
|
|
||||||
extrapolating_c = 0.0 : 0.2 : 1.2
|
|
||||||
|
|
||||||
training_ref = -2.22 # complete list not needed because identification is simple
|
|
||||||
|
|
||||||
exact_E = [4.076662025307587-0.012709842443350328im,
|
|
||||||
3.613318119833891-0.007335804709990623im,
|
|
||||||
3.1453431847006783-0.004030580410326795im,
|
|
||||||
2.672967129943755-0.00211498327461944im,
|
|
||||||
2.196542557810288-0.0010719835443437104im,
|
|
||||||
1.7164583929199813-0.0005455212208182736im,
|
|
||||||
1.233088227541505-0.0003070320106485624im]
|
|
||||||
|
|
||||||
EC = affine_EC(H0, Vp, weights)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=true)
|
|
||||||
extrapolate!(EC, extrapolating_c; precalculated_exact_E=exact_E)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/CSM_B2R.csv")
|
|
||||||
plot(EC, "temp/CSM_B2R.pdf")
|
|
||||||
|
|
@ -1,68 +0,0 @@
|
||||||
using Roots, LinearAlgebra, Plots
|
|
||||||
|
|
||||||
include("../EC.jl")
|
|
||||||
include("../common.jl")
|
|
||||||
include("../ho_basis.jl")
|
|
||||||
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
|
|
||||||
ϕ = 0.1
|
|
||||||
μω_global = 0.5 * exp(-2im * ϕ)
|
|
||||||
E_max = 40
|
|
||||||
|
|
||||||
H0 = get_3b_H_matrix(jacobi, V_of_r, μω_global, E_max, Λ, m, true, true)
|
|
||||||
|
|
||||||
# Vp = perturbation to make the state artificially bound
|
|
||||||
Vp_of_r(r) = -exp(-(r/3)^2)
|
|
||||||
@time "Vp" Vp = get_3b_H_matrix(jacobi, Vp_of_r, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
training_ref = -2.22
|
|
||||||
extrapolating_ref = [4.076662025307587-0.012709842443350328im,
|
|
||||||
3.613318119833891-0.007335804709990623im,
|
|
||||||
3.1453431847006783-0.004030580410326795im,
|
|
||||||
2.672967129943755-0.00211498327461944im,
|
|
||||||
2.196542557810288-0.0010719835443437104im,
|
|
||||||
1.7164583929199813-0.0005455212208182736im,
|
|
||||||
1.233088227541505-0.0003070320106485624im]
|
|
||||||
|
|
||||||
training_c = range(2.8, 1.8, 5)
|
|
||||||
extrapolating_c = 0.0 : 0.2 : 1.2
|
|
||||||
|
|
||||||
EC = affine_EC(H0, Vp)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=true)
|
|
||||||
extrapolate!(EC, extrapolating_c; ref_eval=extrapolating_ref)
|
|
||||||
|
|
||||||
# determining c0 with EC
|
|
||||||
approx_c0 = 1.5
|
|
||||||
quick_extrapolate(c) = minimum(abs2, get_extrapolated_evals(EC.H0_EC, EC.H1_EC, EC.N_EC, c, 1e-14))
|
|
||||||
c0 = find_zero(quick_extrapolate, approx_c0)
|
|
||||||
|
|
||||||
order::Int = ceil((length(training_c) - 1) / 2) # order of the Pade approximant
|
|
||||||
|
|
||||||
# Solve coefficients as a linear system
|
|
||||||
training_k = alt_sqrt.(EC.training_E)
|
|
||||||
M_left_element(c, i) = alt_sqrt(c - c0)^i
|
|
||||||
M_left = M_left_element.(training_c, (0:order)')
|
|
||||||
M_right = -training_k .* M_left[:, 2:end] # remove the first column
|
|
||||||
M = hcat(M_left, M_right) # M = [M_left | M_right]
|
|
||||||
sol = M \ training_k
|
|
||||||
a = sol[1:order+1]
|
|
||||||
b = [1; sol[order+2:end]]
|
|
||||||
|
|
||||||
# Pade approximant
|
|
||||||
polynomial(a, c) = sum(i -> a[i+1] * alt_sqrt(c - c0)^i, 0:order)
|
|
||||||
pade_approx(c) = polynomial(a, c) / polynomial(b, c)
|
|
||||||
|
|
||||||
# Extrapolate
|
|
||||||
extrapolated_k = pade_approx.([extrapolating_c; training_c])
|
|
||||||
extrapolated_E = extrapolated_k .^ 2
|
|
||||||
|
|
||||||
# Plotting
|
|
||||||
scatter(real.(EC.training_E), imag.(EC.training_E), label="training")
|
|
||||||
scatter!(real.(EC.exact_E), imag.(EC.exact_E), label="exact")
|
|
||||||
scatter!(real.(EC.extrapolated_E), imag.(EC.extrapolated_E), label="CAEC", m=:x)
|
|
||||||
scatter!(real.(extrapolated_E), imag.(extrapolated_E), label="ACCC", m=:+)
|
|
||||||
title!("3-body extrapolation with $(length(training_c)) training points")
|
|
||||||
savefig("temp/3body_HO_B2R_ACCC-$(length(training_c)).pdf")
|
|
||||||
|
|
@ -1,36 +0,0 @@
|
||||||
include("../ho_basis.jl")
|
|
||||||
include("../EC.jl")
|
|
||||||
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
|
|
||||||
ϕ = 0.1
|
|
||||||
μω_global = 0.5 * exp(-2im * ϕ)
|
|
||||||
E_max = 40
|
|
||||||
|
|
||||||
H0 = get_3b_H_matrix(jacobi, V_of_r, μω_global, E_max, Λ, m, true, true)
|
|
||||||
|
|
||||||
# Vp = perturbation to make the state artificially bound
|
|
||||||
Vp_of_r(r) = -exp(-(r/3)^2)
|
|
||||||
@time "Vp" Vp = get_3b_H_matrix(jacobi, Vp_of_r, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
training_ref = -2.22
|
|
||||||
|
|
||||||
extrapolating_ref = [4.076662025307587-0.012709842443350328im,
|
|
||||||
3.613318119833891-0.007335804709990623im,
|
|
||||||
3.1453431847006783-0.004030580410326795im,
|
|
||||||
2.672967129943755-0.00211498327461944im,
|
|
||||||
2.196542557810288-0.0010719835443437104im,
|
|
||||||
1.7164583929199813-0.0005455212208182736im,
|
|
||||||
1.233088227541505-0.0003070320106485624im]
|
|
||||||
|
|
||||||
training_c = [2.6, 2.4, 2.2, 2.0, 1.8]
|
|
||||||
extrapolating_c = 0.0 : 0.2 : 1.2
|
|
||||||
|
|
||||||
EC = affine_EC(H0, Vp)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=true)
|
|
||||||
extrapolate!(EC, extrapolating_c; ref_eval=extrapolating_ref)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/HO_B2R.csv")
|
|
||||||
plot(EC, "temp/HO_B2R.pdf")
|
|
||||||
|
|
@ -1,24 +0,0 @@
|
||||||
include("../ho_basis.jl")
|
|
||||||
include("../EC.jl")
|
|
||||||
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
|
|
||||||
μω_global = 0.5 * exp(-2im * pi / 9)
|
|
||||||
E_max = 40
|
|
||||||
|
|
||||||
T = get_3b_H_matrix(src, V_of_r, μω_global, E_max, Λ, m, true, false)
|
|
||||||
V = get_3b_H_matrix(src, V_of_r, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
ref_E = 5.9673 - 0.0006im
|
|
||||||
|
|
||||||
training_c = 2.0 : -0.2 : 1.2
|
|
||||||
extrapolating_c = 1.05 .- [0.0 : 0.1 : 0.4; 0.45 : 0.05 : 0.60]
|
|
||||||
|
|
||||||
EC = affine_EC(T, V)
|
|
||||||
train!(EC, training_c; ref_eval=ref_E, CAEC=false)
|
|
||||||
extrapolate!(EC, extrapolating_c)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/HO_R2R.csv")
|
|
||||||
plot(EC, "temp/HO_R2R.pdf")
|
|
||||||
|
|
@ -1,77 +0,0 @@
|
||||||
include("../p_space.jl")
|
|
||||||
include("../EC.jl")
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
|
|
||||||
# Distinguishable particles: V12 = bound and V13 & V23 = resonant
|
|
||||||
Vsubsystem_of_r(r) = -2 * exp(-r^2/4)
|
|
||||||
Vdecay_of_r(r) = -exp(-r^2 / 3) + exp(-r^2 / 10)
|
|
||||||
|
|
||||||
ϕ = 0.1
|
|
||||||
vertices = [0, 6 * exp(-1im * ϕ)]
|
|
||||||
subdivisions = [50]
|
|
||||||
jmax = 4
|
|
||||||
|
|
||||||
E_max = 40
|
|
||||||
μω_global = 0.4
|
|
||||||
|
|
||||||
# Jacobi coordinates
|
|
||||||
μ1ω1 = μω_global * 1/2
|
|
||||||
μ2ω2 = μω_global * 2
|
|
||||||
μ1 = m * 1/2
|
|
||||||
μ2 = m * 2/3
|
|
||||||
|
|
||||||
atol=10^-5; maxevals=10^5; R_cutoff=16; verbose=true;
|
|
||||||
|
|
||||||
###########
|
|
||||||
|
|
||||||
verbose && println("No of threads = ", Threads.nthreads())
|
|
||||||
|
|
||||||
V_l(j, k, kp) = Vl_mat_elem(Vsubsystem_of_r, j, k, kp; atol=atol, maxevals=maxevals, R_cutoff=R_cutoff)
|
|
||||||
|
|
||||||
ks, ws = get_mesh(vertices, subdivisions)
|
|
||||||
weights = repeat(kron(ws, ws), jmax + 1)
|
|
||||||
block_size = length(ks)
|
|
||||||
tri((j1, j2)) = triangle_ineq(j1, j2, Λ)
|
|
||||||
js = collect(Iterators.filter(tri, iter_prod(0:jmax, 0:jmax)))
|
|
||||||
basis = iter_prod(js, zip(ks, ws), zip(ks, ws)) # basis = ((j1, j2), (k1, w1), (k2, w2))
|
|
||||||
basis_size = length(js) * length(ks)^2
|
|
||||||
@assert length(basis) == basis_size "Something wrong with the basis"
|
|
||||||
verbose && println("Basis size = $basis_size")
|
|
||||||
|
|
||||||
@time "Block diagonal part" begin
|
|
||||||
blocks = [kron_sum(get_H_matrix((k, kp) -> V_l(j1, k, kp), ks, ws, μ1), get_T_matrix(ks, μ2)) for (j1, _) in js]
|
|
||||||
Ha = blockdiag(sparse.(blocks)...)
|
|
||||||
end
|
|
||||||
|
|
||||||
basis_ho = ho_basis_2B(E_max, Λ)
|
|
||||||
verbose && println("HO basis size = ", basis_ho.dim)
|
|
||||||
|
|
||||||
@time "V2_HO" V2_HO = get_jacobi_V2_matrix(Vdecay_of_r, basis_ho, μω_global)
|
|
||||||
|
|
||||||
@time "W_right" W_right = get_W_matrix(basis, basis_ho, μ1ω1, μ2ω2; weights=true)
|
|
||||||
@time "W_left" W_left = get_W_matrix(basis, basis_ho, μ1ω1, μ2ω2; weights=false)
|
|
||||||
|
|
||||||
@time "V2" Vb = W_left * V2_HO * transpose(W_right)
|
|
||||||
|
|
||||||
###########
|
|
||||||
|
|
||||||
training_c = [-0.55, -0.7, -0.85, -1, -1.2]
|
|
||||||
extrapolating_c = [0.2, 0.1, 0.0, -0.1, -0.2, -0.3]
|
|
||||||
|
|
||||||
ref_E = -0.5173809356244544
|
|
||||||
|
|
||||||
exact_ref = [-0.31360746615280954-0.07689284936870341im
|
|
||||||
-0.3233372403877718-0.06011323914565665im
|
|
||||||
-0.339615582074795-0.04239442037174759im
|
|
||||||
-0.36333816534241997-0.02648721825958402im
|
|
||||||
-0.39376650561322885-0.014382935339817332im
|
|
||||||
-0.4299479825535172-0.006510710745123606im]
|
|
||||||
|
|
||||||
EC = affine_EC(Ha, Vb)
|
|
||||||
train!(EC, training_c; ref_eval=ref_E, CAEC=true)
|
|
||||||
extrapolate!(EC, extrapolating_c; precalculated_exact_E=exact_ref)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/dis_CSM_B2R.csv")
|
|
||||||
plot(EC, "temp/dis_CSM_B2R.pdf")
|
|
||||||
|
|
@ -1,52 +0,0 @@
|
||||||
include("../ho_basis.jl")
|
|
||||||
include("../EC.jl")
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
|
|
||||||
# Distinguishable particles: V12 = bound and V13 & V23 = resonant
|
|
||||||
Va_of_r(r) = -2 * exp(-r^2/4)
|
|
||||||
Vb_of_r(r) = -exp(-r^2 / 3) + exp(-r^2 / 10)
|
|
||||||
|
|
||||||
E_max = 40
|
|
||||||
μω_global = 0.4 * exp(-2im * pi / 9)
|
|
||||||
|
|
||||||
# due to Jacobi coordinates
|
|
||||||
μ1ω1 = μω_global * 1/2
|
|
||||||
μ2ω2 = μω_global * 2
|
|
||||||
μ1 = m * 1/2
|
|
||||||
μ2 = m * 2/3
|
|
||||||
|
|
||||||
println("No of threads = ", Threads.nthreads())
|
|
||||||
|
|
||||||
basis = ho_basis_2B(E_max, Λ)
|
|
||||||
println("Basis size = ", basis.dim)
|
|
||||||
|
|
||||||
@time "T1" T1 = get_sp_T_matrix(basis.n1s, basis.l1s, [basis.n2s, basis.l2s]; μω_gen=μ1ω1, μ=μ1)
|
|
||||||
@time "T2" T2 = get_sp_T_matrix(basis.n2s, basis.l2s, [basis.n1s, basis.l1s]; μω_gen=μ2ω2, μ=μ2)
|
|
||||||
|
|
||||||
@time "Va" Va = get_jacobi_V1_matrix(Va_of_r, basis, μ1ω1)
|
|
||||||
@time "Vb" Vb = get_jacobi_V2_matrix(Vb_of_r, basis, μω_global)
|
|
||||||
|
|
||||||
@time "Ha" Ha = T1 + T2 + Va
|
|
||||||
@time "Eigenvalues" target_evals, _ = eigs(Ha, nev=5, ncv=50, which=:SR, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
|
||||||
display(target_evals)
|
|
||||||
|
|
||||||
training_c = [-0.55, -0.7, -0.85, -1, -1.2]
|
|
||||||
extrapolating_c = [0.2, 0.1, 0.0, -0.1, -0.2, -0.3]
|
|
||||||
|
|
||||||
ref_E = -0.5173809356244544
|
|
||||||
|
|
||||||
exact_ref = [-0.3136074661528041-0.07689284936868852im
|
|
||||||
-0.323337240387771-0.06011323914564878im
|
|
||||||
-0.33961558207479553-0.04239442037174764im
|
|
||||||
-0.3633381653424224-0.026487218259589693im
|
|
||||||
-0.393766505613234-0.014382935339825854im
|
|
||||||
-0.42994798255352606-0.006510710745131777im]
|
|
||||||
|
|
||||||
EC = affine_EC(Ha, Vb)
|
|
||||||
train!(EC, training_c; ref_eval=ref_E, CAEC=true) # try CAEC=false !!!
|
|
||||||
extrapolate!(EC, extrapolating_c, precalculated_exact_E = exact_ref)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/dis_HO_B2R.csv")
|
|
||||||
plot(EC, "temp/dis_HO_B2R.pdf")
|
|
||||||
|
|
@ -1,50 +0,0 @@
|
||||||
using Roots, LinearAlgebra, Plots
|
|
||||||
|
|
||||||
include("../EC.jl")
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
μ = 0.5
|
|
||||||
V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q)) # ResonanceEC: Eq. (20)
|
|
||||||
|
|
||||||
# determining c0 with EC
|
|
||||||
temp_c = range(1.1, 0.9, 3)
|
|
||||||
p, w = get_mesh([0, 8], [256])
|
|
||||||
H0 = get_T_matrix(p, μ)
|
|
||||||
V = get_V_matrix(V_system(1), p, w)
|
|
||||||
EC = affine_EC(H0, V, w)
|
|
||||||
train!(EC, temp_c; ref_eval=-0.2, CAEC=false, verbose=false)
|
|
||||||
quick_extrapolate(c) = minimum(abs2, get_extrapolated_evals(EC.H0_EC, EC.H1_EC, EC.N_EC, c, 0))
|
|
||||||
c0 = find_zero(quick_extrapolate, 0.85)
|
|
||||||
|
|
||||||
# Calculation of training and extrapolating E
|
|
||||||
training_c = range(1.2, 0.9, 9) # original: range(1.35, 0.9, 5)
|
|
||||||
training_E = [quick_pole_E(V_system(c)) for c in training_c]
|
|
||||||
training_k = alt_sqrt.(2μ .* training_E)
|
|
||||||
|
|
||||||
extrapolating_c = range(0.78, 0.45, 7) # original: range(0.75, 0.40, 8)
|
|
||||||
exact_E = [quick_pole_E(V_system(c)) for c in extrapolating_c]
|
|
||||||
|
|
||||||
order::Int = ceil((length(training_c) - 1) / 2) # order of the Pade approximant
|
|
||||||
|
|
||||||
# Solve coefficients as a linear system
|
|
||||||
M_left_element(c, i) = alt_sqrt(c - c0)^i
|
|
||||||
M_left = M_left_element.(training_c, (0:order)')
|
|
||||||
M_right = -training_k .* M_left[:, 2:end] # remove the first column
|
|
||||||
M = hcat(M_left, M_right) # M = [M_left | M_right]
|
|
||||||
sol = M \ training_k
|
|
||||||
a = sol[1:order+1]
|
|
||||||
b = [1; sol[order+2:end]]
|
|
||||||
|
|
||||||
# Pade approximant
|
|
||||||
polynomial(a, c) = sum(i -> a[i+1] * alt_sqrt(c - c0)^i, 0:order)
|
|
||||||
pade_approx(c) = polynomial(a, c) / polynomial(b, c)
|
|
||||||
|
|
||||||
# Extrapolate
|
|
||||||
extrapolated_k = pade_approx.([training_c; extrapolating_c])
|
|
||||||
extrapolated_E = (extrapolated_k .^ 2) / (2μ)
|
|
||||||
|
|
||||||
# Plotting
|
|
||||||
scatter(real.(training_E), imag.(training_E), label="training")
|
|
||||||
scatter!(real.(exact_E), imag.(exact_E), label="exact")
|
|
||||||
scatter!(real.(extrapolated_E), imag.(extrapolated_E), label="extrapolated", m=:star5)
|
|
||||||
|
|
@ -1,33 +0,0 @@
|
||||||
include("../EC.jl")
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
# contour
|
|
||||||
p, w = get_mesh([0, 0.4 - 0.15im, 0.8, 6], [128, 128, 128])
|
|
||||||
|
|
||||||
μ = 0.5
|
|
||||||
V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q)) # ResonanceEC: Eq. (20)
|
|
||||||
|
|
||||||
# generating a Berggren basis with a pole using the same system
|
|
||||||
basis_c = 0.6
|
|
||||||
basis_E, berg_basis = eigen(get_H_matrix(V_system(basis_c), p, w); permute=false, scale=false)
|
|
||||||
basis_p = sqrt.(basis_E)
|
|
||||||
N_berg = sqrt.(diag(transpose(berg_basis .* w) * berg_basis))
|
|
||||||
berg_basis = berg_basis ./ transpose(N_berg)
|
|
||||||
berg_basis_w = berg_basis .* w
|
|
||||||
|
|
||||||
H0 = transpose(berg_basis_w) * get_T_matrix(p, μ) * berg_basis
|
|
||||||
V = transpose(berg_basis_w) * get_V_matrix(V_system(1), p, w) * berg_basis
|
|
||||||
|
|
||||||
training_c = range(1.1, 0.9, 5) # original: range(1.35, 0.9, 5)
|
|
||||||
extrapolating_c = range(0.78, 0.45, 7) # original: range(0.75, 0.40, 8)
|
|
||||||
|
|
||||||
training_ref = -0.26
|
|
||||||
extrapolating_ref = [quick_pole_E(V_system(c)) for c in extrapolating_c]
|
|
||||||
|
|
||||||
EC = affine_EC(H0, V)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=true)
|
|
||||||
extrapolate!(EC, extrapolating_c; ref_eval=extrapolating_ref)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/2b_GSM_B2R.csv")
|
|
||||||
plot(EC, "temp/2b_GSM_B2R.pdf"; basis_points=basis_E, xlims=(0, 0.3), ylims=(-0.120, 0.020))
|
|
||||||
|
|
@ -1,30 +0,0 @@
|
||||||
include("../EC.jl")
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
berggren_mesh = get_mesh([0, 0.4 - 0.15im, 0.8, 6], [128, 128, 128])
|
|
||||||
csm_mesh = get_mesh([0, 8 - 3im], [512])
|
|
||||||
|
|
||||||
for (mesh, name) in zip((berggren_mesh, csm_mesh), ("beggren", "csm"))
|
|
||||||
p, w = mesh
|
|
||||||
mesh_E = p.*p ./ (2*0.5)
|
|
||||||
|
|
||||||
μ = 0.5
|
|
||||||
V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q)) # ResonanceEC: Eq. (20)
|
|
||||||
|
|
||||||
H0 = get_T_matrix(p, μ)
|
|
||||||
V = get_V_matrix(V_system(1), p, w)
|
|
||||||
|
|
||||||
training_c = range(1.1, 0.9, 5) # original: range(1.35, 0.9, 5)
|
|
||||||
extrapolating_c = range(0.78, 0.45, 7) # original: range(0.75, 0.40, 8)
|
|
||||||
|
|
||||||
training_ref = [quick_pole_E(V_system(c)) for c in training_c]
|
|
||||||
exact_E = [quick_pole_E(V_system(c)) for c in extrapolating_c]
|
|
||||||
|
|
||||||
EC = affine_EC(H0, V, w)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=true)
|
|
||||||
extrapolate!(EC, extrapolating_c; precalculated_exact_E=exact_E)
|
|
||||||
|
|
||||||
#exportCSV(EC, "temp/2b_comparison_$name.csv")
|
|
||||||
plot(EC, "temp/2b_comparison_$name.pdf"; basis_contour=mesh_E, xlims=(-0.3,0.3), ylims=(-0.120,0.020))
|
|
||||||
end
|
|
||||||
|
|
@ -1,81 +0,0 @@
|
||||||
#%%
|
|
||||||
import pandas as pd
|
|
||||||
import torch
|
|
||||||
import numpy as np
|
|
||||||
|
|
||||||
#%%
|
|
||||||
df = pd.read_csv('../temp/2body_data.csv').sort_values(by='c')
|
|
||||||
df['E'] = df['re_E'] + 1j * df['im_E']
|
|
||||||
train_data = df[df['re_E'] < 0]
|
|
||||||
target_data = df[df['re_E'] > 0]
|
|
||||||
|
|
||||||
train_cs = train_data['c'].to_numpy()
|
|
||||||
train_Es = torch.tensor(train_data['E'].to_numpy(), dtype=torch.complex128)
|
|
||||||
|
|
||||||
#%%
|
|
||||||
# hyperparameters
|
|
||||||
N = 9
|
|
||||||
|
|
||||||
# initialize random Hamiltonians
|
|
||||||
H0 = torch.randn(N, N, dtype=torch.complex128)
|
|
||||||
H0 = (H0 + torch.transpose(H0, 0, 1)).requires_grad_() # symmetric
|
|
||||||
H1 = torch.randn(N, N, dtype=torch.complex128)
|
|
||||||
H1 = (H1 + torch.transpose(H1, 0, 1)).requires_grad_() # symmetric
|
|
||||||
|
|
||||||
#%%
|
|
||||||
# training
|
|
||||||
|
|
||||||
# generate a set of c values to follow by subdividing the training cs
|
|
||||||
subdivisions = 3
|
|
||||||
c_steps = np.concatenate([np.linspace(start, stop, subdivisions, endpoint=False) for (start, stop) in zip(train_cs, train_cs[1:])])
|
|
||||||
c_steps = np.append(c_steps, train_cs[-1])
|
|
||||||
|
|
||||||
lr = 0.05
|
|
||||||
epochs = 100000
|
|
||||||
for epoch in range(epochs):
|
|
||||||
Es = torch.empty(len(train_data), dtype=torch.complex128)
|
|
||||||
current_E = 0.0 # start at the threshold
|
|
||||||
for c in c_steps:
|
|
||||||
H = H0 + c * H1
|
|
||||||
evals = torch.linalg.eigvals(H)
|
|
||||||
current_E = evals[torch.argmin(torch.abs(evals - current_E))]
|
|
||||||
if np.any(c == train_cs):
|
|
||||||
index = np.where(c == train_cs)[0][0]
|
|
||||||
Es[index] = current_E
|
|
||||||
|
|
||||||
loss = ((Es - train_Es).abs() ** 2).sum()
|
|
||||||
|
|
||||||
if epoch % 1000 == 0:
|
|
||||||
print(f"Training {(epoch+1)/epochs:.1%} \t Loss: {loss}")
|
|
||||||
|
|
||||||
if H0.grad is not None:
|
|
||||||
H0.grad.zero_()
|
|
||||||
if H1.grad is not None:
|
|
||||||
H1.grad.zero_()
|
|
||||||
loss.backward()
|
|
||||||
|
|
||||||
with torch.no_grad():
|
|
||||||
H0 -= lr * H0.grad
|
|
||||||
H1 -= lr * H1.grad
|
|
||||||
|
|
||||||
# %%
|
|
||||||
# evaluate for all points
|
|
||||||
all_c = torch.tensor(df['c'].values, dtype=torch.float64)
|
|
||||||
exact_E = torch.tensor(df['E'].values, dtype=torch.complex128)
|
|
||||||
pred_Es = torch.empty(len(df), dtype=torch.complex128)
|
|
||||||
with torch.no_grad():
|
|
||||||
for (index, (c, E)) in enumerate(zip(all_c, exact_E)):
|
|
||||||
H = H0 + c * H1
|
|
||||||
evals = torch.linalg.eigvals(H)
|
|
||||||
i = torch.argmin(torch.abs(evals - E)) # TODO: more robust way to identify the eigenvector
|
|
||||||
pred_Es[index]= evals[i]
|
|
||||||
|
|
||||||
# %%
|
|
||||||
# plot the results
|
|
||||||
import matplotlib.pyplot as plt
|
|
||||||
plt.scatter(train_data['re_E'], train_data['im_E'], label='training')
|
|
||||||
plt.scatter(target_data['re_E'], target_data['im_E'], label='target')
|
|
||||||
plt.scatter(pred_Es.real, pred_Es.imag, marker='x', label='predicted')
|
|
||||||
plt.legend()
|
|
||||||
|
|
||||||
# %%
|
|
||||||
|
|
@ -1,28 +0,0 @@
|
||||||
include("../EC.jl")
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
# contour
|
|
||||||
vertices = [0, 0.4 - 0.15im, 0.8, 6]
|
|
||||||
subdivisions = [128, 128, 128]
|
|
||||||
p, w = get_mesh(vertices, subdivisions)
|
|
||||||
mesh_E = p.*p ./ (2*0.5)
|
|
||||||
|
|
||||||
μ = 0.5
|
|
||||||
V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q)) # ResonanceEC: Eq. (20)
|
|
||||||
|
|
||||||
H0 = get_T_matrix(p, μ)
|
|
||||||
V = get_V_matrix(V_system(1), p, w)
|
|
||||||
|
|
||||||
training_c = range(0.75, 0.45, 5)
|
|
||||||
extrapolating_c = range(0.40, 0.25, 5)
|
|
||||||
|
|
||||||
training_ref = [quick_pole_E(V_system(c)) for c in training_c]
|
|
||||||
exact_E = [quick_pole_E(V_system(c)) for c in extrapolating_c]
|
|
||||||
|
|
||||||
EC = affine_EC(H0, V, w)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=false)
|
|
||||||
extrapolate!(EC, extrapolating_c; precalculated_exact_E=exact_E)
|
|
||||||
|
|
||||||
#exportCSV(EC, "temp/2b_R2R.csv")
|
|
||||||
plot(EC, "temp/2b_R2R.pdf"; basis_contour=mesh_E, xlims=(0, 1))
|
|
||||||
|
|
@ -1,33 +0,0 @@
|
||||||
include("../EC.jl")
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
# contour
|
|
||||||
p, w = get_mesh([0, 0.4 - 0.15im, 0.8, 6], [128, 128, 128])
|
|
||||||
|
|
||||||
μ = 0.5
|
|
||||||
V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q)) # ResonanceEC: Eq. (20)
|
|
||||||
|
|
||||||
# generating a Berggren basis with a pole using the same system
|
|
||||||
basis_c = 0.6
|
|
||||||
basis_E, berg_basis = eigen(get_H_matrix(V_system(basis_c), p, w); permute=false, scale=false)
|
|
||||||
basis_p = sqrt.(basis_E)
|
|
||||||
N_berg = sqrt.(diag(transpose(berg_basis .* w) * berg_basis))
|
|
||||||
berg_basis = berg_basis ./ transpose(N_berg)
|
|
||||||
berg_basis_w = berg_basis .* w
|
|
||||||
|
|
||||||
H0 = transpose(berg_basis_w) * get_T_matrix(p, μ) * berg_basis
|
|
||||||
V = transpose(berg_basis_w) * get_V_matrix(V_system(1), p, w) * berg_basis
|
|
||||||
|
|
||||||
training_c = range(0.79, 0.66, 4) # original: range(1.35, 0.9, 5)
|
|
||||||
extrapolating_c = range(0.62, 0.40, 6) # original: range(0.75, 0.40, 8)
|
|
||||||
|
|
||||||
training_ref = [quick_pole_E(V_system(c)) for c in training_c]
|
|
||||||
extrapolating_ref = [quick_pole_E(V_system(c)) for c in extrapolating_c]
|
|
||||||
|
|
||||||
EC = affine_EC(H0, V)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=false)
|
|
||||||
extrapolate!(EC, extrapolating_c; ref_eval=extrapolating_ref)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/2b_GSM_R2R.csv")
|
|
||||||
plot(EC, "temp/2b_GSM_R2R.pdf"; basis_points=basis_E, xlims=(0, 0.3), ylims=(-0.120, 0.020))
|
|
||||||
|
|
@ -1,39 +0,0 @@
|
||||||
using Plots
|
|
||||||
|
|
||||||
include("../../EC.jl")
|
|
||||||
include("../../ho_basis.jl")
|
|
||||||
include("../../p_space.jl")
|
|
||||||
|
|
||||||
angle = 0.25 * pi # DOESN'T WORK WITHOUT ROTATION
|
|
||||||
μω_gen = 0.5 * exp(-2im * angle)
|
|
||||||
μ = 0.5
|
|
||||||
l = 0
|
|
||||||
V1 = -5
|
|
||||||
R1 = sqrt(3)
|
|
||||||
V2 = 2
|
|
||||||
R2 = sqrt(10)
|
|
||||||
n_max = 15
|
|
||||||
|
|
||||||
ns = collect(0:n_max)
|
|
||||||
ls = fill(l, n_max + 1)
|
|
||||||
|
|
||||||
T = get_sp_T_matrix(ns, ls; μω_gen=μω_gen, μ=μ)
|
|
||||||
V = V1 .* V_Gaussian.(R1, l, ns, transpose(ns); μω_gen=μω_gen) + V2 .* V_Gaussian.(R2, l, ns, transpose(ns); μω_gen=μω_gen)
|
|
||||||
|
|
||||||
n_EC = 8
|
|
||||||
train_cs = (0.7 .+ 0.05 * randn(n_EC)) - 1im * (0.2 .+ 0.05 * randn(n_EC))
|
|
||||||
target_cs = [0.5]
|
|
||||||
|
|
||||||
near_E = 0.2 + 0.2im
|
|
||||||
exact_E = [0.20845136860234303 - 0.07100640993695649im]
|
|
||||||
|
|
||||||
EC = affine_EC(T, V; ensemble_size=32)
|
|
||||||
train!(EC, train_cs; ref_eval=near_E, CAEC=false)
|
|
||||||
extrapolate!(EC, target_cs; precalculated_exact_E=exact_E)
|
|
||||||
|
|
||||||
plot(EC; xlims=(0,0.3), ylims=(-0.3,0.3))
|
|
||||||
hline!([0], color=:red, label="continuum")
|
|
||||||
xlabel!("Re(E)")
|
|
||||||
ylabel!("Im(E)")
|
|
||||||
plot!(legend=:bottomleft)
|
|
||||||
savefig("temp/2b_HO_XZ.pdf")
|
|
||||||
|
|
@ -1,50 +0,0 @@
|
||||||
using Plots
|
|
||||||
|
|
||||||
include("../../EC.jl")
|
|
||||||
include("../../ho_basis.jl")
|
|
||||||
include("../../p_space.jl")
|
|
||||||
|
|
||||||
# paramters of the system
|
|
||||||
|
|
||||||
angle = 0.0
|
|
||||||
μ = 0.5
|
|
||||||
l = 0
|
|
||||||
V1 = -5
|
|
||||||
R1 = sqrt(3)
|
|
||||||
V2 = 2
|
|
||||||
R2 = sqrt(10)
|
|
||||||
|
|
||||||
n_EC = 8
|
|
||||||
train_cs = (0.7 .+ 0.03 * randn(n_EC)) - 1im * (0.2 .+ 0.03 * randn(n_EC))
|
|
||||||
near_E = 0.2 + 0.2im
|
|
||||||
|
|
||||||
target_c = 0.5
|
|
||||||
exact_E = 0.20845136860234303 - 0.07100640993695649im
|
|
||||||
|
|
||||||
vertices = [0, 4 * exp(-1im * angle)]
|
|
||||||
subdivisions = [256]
|
|
||||||
ks, ws = get_mesh(vertices, subdivisions)
|
|
||||||
|
|
||||||
V_of_r(r) = V1 * exp(-r^2 / R1^2) + V2 * exp(-r^2 / R2^2)
|
|
||||||
V_mat_elem(k, kp) = Vl_mat_elem(V_of_r, l, k, kp; atol=10^-5, maxevals=10^5, R_cutoff=16)
|
|
||||||
V = get_V_matrix(V_mat_elem, ks, ws)
|
|
||||||
T = get_T_matrix(ks, μ)
|
|
||||||
|
|
||||||
EC_p_space = affine_EC(T, V)
|
|
||||||
train!(EC_p_space, train_cs; ref_eval=near_E, CAEC=false)
|
|
||||||
extrapolate!(EC_p_space, [target_c]; precalculated_exact_E=[exact_E])
|
|
||||||
|
|
||||||
# Plotting
|
|
||||||
|
|
||||||
theme(:dark) # Set the global theme to dark
|
|
||||||
|
|
||||||
scatter([real(exact_E)], [imag(exact_E)], label="exact", marker=:circle, markercolor=:white, bg = :black) # black background
|
|
||||||
scatter!(real.(EC_p_space.training_E), imag.(EC_p_space.training_E), label="training", marker=:circle, color=:blue)
|
|
||||||
scatter!(real.(EC_p_space.extrapolated_E), imag.(EC_p_space.extrapolated_E), label="extrapolated", marker=:x, color=:green)
|
|
||||||
hline!([0], color=:red, label="continuum")
|
|
||||||
plot!(legend=:bottomleft)
|
|
||||||
xlabel!("Re(E)")
|
|
||||||
ylabel!("Im(E)")
|
|
||||||
xlims!(0, 0.3)
|
|
||||||
ylims!(-0.3, 0.3)
|
|
||||||
savefig("temp/2body_p_space.pdf")
|
|
||||||
|
|
@ -1,36 +0,0 @@
|
||||||
include("../../ho_basis.jl")
|
|
||||||
include("../../EC.jl")
|
|
||||||
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
|
|
||||||
ϕ = 0.1 # DOESN'T WORK WITHOUT ROTATION
|
|
||||||
μω_global = 0.5 * exp(-2im * ϕ)
|
|
||||||
E_max = 40
|
|
||||||
|
|
||||||
H0 = get_3b_H_matrix(jacobi, V_of_r, μω_global, E_max, Λ, m, true, true)
|
|
||||||
|
|
||||||
# Vp = perturbation to make the state artificially bound
|
|
||||||
Vp_of_r(r) = -exp(-(r/3)^2)
|
|
||||||
@time "Vp" Vp = get_3b_H_matrix(jacobi, Vp_of_r, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
training_ref = 2 + 0.5im
|
|
||||||
|
|
||||||
exact_E = [4.076642792419057-0.012998408352259658im,
|
|
||||||
3.6129849325287-0.007397677539402868im,
|
|
||||||
3.145212908643357-0.0038660337822150753im,
|
|
||||||
2.6729225739451596-0.0021090370393881063im,
|
|
||||||
2.196385760253282-0.0010430088245526555im,
|
|
||||||
1.7162659936896967-0.0004515351140200029,
|
|
||||||
1.2329926791785895-0.00017698044022813525im]
|
|
||||||
|
|
||||||
training_c = [0.6 - 0.16im] .+ 0.04 .* (randn(8) .+ 0.5im * randn(8))
|
|
||||||
extrapolating_c = 0.0 : 0.2 : 1.2
|
|
||||||
|
|
||||||
EC = affine_EC(H0, Vp)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=false)
|
|
||||||
extrapolate!(EC, extrapolating_c; precalculated_exact_E=exact_E)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/3b_HO_XZ.csv")
|
|
||||||
plot(EC, "temp/3b_HO_XZ.pdf")
|
|
||||||
|
|
@ -1,45 +0,0 @@
|
||||||
include("../../p_space.jl")
|
|
||||||
include("../../EC.jl")
|
|
||||||
|
|
||||||
using Arpack
|
|
||||||
|
|
||||||
# target = 4.0766890719636875 - 0.012758927741074495im
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
|
|
||||||
vertices = [0, 2 - 0.2im, 3, 4] # TODO: real contour instead of Berggren basis
|
|
||||||
subdivisions = [15, 10, 10]
|
|
||||||
jmax = 4
|
|
||||||
|
|
||||||
E_max = 40
|
|
||||||
μω_global = 0.5
|
|
||||||
|
|
||||||
@time "H0" H0, _ = get_3b_H_matrix(jacobi, V_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m, true, true)
|
|
||||||
|
|
||||||
# Vp = perturbation to make the state artificially bound
|
|
||||||
Vp_of_r(r) = -exp(-(r/3)^2)
|
|
||||||
@time "Vp" Vp, _ = get_3b_H_matrix(jacobi, Vp_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m, false, true)
|
|
||||||
|
|
||||||
training_ref = 2 + 0.5im
|
|
||||||
|
|
||||||
exact_E = [4.076642792419057-0.012998408352259658im,
|
|
||||||
3.6129849325287-0.007397677539402868im,
|
|
||||||
3.145212908643357-0.0038660337822150753im,
|
|
||||||
2.6729225739451596-0.0021090370393881063im,
|
|
||||||
2.196385760253282-0.0010430088245526555im,
|
|
||||||
1.7162659936896967-0.0004515351140200029,
|
|
||||||
1.2329926791785895-0.00017698044022813525im]
|
|
||||||
|
|
||||||
training_c = [-1.5 - 0.5im] .+ (randn(8) .+ 0.05im * randn(8))
|
|
||||||
extrapolating_c = 0.0 : 0.2 : 1.2
|
|
||||||
|
|
||||||
EC = affine_EC(H0, Vp)
|
|
||||||
train!(EC, training_c; ref_eval=training_ref, CAEC=false)
|
|
||||||
extrapolate!(EC, extrapolating_c; precalculated_exact_E=exact_E)
|
|
||||||
|
|
||||||
exportCSV(EC, "temp/3b_p_space_XZ.csv")
|
|
||||||
plot(EC, "temp/3b_p_space_XZ.pdf")
|
|
||||||
|
|
||||||
# Results: training points are all over the place, and extrapolated values are garbage.
|
|
||||||
|
|
@ -1,71 +0,0 @@
|
||||||
using Plots
|
|
||||||
|
|
||||||
include("../../EC.jl")
|
|
||||||
include("../../ho_basis.jl")
|
|
||||||
include("../../p_space.jl")
|
|
||||||
|
|
||||||
# paramters of the system
|
|
||||||
|
|
||||||
angle = 0.0 * pi
|
|
||||||
μ = 0.5
|
|
||||||
l = 0
|
|
||||||
V1 = -5
|
|
||||||
R1 = sqrt(3)
|
|
||||||
V2 = 2
|
|
||||||
R2 = sqrt(10)
|
|
||||||
|
|
||||||
n_EC = 8
|
|
||||||
train_cs = (0.7 .+ 0.03 * randn(n_EC)) - 1im * (0.2 .+ 0.03 * randn(n_EC))
|
|
||||||
near_E = 0.2 + 0.2im
|
|
||||||
|
|
||||||
target_c = 0.5
|
|
||||||
exact_E = 0.20845136860234303 - 0.07100640993695649im
|
|
||||||
|
|
||||||
# HO basis
|
|
||||||
|
|
||||||
global EC_HO
|
|
||||||
begin
|
|
||||||
println("HO basis calculation")
|
|
||||||
μω_gen = 0.5 * exp(-1im * angle)
|
|
||||||
n_max = 40
|
|
||||||
ns = collect(0:n_max)
|
|
||||||
ls = fill(l, n_max + 1)
|
|
||||||
|
|
||||||
T = get_sp_T_matrix(ns, ls; μω_gen=μω_gen, μ=μ)
|
|
||||||
V = V1 .* V_Gaussian.(R1, l, ns, transpose(ns); μω_gen=μω_gen) + V2 .* V_Gaussian.(R2, l, ns, transpose(ns); μω_gen=μω_gen)
|
|
||||||
|
|
||||||
global EC_HO = affine_EC(T, V)
|
|
||||||
train!(EC_HO, train_cs; ref_eval=near_E, CAEC=false)
|
|
||||||
extrapolate!(EC_HO, [target_c]; precalculated_exact_E=[exact_E])
|
|
||||||
end
|
|
||||||
|
|
||||||
# p-space
|
|
||||||
|
|
||||||
global EC_p_space
|
|
||||||
begin
|
|
||||||
println("p-space calculation")
|
|
||||||
vertices = [0, 4 * exp(-1im * angle)]
|
|
||||||
subdivisions = [256]
|
|
||||||
ks, ws = get_mesh(vertices, subdivisions)
|
|
||||||
|
|
||||||
V_of_r(r) = V1 * exp(-r^2 / R1^2) + V2 * exp(-r^2 / R2^2)
|
|
||||||
V_mat_elem(k, kp) = Vl_mat_elem(V_of_r, l, k, kp; atol=10^-5, maxevals=10^5, R_cutoff=16)
|
|
||||||
V = get_V_matrix(V_mat_elem, ks, ws)
|
|
||||||
T = get_T_matrix(ks, μ)
|
|
||||||
|
|
||||||
global EC_p_space = affine_EC(T, V)
|
|
||||||
train!(EC_p_space, train_cs; ref_eval=near_E, CAEC=false)
|
|
||||||
extrapolate!(EC_p_space, [target_c]; precalculated_exact_E=[exact_E])
|
|
||||||
end
|
|
||||||
|
|
||||||
# Plotting
|
|
||||||
|
|
||||||
scatter([real(exact_E)], [imag(exact_E)], label="Exact", marker=:circle, markercolor=:white)
|
|
||||||
scatter!(real.(EC_HO.training_E), imag.(EC_HO.training_E), label="HO basis training", marker=:circle, color=:blue)
|
|
||||||
scatter!(real.(EC_HO.extrapolated_E), imag.(EC_HO.extrapolated_E), label="HO basis extrapolated", marker=:x, color=:blue)
|
|
||||||
scatter!(real.(EC_p_space.training_E), imag.(EC_p_space.training_E), label="p-space training", marker=:circle, color=:red)
|
|
||||||
scatter!(real.(EC_p_space.extrapolated_E), imag.(EC_p_space.extrapolated_E), label="p-space extrapolated", marker=:x, color=:red)
|
|
||||||
plot!(legend=:bottomleft)
|
|
||||||
xlabel!("Re(E)")
|
|
||||||
ylabel!("Im(E)")
|
|
||||||
savefig("temp/2b_p_space_vs_HO.pdf")
|
|
||||||
106
common.jl
106
common.jl
|
|
@ -1,106 +0,0 @@
|
||||||
using LinearAlgebra, DelimitedFiles, SparseArrays
|
|
||||||
|
|
||||||
@enum coordinate_system jacobi src
|
|
||||||
|
|
||||||
"Square root function with the branch cut along the postive imaginary axis"
|
|
||||||
alt_sqrt(x::Number)::ComplexF64 = sqrt(im * x) / sqrt(im)
|
|
||||||
|
|
||||||
"Sum over array while minimizing catastrophic cancellation as much as possible"
|
|
||||||
function better_sum(arr::Array{T}) where T<:Real
|
|
||||||
pos_arr = arr[arr .> 0]
|
|
||||||
neg_arr = arr[arr .< 0]
|
|
||||||
|
|
||||||
sort!(pos_arr)
|
|
||||||
sort!(neg_arr, rev=true)
|
|
||||||
|
|
||||||
return sum(pos_arr) + sum(neg_arr)
|
|
||||||
end
|
|
||||||
|
|
||||||
better_sum(arr::Array{ComplexF64}) = better_sum(real.(arr)) + 1im * better_sum(imag.(arr))
|
|
||||||
|
|
||||||
"The triangle inequality for angular momenta"
|
|
||||||
triangle_ineq(l1, l2, L) = abs(l1 - l2) ≤ L && L ≤ (l1 + l2)
|
|
||||||
|
|
||||||
"Index of the nearest value in a list to a given reference point"
|
|
||||||
nearestIndex(list::Array, ref) = argmin(norm.(list .- ref))
|
|
||||||
|
|
||||||
"Nearest value in a list to a given reference point"
|
|
||||||
nearest(list::Array, ref) = list[nearestIndex(list, ref)]
|
|
||||||
|
|
||||||
"Simple implementation of the Kronecker sum"
|
|
||||||
function kron_sum(A::AbstractMatrix, B::AbstractMatrix)
|
|
||||||
@assert size(A, 1) == size(A, 2) && size(B, 1) == size(B, 2) "Matrices should be square"
|
|
||||||
return kron(A, I(size(B, 1))) + kron(I(size(A, 1)), B)
|
|
||||||
end
|
|
||||||
|
|
||||||
"Flattened vector version of Iterators.product(...) with index hierachy reversed -- leftmost index has the highest hierachy"
|
|
||||||
iter_prod(args...) = reverse.(collect(Iterators.product(reverse(args)...))[:])
|
|
||||||
|
|
||||||
"Export CSV data for a scatter plot taking in data as a list of complex vectors (x=Re, y=Im), and a list of corresponding labels, typically used for EC results"
|
|
||||||
function exportCSV(file::String, data, labels=nothing)
|
|
||||||
columns = ["x" "y" "label"]
|
|
||||||
|
|
||||||
if isnothing(labels)
|
|
||||||
columns[end] = ""
|
|
||||||
labels = fill("", length(data))
|
|
||||||
end
|
|
||||||
|
|
||||||
open(file, "w") do f
|
|
||||||
writedlm(f, columns)
|
|
||||||
for (d, label) in zip(data, labels)
|
|
||||||
l = fill(label, length(d))
|
|
||||||
writedlm(f, hcat(reim(d)..., l))
|
|
||||||
end
|
|
||||||
end
|
|
||||||
end
|
|
||||||
|
|
||||||
"In-place c-orthogonalization via (modified) Gram-Schmidt. Only significant vectors are returned (c-normalized).
|
|
||||||
The number of significant vectors to return are determined by the original singular values (compared to the threshold)."
|
|
||||||
function gram_schmidt!(vecs::Vector{Vector{T}}, ws=ones(length(vecs[1])), threshold=1e-5; verbose=false) where T<: Number
|
|
||||||
c_product(i, j) = sum(vecs[i] .* ws .* vecs[j])
|
|
||||||
norm(i) = c_product(i, i)
|
|
||||||
proj(i, j) = (c_product(i, j) / norm(i)) .* vecs[i] # component of vec[i] in vec[j]
|
|
||||||
|
|
||||||
# initial normalization
|
|
||||||
for i in eachindex(vecs)
|
|
||||||
vecs[i] ./= sqrt(norm(i))
|
|
||||||
end
|
|
||||||
|
|
||||||
verbose && println("Absolute singular values = $(round.(c_singular_values(vecs, ws); sigdigits=1))")
|
|
||||||
target_dim = c_rank(vecs, ws, threshold)
|
|
||||||
verbose && println("Target dimensionality = $target_dim")
|
|
||||||
|
|
||||||
selected_vecs_i = Integer[]
|
|
||||||
|
|
||||||
while length(selected_vecs_i) < target_dim
|
|
||||||
i = argmax(i -> abs(norm(i)), setdiff(eachindex(vecs), selected_vecs_i)) # find the largest vector from the remaining
|
|
||||||
push!(selected_vecs_i, i)
|
|
||||||
for j in setdiff(eachindex(vecs), selected_vecs_i)
|
|
||||||
vecs[j] .-= proj(i, j)
|
|
||||||
end
|
|
||||||
end
|
|
||||||
|
|
||||||
verbose && println("Absolute norms of selected vectors = $(round.(selected_vecs_i .|> norm .|> abs; sigdigits=1))")
|
|
||||||
verbose && println("Absolute norms of dropped vectors = $(round.(setdiff(eachindex(vecs), selected_vecs_i) .|> norm .|> abs; sigdigits=1))")
|
|
||||||
|
|
||||||
# final normalization
|
|
||||||
for i in selected_vecs_i
|
|
||||||
vecs[i] ./= sqrt(norm(i))
|
|
||||||
end
|
|
||||||
|
|
||||||
return vecs[selected_vecs_i]
|
|
||||||
end
|
|
||||||
|
|
||||||
"Same as above but the basis is provided as a matrix instead of an array of vectors"
|
|
||||||
gram_schmidt!(vecs::Matrix{T}, ws=ones(size(vecs, 1)), threshold=1e-5; verbose=false) where T<: Number = hcat(gram_schmidt!([vecs[:, i] for i in axes(vecs, 2)], ws, threshold; verbose=verbose)...)
|
|
||||||
|
|
||||||
"Rank of a basis set (pre-normalized) under c-product"
|
|
||||||
c_rank(vecs, ws, threshold=1e-8) = count(c_singular_values(vecs, ws) .> threshold)
|
|
||||||
|
|
||||||
"Singular values (magnitudes) of a basis set (pre-normalized) under c-product"
|
|
||||||
function c_singular_values(vecs, ws)
|
|
||||||
basis = hcat(vecs...)
|
|
||||||
N = transpose(basis) * spdiagm(ws) * basis
|
|
||||||
singular_values = eigvals(N) .|> abs .|> sqrt
|
|
||||||
return singular_values
|
|
||||||
end
|
|
||||||
|
|
@ -0,0 +1,15 @@
|
||||||
|
"Sum over array while minimizing catastrophic cancellation as much as possible"
|
||||||
|
function better_sum(arr::Array{Float64})
|
||||||
|
pos_arr = arr[arr .> 0]
|
||||||
|
neg_arr = arr[arr .< 0]
|
||||||
|
|
||||||
|
sort!(pos_arr)
|
||||||
|
sort!(neg_arr, rev=true)
|
||||||
|
|
||||||
|
return sum(pos_arr) + sum(neg_arr)
|
||||||
|
end
|
||||||
|
|
||||||
|
better_sum(arr::Array{ComplexF64}) = better_sum(real.(arr)) + 1im * better_sum(imag.(arr))
|
||||||
|
|
||||||
|
"The triangle inequality for angular momenta"
|
||||||
|
triangle_ineq(l1, l2, L) = abs(l1 - l2) ≤ L && L ≤ (l1 + l2)
|
||||||
273
ho_basis.jl
273
ho_basis.jl
|
|
@ -1,74 +1,64 @@
|
||||||
using SparseArrays
|
using SparseArrays
|
||||||
using QuadGK
|
using NuclearToolkit
|
||||||
using LRUCache
|
using SpecialFunctions
|
||||||
include("common.jl")
|
include("helper.jl")
|
||||||
include("math.jl")
|
|
||||||
|
|
||||||
"2-body HO basis (used for 3-body systems without the CM DOF)"
|
# Gaussian potentials in HO space
|
||||||
struct ho_basis_2B
|
inv_factorial(n) = Iterators.prod(inv.(1:n))
|
||||||
E_max::Int
|
sqrt_factorial(n) = Iterators.prod(sqrt.(n:-1:1))
|
||||||
Λ::Int
|
N_lnk(l, n, k) = 1/sqrt_factorial(n) * binomial(n, k) * sqrt(gamma(n + l + 3/2)) / gamma(k + l + 3/2)
|
||||||
dim::Int # dimensionality of the basis
|
Talmi(l, R, k1, k2; μω_gen=1.0) = (-1)^(k1 + k2) * (1 + 1/(μω_gen * R^2))^-(3/2 + l + k1 + k2) * gamma(3/2 + l + k1 + k2)
|
||||||
Es::Vector{Int}
|
V_Gaussian(R, l, n1, n2; μω_gen=1.0) = (-1)^(n1 + n2) * better_sum([N_lnk(l, n1, k1) * N_lnk(l, n2, k2) * Talmi(l, R, k1, k2; μω_gen=μω_gen) for (k1, k2) in Iterators.product(0:n1, 0:n2)])
|
||||||
n1s::Vector{Int}
|
|
||||||
l1s::Vector{Int}
|
|
||||||
n2s::Vector{Int}
|
|
||||||
l2s::Vector{Int}
|
|
||||||
|
|
||||||
function ho_basis_2B(E_max, Λ=-1)
|
function get_sp_basis(E_max)
|
||||||
Es = Int[]
|
Es = Int[]
|
||||||
n1s = Int[]
|
ns = Int[]
|
||||||
l1s = Int[]
|
ls = Int[]
|
||||||
n2s = Int[]
|
|
||||||
l2s = Int[]
|
|
||||||
|
|
||||||
# E = 2*n1 + l1 + 2*n2 + l2
|
# E = 2*n + l
|
||||||
for E in E_max : -2 : 0 # same parity states only
|
for E in 0 : E_max
|
||||||
for n1 in 0 : E ÷ 2
|
for n in 0 : E ÷ 2
|
||||||
for n2 in 0 : (E - 2*n1) ÷ 2
|
l = E - 2*n
|
||||||
for l1 in 0 : (E - 2*n1 - 2*n2)
|
push!(Es, E)
|
||||||
l2 = E - 2*n1 - 2*n2 - l1
|
push!(ns, n)
|
||||||
if Λ≥0 && !triangle_ineq(l1, l2, Λ); continue; end
|
push!(ls, l)
|
||||||
push!(Es, E)
|
end
|
||||||
push!(n1s, n1)
|
end
|
||||||
push!(l1s, l1)
|
|
||||||
push!(n2s, n2)
|
return (Es, ns, ls)
|
||||||
push!(l2s, l2)
|
end
|
||||||
end
|
|
||||||
|
function get_2p_basis(E_max)
|
||||||
|
Es = Int[]
|
||||||
|
n1s = Int[]
|
||||||
|
l1s = Int[]
|
||||||
|
n2s = Int[]
|
||||||
|
l2s = Int[]
|
||||||
|
|
||||||
|
# E = 2*n1 + l1 + 2*n2 + l2
|
||||||
|
for E in 2*E_max : -2 : 0
|
||||||
|
for n1 in 0 : E ÷ 2
|
||||||
|
for n2 in 0 : (E - 2*n1) ÷ 2
|
||||||
|
for l1 in 0 : (E - 2*n1 - 2*n2)
|
||||||
|
l2 = E - 2*n1 - 2*n2 - l1
|
||||||
|
push!(Es, E)
|
||||||
|
push!(n1s, n1)
|
||||||
|
push!(l1s, l1)
|
||||||
|
push!(n2s, n2)
|
||||||
|
push!(l2s, l2)
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
return new(E_max, Λ, length(Es), Es, n1s, l1s, n2s, l2s)
|
|
||||||
end
|
end
|
||||||
|
|
||||||
|
return (Es, n1s, l1s, n2s, l2s)
|
||||||
end
|
end
|
||||||
|
|
||||||
"Number of possible distinct matrix elements for a given l."
|
function sp_T_matrix(ns, ls; mask=trues(length(ns),length(ns)), μω_gen=1.0, μ=1.0)
|
||||||
function cache_size_l(E_max::Int, l::Int)::Int
|
|
||||||
n_max = (E_max - l) ÷ 2
|
|
||||||
return (n_max * (n_max + 1)) ÷ 2
|
|
||||||
end
|
|
||||||
|
|
||||||
"Number of possible distinct matrix elements for all l."
|
|
||||||
cache_size(E_max::Int)::Int = sum(l -> cache_size_l(E_max, l), 0 : E_max)
|
|
||||||
|
|
||||||
"Preallocation of cache for PE matrix elements."
|
|
||||||
prealloc_V_cache(E_max::Int, dtype::DataType=Float64) = LRU{Tuple{UInt8, UInt8, UInt8}, dtype}(maxsize=cache_size(E_max))
|
|
||||||
|
|
||||||
function V_numerical(V_of_r, l, n1, n2; μω_gen=1.0, atol=0, maxevals=10^7)
|
|
||||||
const_part = sqrt(μω_gen) * ho_basis_const(l, n1) * ho_basis_const(l, n2)
|
|
||||||
integrand(r) = ho_basis_func(l, n1, sqrt(μω_gen) * r) * ho_basis_func(l, n2, sqrt(μω_gen) * r) * V_of_r(r)
|
|
||||||
(integral, _) = quadgk(integrand, 0, Inf; atol=atol, maxevals=maxevals)
|
|
||||||
return const_part * integral
|
|
||||||
end
|
|
||||||
|
|
||||||
"KE matrix for a single DOF. Set kron_deltas=[other quantum numbers] for other DOFs which the operator does not act on.
|
|
||||||
E.g. get_sp_T_matrix(n1s, l1s, kron_deltas=[n2s l2s])"
|
|
||||||
function get_sp_T_matrix(ns, ls, kron_deltas=[]; μω_gen=1.0, μ=1.0)
|
|
||||||
mat = spzeros(length(ns), length(ns))
|
mat = spzeros(length(ns), length(ns))
|
||||||
for idx in CartesianIndices(mat)
|
for idx in CartesianIndices(mat)
|
||||||
|
if !mask[idx]; continue; end
|
||||||
(i, j) = Tuple(idx)
|
(i, j) = Tuple(idx)
|
||||||
all(arr -> arr[i]==arr[j], kron_deltas) || continue # check if all Kronecker deltas are non-zero
|
|
||||||
if ls[i] == ls[j]
|
if ls[i] == ls[j]
|
||||||
if ns[i] == ns[j]
|
if ns[i] == ns[j]
|
||||||
mat[idx] = ns[j] + ls[i]/2 + 3/4
|
mat[idx] = ns[j] + ls[i]/2 + 3/4
|
||||||
|
|
@ -81,171 +71,38 @@ function get_sp_T_matrix(ns, ls, kron_deltas=[]; μω_gen=1.0, μ=1.0)
|
||||||
return (μω_gen / μ) .* mat
|
return (μω_gen / μ) .* mat
|
||||||
end
|
end
|
||||||
|
|
||||||
"PE matrix for a single DOF. Set kron_deltas=[other quantum numbers] for other DOFs which the operator does not act on.
|
function sp_V_matrix(V_l, ns, ls; mask=trues(length(ns),length(ns)), dtype=Float64)
|
||||||
E.g. get_sp_V_matrix(n1s, l1s, kron_deltas=[n2s l2s])
|
|
||||||
Providing a preallocated cache is optional. Otherwise, provided E_max value is used to initialize one (defaults to 100)."
|
|
||||||
function get_sp_V_matrix(V_l, ns, ls, kron_deltas=[]; dtype=Float64, E_max=100, cache=prealloc_V_cache(E_max, dtype))
|
|
||||||
mat = zeros(dtype, length(ns), length(ns))
|
mat = zeros(dtype, length(ns), length(ns))
|
||||||
Threads.@threads for idx in CartesianIndices(mat)
|
Threads.@threads for idx in CartesianIndices(mat)
|
||||||
|
if !mask[idx]; continue; end
|
||||||
(i, j) = Tuple(idx)
|
(i, j) = Tuple(idx)
|
||||||
all(arr -> arr[i]==arr[j], kron_deltas) || continue # check if all Kronecker deltas are non-zero
|
|
||||||
if ls[i] == ls[j]
|
if ls[i] == ls[j]
|
||||||
l = UInt8(ls[i])
|
mat[idx] = V_l(ls[i], ns[i], ns[j])
|
||||||
n1, n2 = UInt8.(minmax(ns[i], ns[j])) # assuming transpose symmetry
|
|
||||||
mat[idx] = (get!(cache, (l, n1, n2)) do; V_l(l, n1, n2); end)
|
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
return sparse(mat)
|
return sparse(mat)
|
||||||
end
|
end
|
||||||
|
|
||||||
function Moshinsky_transform(basis::ho_basis_2B)
|
function Moshinsky_transform(Es, n1s, l1s, n2s, l2s, Λ)
|
||||||
NQMAX = maximum(basis.Es)
|
E_max = maximum(Es)
|
||||||
@assert all(mod.(basis.Es, 2) .== mod(NQMAX, 2)) "Can only admit basis states with same parity"
|
j_max = 2 * E_max + 1
|
||||||
|
l_max = j_max
|
||||||
|
to = 0 # unused
|
||||||
|
|
||||||
LMIN = basis.Λ
|
dtri = NuclearToolkit.prep_dtri(l_max + 1);
|
||||||
LMAX = basis.Λ
|
dcgm0 = NuclearToolkit.prep_dcgm0(l_max);
|
||||||
CO = 1/sqrt(2)
|
d6j = nothing # NuclearToolkit.prep_d6j_int(E_max, j_max, to);
|
||||||
SI = 1/sqrt(2)
|
|
||||||
|
|
||||||
# dimensions BRAC(0:LMAX,0:(NQMAX-LMIN)/2,0:(NQMAX-LMIN)/2,0:(NQMAX-LMIN)/2,0:(NQMAX-LMIN)/2,0:LMAX,0:(NQMAX-LMIN)/2,LMIN:LMAX)
|
mat = spzeros(length(Es), length(Es))
|
||||||
BRAC = zeros(Float64, 1 + LMAX, 1 + (NQMAX - LMIN) ÷ 2, 1 + (NQMAX - LMIN) ÷ 2, 1 + (NQMAX - LMIN) ÷ 2, 1 + (NQMAX - LMIN) ÷ 2, 1 + LMAX, 1 + (NQMAX-LMIN) ÷ 2, 1 + LMAX-LMIN)
|
s = hcat(Es, n1s, l1s, n2s, l2s)
|
||||||
@ccall "../OSBRACKETS/allosbrac.so".allosbrac_(NQMAX::Ref{Int32},LMIN::Ref{Int32},LMAX::Ref{Int32},CO::Ref{Float64},SI::Ref{Float64},BRAC::Ptr{Array{Float64}})::Cvoid
|
for idx in CartesianIndices(mat)
|
||||||
|
|
||||||
mat = zeros(basis.dim, basis.dim)
|
|
||||||
|
|
||||||
s = hcat(basis.Es, basis.n1s, basis.l1s, basis.n2s, basis.l2s)
|
|
||||||
Threads.@threads for idx in CartesianIndices(mat)
|
|
||||||
(i, j) = Tuple(idx)
|
(i, j) = Tuple(idx)
|
||||||
(Elhs, N, L, n, l) = s[i, :]
|
(Elhs, N, L, n, l) = s[i, :]
|
||||||
(Erhs, n1, l1, n2, l2) = s[j, :]
|
(Erhs, n1, l1, n2, l2) = s[j, :]
|
||||||
if Elhs == Erhs && triangle_ineq(L, l, basis.Λ) && triangle_ineq(l1, l2, basis.Λ)
|
if Elhs == Erhs && triangle_ineq(L, l, Λ) && triangle_ineq(l1, l2, Λ)
|
||||||
mat[i, j] = (-1)^(n1 + n2 + N + n) * pick_Moshinsky_bracket(BRAC, n1, l1, n2, l2, N, L, n, l, basis.Λ)
|
mat[i, j] = NuclearToolkit.HObracket_d6j(N, L, n, l, n1, l1, n2, l2, Λ, 1.0, dtri, dcgm0, d6j)
|
||||||
end
|
end
|
||||||
end
|
end
|
||||||
|
|
||||||
return sparse(mat)
|
return mat
|
||||||
end
|
|
||||||
|
|
||||||
function pick_Moshinsky_bracket(BRAC, n1′, l1′, n2′, l2′, n1, l1, n2, l2, Λ) # Efros notation -- don't confuse
|
|
||||||
ϵ = (l1 + l2 - Λ) % 2
|
|
||||||
NP = (l1′ - l2′ + Λ - ϵ) ÷ 2
|
|
||||||
MP = (l1′ + l2′ - Λ - ϵ) ÷ 2
|
|
||||||
N = (l1 - l2 + Λ - ϵ) ÷ 2
|
|
||||||
M = (l1 + l2 - Λ - ϵ) ÷ 2
|
|
||||||
|
|
||||||
# BRAC(NP,N1P,MP,N1,N2,N,M,L)
|
|
||||||
return BRAC[1 + NP, 1 + n1′, 1 + MP, 1 + n1, 1 + n2, 1 + N, 1 + M, 1]
|
|
||||||
end
|
|
||||||
|
|
||||||
function get_jacobi_V_matrix(V_of_r, basis::ho_basis_2B, μ1ω1, μω_global; atol=10^-6, maxevals=10^5)
|
|
||||||
V1 = get_jacobi_V1_matrix(V_of_r, basis, μ1ω1; atol=atol, maxevals=maxevals)
|
|
||||||
V2 = get_jacobi_V2_matrix(V_of_r, basis, μω_global; atol=atol, maxevals=maxevals)
|
|
||||||
return V1 + V2
|
|
||||||
end
|
|
||||||
|
|
||||||
function get_jacobi_V1_matrix(V_of_r, basis::ho_basis_2B, μ1ω1; atol=10^-6, maxevals=10^5)
|
|
||||||
V1_elem(l, n1, n2) = V_numerical(V_of_r, l, n1, n2; μω_gen=μ1ω1, atol=atol, maxevals=maxevals)
|
|
||||||
V1 = get_sp_V_matrix(V1_elem, basis.n1s, basis.l1s, [basis.n2s, basis.l2s]; dtype=ComplexF64, E_max=basis.E_max)
|
|
||||||
return V1
|
|
||||||
end
|
|
||||||
|
|
||||||
function get_jacobi_V2_matrix(V_of_r, basis::ho_basis_2B, μω_global; atol=10^-6, maxevals=10^5)
|
|
||||||
V_relative_elem(l, n1, n2) = V_numerical(V_of_r, l, n1, n2; μω_gen=μω_global, atol=atol, maxevals=maxevals)
|
|
||||||
V_relative_cache = prealloc_V_cache(basis.E_max, ComplexF64)
|
|
||||||
|
|
||||||
V_relative = get_sp_V_matrix(V_relative_elem, basis.n1s, basis.l1s, [basis.n2s, basis.l2s]; dtype=ComplexF64, cache=V_relative_cache) + get_sp_V_matrix(V_relative_elem, basis.n2s, basis.l2s, [basis.n1s, basis.l1s]; dtype=ComplexF64, cache=V_relative_cache)
|
|
||||||
U = Moshinsky_transform(basis)
|
|
||||||
V2 = transpose(U) * V_relative * U
|
|
||||||
|
|
||||||
return V2
|
|
||||||
end
|
|
||||||
|
|
||||||
function get_2p_p1p2_matrix(basis::ho_basis_2B, μ1ω1, μ2ω2; dtype=Float64)
|
|
||||||
# TODO: Cache for integrals
|
|
||||||
integral1(np, lp, n, l) = integral_HO(np, lp, n, l, μ1ω1)
|
|
||||||
integral2(np, lp, n, l) = integral_HO(np, lp, n, l, μ2ω2)
|
|
||||||
|
|
||||||
mat = zeros(dtype, basis.dim, basis.dim)
|
|
||||||
Threads.@threads for idx in CartesianIndices(mat)
|
|
||||||
(i, j) = Tuple(idx)
|
|
||||||
val = racahs_reduction_formula(basis.n1s[i], basis.l1s[i], basis.n2s[i], basis.l2s[i], basis.n1s[j], basis.l1s[j], basis.n2s[j], basis.l2s[j], basis.Λ, integral1, integral2)
|
|
||||||
if !(val ≈ 0); mat[idx] = val; end
|
|
||||||
end
|
|
||||||
return sparse(mat)
|
|
||||||
end
|
|
||||||
|
|
||||||
function get_src_V_matrix(V_of_r, basis::ho_basis_2B, μω, μω_global; atol=10^-6, maxevals=10^5)
|
|
||||||
V_elem(l, n1, n2) = V_numerical(V_of_r, l, n1, n2; μω_gen=μω, atol=atol, maxevals=maxevals)
|
|
||||||
V_cache = prealloc_V_cache(basis.E_max, ComplexF64)
|
|
||||||
|
|
||||||
V1 = get_sp_V_matrix(V_elem, basis.n1s, basis.l1s, [basis.n2s, basis.l2s]; dtype=ComplexF64, cache=V_cache)
|
|
||||||
V2 = get_sp_V_matrix(V_elem, basis.n2s, basis.l2s, [basis.n1s, basis.l1s]; dtype=ComplexF64, cache=V_cache)
|
|
||||||
|
|
||||||
V12 = get_src_V12_matrix(V_of_r, basis, μω_global; atol=atol, maxevals=maxevals)
|
|
||||||
|
|
||||||
return V1 + V2 + V12
|
|
||||||
end
|
|
||||||
|
|
||||||
function get_src_V12_matrix(V_of_r, basis::ho_basis_2B, μω_global; atol=10^-6, maxevals=10^5)
|
|
||||||
V_relative_elem(l, n1, n2) = V_numerical(V_of_r, l, n1, n2; μω_gen=μω_global, atol=atol, maxevals=maxevals)
|
|
||||||
V_relative = get_sp_V_matrix(V_relative_elem, basis.n1s, basis.l1s, [basis.n2s, basis.l2s]; dtype=ComplexF64, E_max=basis.E_max)
|
|
||||||
|
|
||||||
U = Moshinsky_transform(basis)
|
|
||||||
V12 = transpose(U) * V_relative * U
|
|
||||||
|
|
||||||
return V12
|
|
||||||
end
|
|
||||||
|
|
||||||
"Basis transformation from HO to momentum space"
|
|
||||||
function get_W_matrix(basis_p, basis::ho_basis_2B, μ1ω1, μ2ω2=μ1ω1; weights=true)
|
|
||||||
W = zeros(ComplexF64, length(basis_p), basis.dim)
|
|
||||||
Threads.@threads for idx in CartesianIndices(W)
|
|
||||||
(i1, i2) = Tuple(idx)
|
|
||||||
((j1, j2), (k1, w1), (k2, w2)) = basis_p[i1]
|
|
||||||
if j1 == basis.l1s[i2] && j2 == basis.l2s[i2]
|
|
||||||
elem1 = 1/sqrt(sqrt(μ1ω1)) * (-1)^basis.n1s[i2] * ho_basis(j1, basis.n1s[i2], 1/sqrt(μ1ω1) * k1)
|
|
||||||
elem2 = 1/sqrt(sqrt(μ2ω2)) * (-1)^basis.n2s[i2] * ho_basis(j2, basis.n2s[i2], 1/sqrt(μ2ω2) * k2)
|
|
||||||
W[idx] = elem1 * elem2 * (weights ? w1 * w2 : 1)
|
|
||||||
end
|
|
||||||
end
|
|
||||||
return sparse(W)
|
|
||||||
end
|
|
||||||
|
|
||||||
function get_3b_H_matrix(coord_system::coordinate_system, V_of_r, μω_global, E_max, Λ, m=1.0, kinetic_part=true, potential_part=true; atol=10^-5, maxevals=10^5, verbose=true)
|
|
||||||
if coord_system == jacobi
|
|
||||||
μ1ω1 = μω_global * 1/2
|
|
||||||
μ2ω2 = μω_global * 2
|
|
||||||
μ1 = m * 1/2
|
|
||||||
μ2 = m * 2/3
|
|
||||||
elseif coord_system == src
|
|
||||||
μ1ω1 = μ2ω2 = μω = μω_global * 2
|
|
||||||
μ1 = μ2 = μ = m/2
|
|
||||||
end
|
|
||||||
|
|
||||||
verbose && println("No of threads = ", Threads.nthreads())
|
|
||||||
|
|
||||||
@time "Basis" basis = ho_basis_2B(E_max, Λ)
|
|
||||||
verbose && println("Basis size = ", basis.dim)
|
|
||||||
|
|
||||||
out = spzeros(basis.dim, basis.dim)
|
|
||||||
|
|
||||||
if kinetic_part
|
|
||||||
verbose && println("Constructing KE matrices")
|
|
||||||
@time "T1" out += get_sp_T_matrix(basis.n1s, basis.l1s, [basis.n2s, basis.l2s]; μω_gen=μ1ω1, μ=μ1)
|
|
||||||
@time "T2" out += get_sp_T_matrix(basis.n2s, basis.l2s, [basis.n1s, basis.l1s]; μω_gen=μ2ω2, μ=μ2)
|
|
||||||
if coord_system == src
|
|
||||||
@time "T_cross" out += get_2p_p1p2_matrix(basis, μ1ω1, μ2ω2; dtype=ComplexF64) ./ (2*μ)
|
|
||||||
end
|
|
||||||
end
|
|
||||||
|
|
||||||
if potential_part
|
|
||||||
verbose && println("Constructing PE matrices")
|
|
||||||
if coord_system == jacobi
|
|
||||||
@time "V" out += get_jacobi_V_matrix(V_of_r, basis, μ1ω1, μω_global; atol=atol, maxevals=maxevals)
|
|
||||||
elseif coord_system == src
|
|
||||||
@time "V" out += get_src_V_matrix(V_of_r, basis, μω, μω_global; atol=atol, maxevals=maxevals)
|
|
||||||
end
|
|
||||||
end
|
|
||||||
|
|
||||||
return out
|
|
||||||
end
|
end
|
||||||
|
|
@ -0,0 +1,31 @@
|
||||||
|
using Arpack, SparseArrays
|
||||||
|
include("ho_basis.jl")
|
||||||
|
include("p_space.jl")
|
||||||
|
|
||||||
|
E_max = 30
|
||||||
|
μω_gen = 0.2
|
||||||
|
Λ = 0
|
||||||
|
m = 1.0
|
||||||
|
|
||||||
|
Va = -2
|
||||||
|
Ra = 2
|
||||||
|
|
||||||
|
μ1 = m * 1/2
|
||||||
|
|
||||||
|
println("No of threads = ", Threads.nthreads())
|
||||||
|
|
||||||
|
Es, n1s, l1s = get_sp_basis(E_max)
|
||||||
|
println("Basis size = ", length(Es))
|
||||||
|
|
||||||
|
println("Constructing KE matrices")
|
||||||
|
@time "T1" T1 = sp_T_matrix(n1s, l1s; μω_gen=μω_gen, μ=μ1)
|
||||||
|
|
||||||
|
println("Constructing PE matrices")
|
||||||
|
V1_elem(l, n1, n2) = Va * V_Gaussian(Ra, l, n1, n2; μω_gen=μω_gen)
|
||||||
|
@time "V1" V1 = sp_V_matrix(V1_elem, n1s, l1s)
|
||||||
|
|
||||||
|
println("Calculating spectrum")
|
||||||
|
@time "H" H = T1 + V1
|
||||||
|
@time "Eigenvalues" evals, _ = eigs(H, nev=3, ncv=30, which=:SR, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
||||||
|
|
||||||
|
display(evals)
|
||||||
|
|
@ -1,14 +1,43 @@
|
||||||
using Arpack
|
using Arpack, SparseArrays
|
||||||
include("ho_basis.jl")
|
include("ho_basis.jl")
|
||||||
|
include("p_space.jl")
|
||||||
|
|
||||||
|
E_max = 20
|
||||||
|
μ1ω1_gen = 1/2
|
||||||
|
μ2ω2_gen = 2
|
||||||
Λ = 0
|
Λ = 0
|
||||||
m = 1.0
|
m = 1.0
|
||||||
V_of_r(r) = -2 * exp(-r^2 / 4)
|
|
||||||
|
|
||||||
E_max = 40
|
Va = -2
|
||||||
μω_global = 0.3
|
Ra = 2
|
||||||
|
|
||||||
H = get_3b_H_matrix(src, V_of_r, μω_global, E_max, Λ, m)
|
μ1 = m * 1/2
|
||||||
|
μ2 = m * 2/3
|
||||||
|
|
||||||
|
println("No of threads = ", Threads.nthreads())
|
||||||
|
|
||||||
|
@time "Basis" Es, n1s, l1s, n2s, l2s = get_2p_basis(E_max)
|
||||||
|
@time "Masks" begin
|
||||||
|
mask1 = (n2s .== n2s') .&& (l2s .== l2s')
|
||||||
|
mask2 = (n1s .== n1s') .&& (l1s .== l1s')
|
||||||
|
end
|
||||||
|
|
||||||
|
println("Basis size = ", length(Es))
|
||||||
|
|
||||||
|
println("Constructing KE matrices")
|
||||||
|
@time "T1" T1 = sp_T_matrix(n1s, l1s; mask=mask1, μω_gen=μ1ω1_gen, μ=μ1)
|
||||||
|
@time "T2" T2 = sp_T_matrix(n2s, l2s; mask=mask2, μω_gen=μ2ω2_gen, μ=μ2)
|
||||||
|
|
||||||
|
println("Constructing PE matrices")
|
||||||
|
V1_elem(l, n1, n2) = Va * V_Gaussian(Ra, l, n1, n2; μω_gen=μ1ω1_gen)
|
||||||
|
V_relative_elem(l, n1, n2) = Va * V_Gaussian(Ra, l, n1, n2; μω_gen=1.0)
|
||||||
|
@time "V1" V1 = sp_V_matrix(V1_elem, n1s, l1s; mask=mask1)
|
||||||
|
@time "V relative" V_relative = sp_V_matrix(V_relative_elem, n1s, l1s; mask=mask1) + sp_V_matrix(V_relative_elem, n2s, l2s; mask=mask2)
|
||||||
|
@time "Moshinsky brackets" U = Moshinsky_transform(Es, n1s, l1s, n2s, l2s, Λ)
|
||||||
|
@time "V2" V2 = U' * V_relative * U
|
||||||
|
|
||||||
|
println("Calculating spectrum")
|
||||||
|
@time "H" H = T1 + T2 + V1 + V2
|
||||||
@time "Eigenvalues" evals, _ = eigs(H, nev=3, ncv=30, which=:SR, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
@time "Eigenvalues" evals, _ = eigs(H, nev=3, ncv=30, which=:SR, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
||||||
|
|
||||||
display(evals)
|
display(evals)
|
||||||
|
|
@ -1,15 +0,0 @@
|
||||||
using Arpack
|
|
||||||
include("ho_basis.jl")
|
|
||||||
|
|
||||||
target_E = 4.07656088827514 - 0.012743522750966718im
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
|
|
||||||
μω_global = 0.5 * exp(-2im * pi / 9)
|
|
||||||
E_max = 30
|
|
||||||
|
|
||||||
H = get_3b_H_matrix(src, V_of_r, μω_global, E_max, Λ, m)
|
|
||||||
|
|
||||||
@time "Eigenvalues" evals, _ = eigs(H, nev=5, ncv=50, sigma=target_E, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
|
||||||
display(evals)
|
|
||||||
|
|
@ -14,10 +14,10 @@ n_max = 15
|
||||||
ns = collect(0:n_max)
|
ns = collect(0:n_max)
|
||||||
ls = fill(l, n_max + 1)
|
ls = fill(l, n_max + 1)
|
||||||
|
|
||||||
T = get_sp_T_matrix(ns, ls; μω_gen=μω_gen, μ=μ)
|
T = sp_T_matrix(ns, ls; μω_gen=μω_gen, μ=μ)
|
||||||
|
|
||||||
V_l(l, n1, n2) = V1 * V_Gaussian(R1, l, n1, n2; μω_gen=μω_gen) + V2 * V_Gaussian(R2, l, n1, n2; μω_gen=μω_gen)
|
V_l(l, n1, n2) = V1 * V_Gaussian(R1, l, n1, n2; μω_gen=μω_gen) + V2 * V_Gaussian(R2, l, n1, n2; μω_gen=μω_gen)
|
||||||
V = get_sp_V_matrix(V_l, ns, ls; dtype=ComplexF64)
|
V = sp_V_matrix(V_l, ns, ls; dtype=ComplexF64)
|
||||||
|
|
||||||
cs = range(1.25, 0.25, 10)
|
cs = range(1.25, 0.25, 10)
|
||||||
|
|
||||||
|
|
|
||||||
|
|
@ -1,5 +1,5 @@
|
||||||
using LinearAlgebra
|
using LinearAlgebra
|
||||||
include("../ho_basis.jl")
|
include("ho_basis.jl")
|
||||||
|
|
||||||
l = 0
|
l = 0
|
||||||
V0 = -10
|
V0 = -10
|
||||||
|
|
@ -9,10 +9,10 @@ n_max = 20
|
||||||
ns = collect(0:n_max)
|
ns = collect(0:n_max)
|
||||||
ls = fill(l, n_max + 1)
|
ls = fill(l, n_max + 1)
|
||||||
|
|
||||||
T = get_sp_T_matrix(ns, ls)
|
T = sp_T_matrix(ns, ls)
|
||||||
|
|
||||||
V_l(l, n1, n2) = V0 * V_Gaussian(R, l, n1, n2)
|
V_l(l, n1, n2) = V0 * V_Gaussian(R, l, n1, n2)
|
||||||
V = get_sp_V_matrix(V_l, ns, ls)
|
V = sp_V_matrix(V_l, ns, ls)
|
||||||
|
|
||||||
H = T + V
|
H = T + V
|
||||||
|
|
||||||
47
math.jl
47
math.jl
|
|
@ -1,47 +0,0 @@
|
||||||
using SpecialFunctions, WignerSymbols
|
|
||||||
include("common.jl")
|
|
||||||
|
|
||||||
# Gaussian potentials in HO space
|
|
||||||
inv_factorial(n) = Iterators.prod(inv.(1:n))
|
|
||||||
sqrt_factorial(n) = Iterators.prod(sqrt.(n:-1:1))
|
|
||||||
N_lnk(l, n, k) = 1/sqrt_factorial(n) * binomial(n, k) * sqrt(gamma(n + l + 3/2)) / gamma(k + l + 3/2)
|
|
||||||
Talmi(l, R, k1, k2; μω_gen=1.0) = (-1)^(k1 + k2) * (1 + 1/(μω_gen * R^2))^-(3/2 + l + k1 + k2) * gamma(3/2 + l + k1 + k2)
|
|
||||||
V_Gaussian(R, l, n1, n2; μω_gen=1.0) = (-1)^(n1 + n2) * better_sum([N_lnk(l, n1, k1) * N_lnk(l, n2, k2) * Talmi(l, R, k1, k2; μω_gen=μω_gen) for (k1, k2) in Iterators.product(0:n1, 0:n2)])
|
|
||||||
|
|
||||||
# for numerical evaluation of V matrix elements
|
|
||||||
sqrt_double_factorial(n) = Iterators.prod(sqrt.(n:-2:1))
|
|
||||||
sqrt_sqrt_pi = sqrt(sqrt(pi))
|
|
||||||
laguerre(l, n, x) = gamma(n + l + 3/2) * better_sum([(-x * x)^k / gamma(k + l + 3/2) * inv_factorial(n - k) * inv_factorial(k) for k in 0:n])
|
|
||||||
ho_basis_const(l, n) = (-1)^n / sqrt_sqrt_pi * (2.0)^((n + l + 2) / 2) * sqrt_factorial(n) / sqrt_double_factorial(2*n + 2*l + 1)
|
|
||||||
ho_basis_func(l, n, x) = x^(l + 1) * exp(-x^2 / 2) * laguerre(l, n, x)
|
|
||||||
ho_basis(l, n, x) = ho_basis_const(l, n) * ho_basis_func(l, n, x)
|
|
||||||
|
|
||||||
# for implementation of simple relative coordinates
|
|
||||||
double_factorial(n::Int) = Iterators.prod(big, n:-2:1)
|
|
||||||
|
|
||||||
"Gaussian integral for n ∈ Integers (Ref: Wolfram MathWorld + simplifications)"
|
|
||||||
gauss_int(a, n) = double_factorial(n - 1) / (2.0 * a)^((n + 1)/2) * (iseven(n) ? sqrt(π / 2) : 1)
|
|
||||||
|
|
||||||
"Gives ∫dp p u' u where u' and u are HO functions with different l (Ref: worked out in Mathematica)"
|
|
||||||
function integral_HO(np, lp, n, l, μω)
|
|
||||||
s = [(-1)^(m + mp) * gauss_int(1, 2 * m + 2 * mp + l + lp + 3) * N_lnk(l, n, m) * N_lnk(lp, np, mp) for (m, mp) in Iterators.product(0:n, 0:np)]
|
|
||||||
return 2 * sqrt(μω) * better_sum(s)
|
|
||||||
end
|
|
||||||
|
|
||||||
"Gives <n' l'|| p ||n l> for the HO basis, where integral(np, lp, n, l) is a function that returns ∫dp p u' u"
|
|
||||||
function reduced_matrix_element(np, lp, n, l, integral)::ComplexF64
|
|
||||||
wig::Float64 = wigner3j(Float64, lp, 1, l, 0, 0, 0)
|
|
||||||
if wig == 0
|
|
||||||
return 0
|
|
||||||
else
|
|
||||||
return (-1)^lp * sqrt(2*lp + 1) * sqrt(2*l + 1) * wig * integral(np, lp, n, l)
|
|
||||||
end
|
|
||||||
end
|
|
||||||
|
|
||||||
"Matrix element <n1p l1p n2p l2p| p1⋅p2 |n1 l1 n2 l2> (Ref: de-Shalit & Talmi, Eq 15.5), where integral(np, lp, n, l) is a function that returns ∫dp p u' u"
|
|
||||||
function racahs_reduction_formula(n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ, integral1, integral2)
|
|
||||||
val = wigner6j(Float64, l1p, l2p, Λ, l2, l1, 1)
|
|
||||||
if val != 0; val *= reduced_matrix_element(n1p, l1p, n1, l1, integral1); end
|
|
||||||
if val != 0; val *= reduced_matrix_element(n2p, l2p, n2, l2, integral2); end
|
|
||||||
return (-1)^(l1 + l2p + Λ) * val
|
|
||||||
end
|
|
||||||
81
p_space.jl
81
p_space.jl
|
|
@ -1,6 +1,8 @@
|
||||||
using LinearAlgebra
|
using FastGaussQuadrature
|
||||||
using SpecialFunctions, FastGaussQuadrature, QuadGK
|
|
||||||
include("ho_basis.jl")
|
# Gaussian potentials in momentum space
|
||||||
|
g0(R, p, q) = (exp(-(1/4)*(p + q)^2*R^2)*(-1 + exp(p*q*R^2))*R)/(2*sqrt(π))
|
||||||
|
g1(R, p, q) = (exp(-(1/4)*(p + q)^2*R^2)*(2 + p*q*R^2 + exp(p*q*R^2)*(-2 + p*q*R^2)))/(2*p*sqrt(π)*q*R)
|
||||||
|
|
||||||
function gausslegendre_shifted(a, b, n)
|
function gausslegendre_shifted(a, b, n)
|
||||||
scale = (b - a) / 2
|
scale = (b - a) / 2
|
||||||
|
|
@ -11,11 +13,11 @@ function gausslegendre_shifted(a, b, n)
|
||||||
return (p, w)
|
return (p, w)
|
||||||
end
|
end
|
||||||
|
|
||||||
function get_mesh(vertices::Vector, subdivs::Vector)
|
function get_mesh(vertices, subdivisions)
|
||||||
p = Vector{ComplexF64}()
|
p = Vector{ComplexF64}()
|
||||||
w = Vector{ComplexF64}()
|
w = Vector{ComplexF64}()
|
||||||
for (a, b, subdiv) in zip(vertices, vertices[2:end], subdivs)
|
for (a, b) in zip(vertices, vertices[2:end])
|
||||||
p_new, w_new = gausslegendre_shifted(a, b, subdiv)
|
p_new, w_new = gausslegendre_shifted(a, b, subdivisions)
|
||||||
append!(p, p_new)
|
append!(p, p_new)
|
||||||
append!(w, w_new)
|
append!(w, w_new)
|
||||||
end
|
end
|
||||||
|
|
@ -44,72 +46,7 @@ function identify_pole_i(p, evals, μ=0.5)
|
||||||
end
|
end
|
||||||
|
|
||||||
function quick_pole_E(V_pq, μ=0.5; cs_angle=0.4, cutoff=8.0, meshpoints=256)
|
function quick_pole_E(V_pq, μ=0.5; cs_angle=0.4, cutoff=8.0, meshpoints=256)
|
||||||
p, w = get_mesh([0, cutoff * exp(-1im * cs_angle)], [meshpoints])
|
p, w = get_mesh([0, cutoff * exp(-1im * cs_angle)], meshpoints)
|
||||||
evals = eigvals(get_H_matrix(V_pq, p, w, μ))
|
evals = eigvals(get_H_matrix(V_pq, p, w, μ))
|
||||||
return evals[identify_pole_i(p, evals, μ)]
|
return evals[identify_pole_i(p, evals, μ)]
|
||||||
end
|
end
|
||||||
|
|
||||||
# Gaussian potentials in momentum space
|
|
||||||
g0(R, p, q) = (exp(-(1/4)*(p + q)^2*R^2)*(-1 + exp(p*q*R^2))*R)/(2*sqrt(π))
|
|
||||||
g1(R, p, q) = (exp(-(1/4)*(p + q)^2*R^2)*(2 + p*q*R^2 + exp(p*q*R^2)*(-2 + p*q*R^2)))/(2*p*sqrt(π)*q*R)
|
|
||||||
|
|
||||||
# general potential (numerical integration)
|
|
||||||
jHat(l, z) = z * sphericalbesselj(l, z)
|
|
||||||
function Vl_mat_elem(V_of_r, l, p, q; atol=0, maxevals=10^7, R_cutoff=Inf)
|
|
||||||
integrand(r) = jHat(l, p * r) * V_of_r(r) * jHat(l, q * r)
|
|
||||||
(integral, _) = quadgk(integrand, 0, R_cutoff; atol=atol, maxevals=maxevals)
|
|
||||||
return (2 / pi) * integral
|
|
||||||
end
|
|
||||||
|
|
||||||
"Return the Hamiltonian matrix (and the array of weights) for the given system."
|
|
||||||
function get_3b_H_matrix(coord_system::coordinate_system, V_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m=1.0, kinetic_part=true, potential_part=true; atol=10^-5, maxevals=10^5, R_cutoff=16, verbose=true)
|
|
||||||
if coord_system == jacobi
|
|
||||||
μ1ω1 = μω_global * 1/2
|
|
||||||
μ2ω2 = μω_global * 2
|
|
||||||
μ1 = m * 1/2
|
|
||||||
μ2 = m * 2/3
|
|
||||||
else
|
|
||||||
error("Only Jacobi coordinates are implemented")
|
|
||||||
end
|
|
||||||
|
|
||||||
verbose && println("No of threads = ", Threads.nthreads())
|
|
||||||
|
|
||||||
V_l(j, k, kp) = Vl_mat_elem(V_of_r, j, k, kp; atol=atol, maxevals=maxevals, R_cutoff=R_cutoff)
|
|
||||||
|
|
||||||
ks, ws = get_mesh(vertices, subdivisions)
|
|
||||||
weights = repeat(kron(ws, ws), jmax + 1)
|
|
||||||
block_size = length(ks)
|
|
||||||
tri((j1, j2)) = triangle_ineq(j1, j2, Λ)
|
|
||||||
js = collect(Iterators.filter(tri, iter_prod(0:jmax, 0:jmax)))
|
|
||||||
basis = iter_prod(js, zip(ks, ws), zip(ks, ws)) # basis = ((j1, j2), (k1, w1), (k2, w2))
|
|
||||||
basis_size = length(js) * length(ks)^2
|
|
||||||
@assert length(basis) == basis_size "Something wrong with the basis"
|
|
||||||
verbose && println("Basis size = $basis_size")
|
|
||||||
|
|
||||||
out = spzeros(basis_size, basis_size)
|
|
||||||
|
|
||||||
@time "Block diagonal part" begin
|
|
||||||
if kinetic_part & potential_part
|
|
||||||
Hb_blocks = [kron_sum(get_H_matrix((k, kp) -> V_l(j1, k, kp), ks, ws, μ1), get_T_matrix(ks, μ2)) for (j1, _) in js]
|
|
||||||
elseif kinetic_part
|
|
||||||
Hb_blocks = [kron_sum(get_T_matrix(ks, μ1), get_T_matrix(ks, μ2)) for _ in js]
|
|
||||||
elseif potential_part
|
|
||||||
Hb_blocks = [kron_sum(get_V_matrix((k, kp) -> V_l(j1, k, kp), ks, ws), spzeros(block_size, block_size)) for (j1, _) in js]
|
|
||||||
end
|
|
||||||
out += blockdiag(sparse.(Hb_blocks)...)
|
|
||||||
end
|
|
||||||
|
|
||||||
if potential_part
|
|
||||||
basis_ho = ho_basis_2B(E_max, Λ)
|
|
||||||
verbose && println("HO basis size = ", basis_ho.dim)
|
|
||||||
|
|
||||||
@time "V2_HO" V2_HO = get_jacobi_V2_matrix(V_of_r, basis_ho, μω_global)
|
|
||||||
|
|
||||||
@time "W_right" W_right = get_W_matrix(basis, basis_ho, μ1ω1, μ2ω2; weights=true)
|
|
||||||
@time "W_left" W_left = get_W_matrix(basis, basis_ho, μ1ω1, μ2ω2; weights=false)
|
|
||||||
|
|
||||||
@time "V2" out += W_left * V2_HO * transpose(W_right)
|
|
||||||
end
|
|
||||||
|
|
||||||
return (out, weights)
|
|
||||||
end
|
|
||||||
|
|
@ -1,25 +0,0 @@
|
||||||
using SparseArrays, Arpack
|
|
||||||
include("common.jl")
|
|
||||||
include("p_space.jl")
|
|
||||||
|
|
||||||
E_target = -0.3919
|
|
||||||
|
|
||||||
μ = 0.5
|
|
||||||
Va = -2
|
|
||||||
Ra = 2
|
|
||||||
V_of_r(r) = Va * exp(-r^2 / Ra^2)
|
|
||||||
|
|
||||||
vertices = [0, 0.5 - 0.3im, 1, 4]
|
|
||||||
subdivisions = [16, 16, 16]
|
|
||||||
ks, ws = get_mesh(vertices, subdivisions)
|
|
||||||
ls = collect(0:4)
|
|
||||||
|
|
||||||
V_l(l, k, kp) = Vl_mat_elem(V_of_r, l, k, kp; atol=10^-5, maxevals=10^5, R_cutoff=16)
|
|
||||||
|
|
||||||
H_l = [get_H_matrix((k, kp) -> V_l(l, k, kp), ks, ws, μ) for l in ls]
|
|
||||||
|
|
||||||
H1 = blockdiag(sparse.(H_l)...)
|
|
||||||
|
|
||||||
H = H1
|
|
||||||
@time "Eigenvalues" evals, _ = eigs(H, nev=10, ncv=50, which=:SR, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
|
||||||
E = nearest(evals, E_target)
|
|
||||||
|
|
@ -1,18 +0,0 @@
|
||||||
using Arpack
|
|
||||||
include("p_space.jl")
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
V_of_r(r) = -2 * exp(-r^2 / 4)
|
|
||||||
|
|
||||||
vertices = [0, 0.5 - 0.3im, 1, 4]
|
|
||||||
subdivisions = [10, 10, 10]
|
|
||||||
jmax = 4
|
|
||||||
|
|
||||||
E_max = 30
|
|
||||||
μω_global = 0.5
|
|
||||||
|
|
||||||
H, _ = get_3b_H_matrix(jacobi, V_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m)
|
|
||||||
|
|
||||||
@time "Eigenvalues" evals, _ = eigs(H, nev=3, ncv=24, which=:SR, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
|
||||||
display(evals)
|
|
||||||
|
|
@ -1,20 +0,0 @@
|
||||||
using Arpack
|
|
||||||
include("p_space.jl")
|
|
||||||
|
|
||||||
target = 4.0766890719636875 - 0.012758927741074495im
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
|
|
||||||
vertices = [0, 2 - 0.2im, 3, 4]
|
|
||||||
subdivisions = [15, 10, 10]
|
|
||||||
jmax = 4
|
|
||||||
|
|
||||||
E_max = 40
|
|
||||||
μω_global = 0.5
|
|
||||||
|
|
||||||
H, _ = get_3b_H_matrix(jacobi, V_of_r, vertices, subdivisions, jmax, μω_global, E_max, Λ, m)
|
|
||||||
|
|
||||||
@time "Eigenvalues" evals, _ = eigs(H, sigma=target, maxiter=5000, tol=1e-5, ritzvec=false, check=1)
|
|
||||||
display(evals)
|
|
||||||
|
|
@ -0,0 +1,61 @@
|
||||||
|
{
|
||||||
|
"cells": [
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"using Plots, LinearAlgebra\n",
|
||||||
|
"include(\"p_space.jl\")"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"vertices = (0, 1 - 0.5im, 2, 6)\n",
|
||||||
|
"subdivisions = 64\n",
|
||||||
|
"p, w = get_mesh(vertices, subdivisions)\n",
|
||||||
|
"\n",
|
||||||
|
"scatter(real.(p), imag.(p))"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# ComplexScaling-FV: Eq. (54)\n",
|
||||||
|
"V_pq(p, q) = -10 * g1(1, p, q)\n",
|
||||||
|
"\n",
|
||||||
|
"H = get_H_matrix(V_pq, p, w)\n",
|
||||||
|
"evals = eigen(H).values\n",
|
||||||
|
"\n",
|
||||||
|
"E_target = 0.258 - 0.164im\n",
|
||||||
|
"E = evals[argmin(norm.(evals .- E_target))]\n",
|
||||||
|
"\n",
|
||||||
|
"print(\"E = $E\")\n",
|
||||||
|
"scatter(real.(evals), imag.(evals), xlim = (0,1))"
|
||||||
|
]
|
||||||
|
}
|
||||||
|
],
|
||||||
|
"metadata": {
|
||||||
|
"kernelspec": {
|
||||||
|
"display_name": "Julia 1.9.0",
|
||||||
|
"language": "julia",
|
||||||
|
"name": "julia-1.9"
|
||||||
|
},
|
||||||
|
"language_info": {
|
||||||
|
"file_extension": ".jl",
|
||||||
|
"mimetype": "application/julia",
|
||||||
|
"name": "julia",
|
||||||
|
"version": "1.9.0"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"nbformat": 4,
|
||||||
|
"nbformat_minor": 2
|
||||||
|
}
|
||||||
|
|
@ -0,0 +1,69 @@
|
||||||
|
{
|
||||||
|
"cells": [
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"using Plots, LinearAlgebra\n",
|
||||||
|
"include(\"p_space.jl\")"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"vertices = (0, 0.5 - 0.3im, 1, 6)\n",
|
||||||
|
"subdivisions = 64\n",
|
||||||
|
"p, w = get_mesh(vertices, subdivisions)\n",
|
||||||
|
"\n",
|
||||||
|
"# resonance example from my thesis\n",
|
||||||
|
"V_basis(p, q) = 2*g0(4, p, q) - 3*g0(2, p, q)\n",
|
||||||
|
"\n",
|
||||||
|
"basis_eig = eigen(get_H_matrix(V_basis, p, w))\n",
|
||||||
|
"basis = basis_eig.vectors .* w\n",
|
||||||
|
"\n",
|
||||||
|
"evals = basis_eig.values\n",
|
||||||
|
"E_target = 0.7\n",
|
||||||
|
"E = evals[argmin(norm.(evals .- E_target))]\n",
|
||||||
|
"\n",
|
||||||
|
"print(\"E = $E\")\n",
|
||||||
|
"scatter(real.(evals), imag.(evals), xlim = (0,2))"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# ResonanceEC: Eq. (20)\n",
|
||||||
|
"V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q))\n",
|
||||||
|
"\n",
|
||||||
|
"H = get_H_matrix(V_system(0.45), p, w)\n",
|
||||||
|
"H_berggren = transpose(basis) * H * basis\n",
|
||||||
|
"\n",
|
||||||
|
"evals = eigvals(H)\n",
|
||||||
|
"scatter(real.(evals), imag.(evals), xlim = (0, 0.5))"
|
||||||
|
]
|
||||||
|
}
|
||||||
|
],
|
||||||
|
"metadata": {
|
||||||
|
"kernelspec": {
|
||||||
|
"display_name": "Julia 1.9.0",
|
||||||
|
"language": "julia",
|
||||||
|
"name": "julia-1.9"
|
||||||
|
},
|
||||||
|
"language_info": {
|
||||||
|
"file_extension": ".jl",
|
||||||
|
"mimetype": "application/julia",
|
||||||
|
"name": "julia",
|
||||||
|
"version": "1.9.0"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"nbformat": 4,
|
||||||
|
"nbformat_minor": 2
|
||||||
|
}
|
||||||
|
|
@ -1,8 +0,0 @@
|
||||||
include("../p_space_3body_resonance.jl")
|
|
||||||
|
|
||||||
@time "Eigenvectors" evals, evecs = eigs(H, sigma=target, maxiter=5000, tol=1e-5, ritzvec=true, check=1)
|
|
||||||
|
|
||||||
weights_mat = spdiagm(repeat(kron(ws, ws), jmax + 1))
|
|
||||||
N = transpose(evecs) * weights_mat * evecs
|
|
||||||
|
|
||||||
display(abs.(N))
|
|
||||||
|
|
@ -2,35 +2,27 @@
|
||||||
# gfortran -shared -fPIC osbrac.f90 -o osbrac.so
|
# gfortran -shared -fPIC osbrac.f90 -o osbrac.so
|
||||||
# gfortran -shared -fPIC allosbrac.f90 -o allosbrac.so
|
# gfortran -shared -fPIC allosbrac.f90 -o allosbrac.so
|
||||||
|
|
||||||
println("OSBRACKETS test against Buck et al.")
|
println("OSBRACKETS test against Brody et al.")
|
||||||
|
|
||||||
n1 = [0,0,0,0,0,0,2,2,2,2];
|
Emax = 3
|
||||||
l1 = [0,1,1,2,2,2,2,2,2,2];
|
Λ = 1
|
||||||
n2 = [0,0,0,0,0,0,1,1,1,1];
|
l1 = 1
|
||||||
l2 = [0,3,5,2,4,5,3,3,4,4];
|
l2 = 2
|
||||||
N = [0,0,0,0,1,0,0,1,0,3];
|
|
||||||
L = [0,2,1,1,3,5,3,0,2,2];
|
|
||||||
n = [0,1,0,0,0,0,1,2,4,0];
|
|
||||||
l = [0,0,5,3,1,2,6,5,2,4];
|
|
||||||
Λ = [0,2,6,4,3,4,4,5,2,4];
|
|
||||||
|
|
||||||
function calculate_single_bracket(n1′, l1′, n2′, l2′, n1, l1, n2, l2, Λ) # Efros notation -- DON'T CONFUSE
|
ϵ = (Emax - Λ) % 2
|
||||||
|
N = (l1 - l2 + Λ - ϵ) ÷ 2
|
||||||
|
M = (l1 + l2 - Λ - ϵ) ÷ 2
|
||||||
|
L = Λ
|
||||||
|
NQMAX = Emax
|
||||||
|
CO = 1/sqrt(2)
|
||||||
|
SI = 1/sqrt(2)
|
||||||
|
FIRSTCALL = true
|
||||||
|
# from source: BRAC(NP,N1P,MP,N1,N2) with dimensions BRAC(0:L,0:(NQMAX-L)/2,0:(NQMAX-L)/2,0:(NQMAX-L)/2,0:(NQMAX-L)/2)
|
||||||
|
BRAC = zeros(Float64, L + 1, (NQMAX - L) ÷ 2 + 1,(NQMAX - L) ÷ 2 + 1,(NQMAX - L) ÷ 2 + 1,(NQMAX - L) ÷ 2 + 1)
|
||||||
|
|
||||||
Emax = max(l1′ + l2′ + 2 * (n1′ + n2′), l1 + l2 + 2 * (n1 + n2))
|
@ccall "../OSBRACKETS/osbrac.so".osbrac_(N::Ref{Int32},M::Ref{Int32},L::Ref{Int32},NQMAX::Ref{Int32},CO::Ref{Float64},SI::Ref{Float64},FIRSTCALL::Ref{UInt8},BRAC::Ptr{Array{Float64}})::Cvoid
|
||||||
|
|
||||||
ϵ = (Emax - Λ) % 2
|
|
||||||
N = (l1 - l2 + Λ - ϵ) ÷ 2
|
|
||||||
M = (l1 + l2 - Λ - ϵ) ÷ 2
|
|
||||||
L = Λ
|
|
||||||
NQMAX = Emax
|
|
||||||
CO = 1/sqrt(2)
|
|
||||||
SI = 1/sqrt(2)
|
|
||||||
FIRSTCALL = true
|
|
||||||
# from source: BRAC(NP,N1P,MP,N1,N2) with dimensions BRAC(0:L,0:(NQMAX-L)/2,0:(NQMAX-L)/2,0:(NQMAX-L)/2,0:(NQMAX-L)/2)
|
|
||||||
BRAC = zeros(Float64, L + 1, (NQMAX - L) ÷ 2 + 1,(NQMAX - L) ÷ 2 + 1,(NQMAX - L) ÷ 2 + 1,(NQMAX - L) ÷ 2 + 1)
|
|
||||||
|
|
||||||
@ccall "../OSBRACKETS/osbrac.so".osbrac_(N::Ref{Int32},M::Ref{Int32},L::Ref{Int32},NQMAX::Ref{Int32},CO::Ref{Float64},SI::Ref{Float64},FIRSTCALL::Ref{UInt8},BRAC::Ptr{Array{Float64}})::Cvoid
|
|
||||||
|
|
||||||
|
function get_bracket(n1′, l1′, n2′, l2′, n1, n2)
|
||||||
Nq = l1 + l2 + 2 * (n1 + n2)
|
Nq = l1 + l2 + 2 * (n1 + n2)
|
||||||
Nq′ = l1′ + l2′ + 2 * (n1′ + n2′)
|
Nq′ = l1′ + l2′ + 2 * (n1′ + n2′)
|
||||||
if Nq ≠ Nq′; return 0; end
|
if Nq ≠ Nq′; return 0; end
|
||||||
|
|
@ -39,55 +31,13 @@ function calculate_single_bracket(n1′, l1′, n2′, l2′, n1, l1, n2, l2, Λ
|
||||||
M′ = (l1′ + l2′ - Λ - ϵ) ÷ 2
|
M′ = (l1′ + l2′ - Λ - ϵ) ÷ 2
|
||||||
N1 = n1
|
N1 = n1
|
||||||
N2 = n2
|
N2 = n2
|
||||||
|
|
||||||
return BRAC[N′ + 1, N1′ + 1, M′ + 1, N1 + 1, N2 + 1]
|
return BRAC[N′ + 1, N1′ + 1, M′ + 1, N1 + 1, N2 + 1]
|
||||||
end
|
end
|
||||||
|
|
||||||
println("OSBRAC results")
|
test_n = [0 0 0 0 1 1]
|
||||||
osbracs = calculate_single_bracket.(n1, l1, n2, l2, N, L, n, l, Λ)
|
test_l = [0 1 1 2 0 1]
|
||||||
display(osbracs)
|
test_N = [1 0 1 0 0 0]
|
||||||
|
test_L = [1 2 0 1 1 0]
|
||||||
|
|
||||||
function calculate_all_brackets(n1′, l1′, n2′, l2′, n1, l1, n2, l2, Λ) # Efros notation -- DON'T CONFUSE
|
bracs = get_bracket.(test_n, test_l, test_N, test_L, 0, 0)
|
||||||
Emax = max(maximum(l1′ + l2′ + 2 * (n1′ + n2′)), maximum(l1 + l2 + 2 * (n1 + n2)))
|
display(bracs)
|
||||||
|
|
||||||
LMIN = minimum(Λ)
|
|
||||||
LMAX = maximum(Λ)
|
|
||||||
CO = 1/sqrt(2)
|
|
||||||
SI = 1/sqrt(2)
|
|
||||||
|
|
||||||
BRACs = Vector{Any}(undef, 2) # two arrays for both parities
|
|
||||||
for parity in 1:2
|
|
||||||
NQMAX = Emax - mod1(Emax, 2) + parity
|
|
||||||
# dimensions BRAC(0:LMAX,0:(NQMAX-LMIN)/2,0:(NQMAX-LMIN)/2,0:(NQMAX-LMIN)/2,0:(NQMAX-LMIN)/2,0:LMAX,0:(NQMAX-LMIN)/2,LMIN:LMAX)
|
|
||||||
BRACs[parity] = zeros(Float64, 1 + LMAX, 1 + (NQMAX - LMIN) ÷ 2, 1 + (NQMAX - LMIN) ÷ 2, 1 + (NQMAX - LMIN) ÷ 2, 1 + (NQMAX - LMIN) ÷ 2, 1 + LMAX, 1 + (NQMAX-LMIN) ÷ 2, 1 + LMAX-LMIN)
|
|
||||||
@ccall "../OSBRACKETS/allosbrac.so".allosbrac_(NQMAX::Ref{Int32},LMIN::Ref{Int32},LMAX::Ref{Int32},CO::Ref{Float64},SI::Ref{Float64},BRACs[parity]::Ptr{Array{Float64}})::Cvoid
|
|
||||||
end
|
|
||||||
|
|
||||||
out = Float64[]
|
|
||||||
|
|
||||||
for (Λ, n1′, l1′, n2′, l2′, n1, l1, n2, l2) in zip(Λ, n1′, l1′, n2′, l2′, n1, l1, n2, l2)
|
|
||||||
# BRAC(NP,N1P,MP,N1,N2,N,M,L)
|
|
||||||
|
|
||||||
Nq = l1 + l2 + 2 * (n1 + n2)
|
|
||||||
ϵ = (Nq - Λ) % 2
|
|
||||||
NP = (l1′ - l2′ + Λ - ϵ) ÷ 2
|
|
||||||
N1P = n1′
|
|
||||||
MP = (l1′ + l2′ - Λ - ϵ) ÷ 2
|
|
||||||
N1 = n1
|
|
||||||
N2 = n2
|
|
||||||
N = (l1 - l2 + Λ - ϵ) ÷ 2
|
|
||||||
M = (l1 + l2 - Λ - ϵ) ÷ 2
|
|
||||||
|
|
||||||
parity = mod1(l1 + l2, 2)
|
|
||||||
push!(out, BRACs[parity][1 + NP, 1 + N1P, 1 + MP, 1 + N1, 1 + N2, 1 + N, 1 + M, 1 + LMIN + Λ])
|
|
||||||
end
|
|
||||||
|
|
||||||
return out
|
|
||||||
end
|
|
||||||
|
|
||||||
println("ALLOSBRAC results")
|
|
||||||
allosbracs = calculate_all_brackets(n1, l1, n2, l2, N, L, n, l, Λ)
|
|
||||||
display(allosbracs)
|
|
||||||
|
|
||||||
println("Difference")
|
|
||||||
display(abs.(allosbracs - osbracs))
|
|
||||||
65
test/misc.jl
65
test/misc.jl
|
|
@ -1,65 +0,0 @@
|
||||||
using SparseArrays, LinearAlgebra
|
|
||||||
include("../common.jl")
|
|
||||||
|
|
||||||
#### gram_schmidt ####
|
|
||||||
|
|
||||||
n = 64
|
|
||||||
N = 8
|
|
||||||
|
|
||||||
vecs = [rand(n) + 1im .* rand(n) for _ in 1:N]
|
|
||||||
|
|
||||||
ws = rand(n)
|
|
||||||
ws_mat = spdiagm(ws)
|
|
||||||
|
|
||||||
basis = hcat(vecs...)
|
|
||||||
|
|
||||||
H = rand(n, n)
|
|
||||||
H += transpose(H) # complex symmetric
|
|
||||||
|
|
||||||
H_EC = transpose(basis) * ws_mat * H * ws_mat * basis
|
|
||||||
N_EC = transpose(basis) * ws_mat * basis
|
|
||||||
|
|
||||||
evals = eigvals(H_EC, N_EC)
|
|
||||||
println("Eigenvalues with GEVP:")
|
|
||||||
display(evals)
|
|
||||||
|
|
||||||
ortho_basis = hcat(gram_schmidt!(vecs, ws)...)
|
|
||||||
N_ortho = transpose(ortho_basis) * ws_mat * ortho_basis
|
|
||||||
println("Norm matrix after Gram-Schmidt:")
|
|
||||||
display(round.(N_ortho; digits=2)) # should be ≈I
|
|
||||||
@assert N_ortho ≈ I(N) "Gram-Schmidt did not yield an orthogonal basis"
|
|
||||||
|
|
||||||
H_EC_ortho = transpose(ortho_basis) * ws_mat * H * ws_mat * ortho_basis
|
|
||||||
|
|
||||||
evals_ortho = eigvals(H_EC_ortho)
|
|
||||||
println("Eigenvalues after Gram-Schmidt:")
|
|
||||||
display(evals_ortho)
|
|
||||||
|
|
||||||
@assert evals ≈ evals_ortho "Gram-Schmidt did not preserve the eigenvalues"
|
|
||||||
|
|
||||||
############
|
|
||||||
println("\nRepeat with a redundant basis\n")
|
|
||||||
|
|
||||||
println("Original dimensionality = $(length(vecs))")
|
|
||||||
|
|
||||||
for pow in 6:9
|
|
||||||
noise = rand(n) ./ 10^pow
|
|
||||||
new_vec = vecs[1] + noise
|
|
||||||
push!(vecs, new_vec)
|
|
||||||
end
|
|
||||||
|
|
||||||
println("Dimensionality before Gram-Schmidt = $(length(vecs))")
|
|
||||||
|
|
||||||
ortho_vecs = gram_schmidt!(vecs, ws; verbose=true)
|
|
||||||
ortho_basis = hcat(ortho_vecs...)
|
|
||||||
println("Dimensionality after Gram-Schmidt = $(length(ortho_vecs))")
|
|
||||||
|
|
||||||
H_EC_ortho = transpose(ortho_basis) * ws_mat * H * ws_mat * ortho_basis
|
|
||||||
|
|
||||||
evals_ortho = eigvals(H_EC_ortho)
|
|
||||||
println("Eigenvalues after Gram-Schmidt:")
|
|
||||||
display(evals_ortho)
|
|
||||||
|
|
||||||
@assert isapprox(evals, evals_ortho; atol=1e-3) "Gram-Schmidt did not approximately preserve the eigenvalues"
|
|
||||||
|
|
||||||
######################
|
|
||||||
|
|
@ -1,21 +1,21 @@
|
||||||
println("### Test: transpose(U) * U == identity")
|
println("### Test: U' * U == identity")
|
||||||
|
|
||||||
using LinearAlgebra
|
using LinearAlgebra
|
||||||
|
|
||||||
include("../ho_basis.jl")
|
include("../ho_basis.jl")
|
||||||
|
|
||||||
E_max = 30
|
E_max = 15
|
||||||
Λ = 0
|
Λ = 0
|
||||||
|
|
||||||
println("No of threads = ", Threads.nthreads())
|
println("No of threads = ", Threads.nthreads())
|
||||||
|
|
||||||
@time "Basis" basis = ho_basis_2B(E_max, Λ)
|
@time "Basis" Es, n1s, l1s, n2s, l2s = get_2p_basis(E_max)
|
||||||
|
|
||||||
println("Basis size = ", basis.dim)
|
println("Basis size = ", length(Es))
|
||||||
|
|
||||||
@time "Moshinsky brackets" U = Moshinsky_transform(basis)
|
@time "Moshinsky brackets" U = Moshinsky_transform(Es, n1s, l1s, n2s, l2s, Λ)
|
||||||
|
|
||||||
check = transpose(U) * U - I
|
check = U' * U - I
|
||||||
|
|
||||||
println("Maximum = ", maximum(abs.(check)))
|
println("Maximum = ", maximum(abs.(check)))
|
||||||
println("Norm = ", sum(check .* conj(check)))
|
println("Norm = ", sum(check .* conj(check)))
|
||||||
|
|
|
||||||
|
|
@ -1,20 +0,0 @@
|
||||||
using Statistics
|
|
||||||
|
|
||||||
include("../ho_basis.jl")
|
|
||||||
|
|
||||||
ls = 0:4
|
|
||||||
ns = 0:15
|
|
||||||
|
|
||||||
R = 2
|
|
||||||
μω_gen = 0.3
|
|
||||||
V_of_r(r) = exp(-r^2 / R^2)
|
|
||||||
|
|
||||||
for l in ls
|
|
||||||
println("Testing l=", l)
|
|
||||||
|
|
||||||
@time "Analytical" V_ana = [V_Gaussian(R, l, n1, n2; μω_gen=μω_gen) for (n1, n2) in Iterators.product(ns, ns)]
|
|
||||||
@time "Numerical" V_num = [V_numerical(V_of_r, l, n1, n2; μω_gen=μω_gen) for (n1, n2) in Iterators.product(ns, ns)]
|
|
||||||
|
|
||||||
mean_diff = Statistics.mean(abs.((V_num .- V_ana) ./ V_ana))
|
|
||||||
@assert mean_diff < 1e-5 "Mean absoute difference of $(mean_diff * 100)% between analytical and numerical calculations"
|
|
||||||
end
|
|
||||||
|
|
@ -1,57 +0,0 @@
|
||||||
using LinearAlgebra, SparseArrays, Plots
|
|
||||||
include("../p_space.jl")
|
|
||||||
include("../ho_basis.jl")
|
|
||||||
include("../berggren.jl")
|
|
||||||
|
|
||||||
println("No of threads = ", Threads.nthreads())
|
|
||||||
atol = 10^-5
|
|
||||||
maxevals = 10^5
|
|
||||||
R_cutoff = 16
|
|
||||||
|
|
||||||
Λ = 0
|
|
||||||
m = 1.0
|
|
||||||
μ = m/2 # due to simple relative coordinates
|
|
||||||
|
|
||||||
vertices = [0, 2 - 0.2im, 3, 4]
|
|
||||||
subdivisions = [15, 10, 10]
|
|
||||||
ks, ws = get_mesh(vertices, subdivisions)
|
|
||||||
|
|
||||||
jmax = 4
|
|
||||||
tri((j1, j2)) = triangle_ineq(j1, j2, Λ)
|
|
||||||
js = collect(Iterators.filter(tri, iter_prod(0:jmax, 0:jmax)))
|
|
||||||
|
|
||||||
basis = iter_prod(js, zip(ks, ws), zip(ks, ws)) # basis = ((j1, j2), (k1, w1), (k2, w2))
|
|
||||||
basis_size = length(js) * length(ks)^2
|
|
||||||
@assert length(basis) == basis_size "Something wrong with the basis"
|
|
||||||
println("Basis size = $basis_size")
|
|
||||||
|
|
||||||
V_of_r(r) = 2 * exp(-(r-3)^2 / (1.5)^2)
|
|
||||||
V_l(j, k, kp) = Vl_mat_elem(V_of_r, j, k, kp; atol=atol, maxevals=maxevals, R_cutoff=R_cutoff)
|
|
||||||
|
|
||||||
# generate p-space bases (actually identity matrices)
|
|
||||||
@time "p-space bases" ps_bases = [Matrix(spdiagm(1 ./ sqrt.(ws))) for _ in 0:jmax]
|
|
||||||
|
|
||||||
# generate Berggren bases
|
|
||||||
@time "Berggren bases" begin
|
|
||||||
berg_bases = Vector{Matrix{ComplexF64}}(undef, jmax + 1)
|
|
||||||
for j in 0:jmax
|
|
||||||
_, berg_basis = eigen(get_H_matrix((k, kp) -> V_l(j, k, kp), ks, ws, μ); permute=false, scale=false)
|
|
||||||
N_berg = sum(berg_basis.^2 .* ws, dims=1)
|
|
||||||
berg_bases[1 + j] = berg_basis ./ transpose(sqrt.(N_berg))
|
|
||||||
end
|
|
||||||
end
|
|
||||||
|
|
||||||
@time "BB tranform matrix" begin
|
|
||||||
U_blocks = [kron(berg_bases[1 + j1] .* ws, berg_bases[1 + j2] .* ws) for (j1, j2) in js]
|
|
||||||
U = blockdiag(sparse.(U_blocks)...)
|
|
||||||
end
|
|
||||||
|
|
||||||
@time "In p-space" T_cross_PS = get_2p_p1p2_matrix(length(ks), js, Λ, ps_bases, ps_bases, ws)
|
|
||||||
@time "In BB" T_cross_BB = get_2p_p1p2_matrix(length(ks), js, Λ, berg_bases, berg_bases, ws)
|
|
||||||
|
|
||||||
@time "Basis transform" T_cross_transformed = transpose(U) * T_cross_PS * U
|
|
||||||
|
|
||||||
diff = abs.(T_cross_transformed .- T_cross_BB)
|
|
||||||
println("Max error = $(maximum(diff))")
|
|
||||||
|
|
||||||
#@time "In in HO" T_cross_HO = get_2p_p1p2_matrix(n1s, l1s, n2s, l2s, Λ, μω, μω; dtype=ComplexF64)
|
|
||||||
|
|
@ -1,21 +0,0 @@
|
||||||
using Plots
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
vertices = [0, 1 - 0.5im, 2, 6]
|
|
||||||
subdivisions = [64, 64, 64]
|
|
||||||
p, w = get_mesh(vertices, subdivisions)
|
|
||||||
|
|
||||||
scatter(real.(p), imag.(p))
|
|
||||||
|
|
||||||
# ComplexScaling-FV: Eq. (54)
|
|
||||||
V_pq(p, q) = -10 * g1(1, p, q)
|
|
||||||
|
|
||||||
H = get_H_matrix(V_pq, p, w)
|
|
||||||
evals = eigen(H).values
|
|
||||||
|
|
||||||
E_target = 0.258 - 0.164im
|
|
||||||
E = nearest(evals, E_target)
|
|
||||||
|
|
||||||
print("E = $E")
|
|
||||||
scatter(real.(evals), imag.(evals), xlim = (0,1))
|
|
||||||
|
|
@ -1,384 +0,0 @@
|
||||||
2,0,2,3,7,3,5,1,3,2.2274833794413214,0.45049378982090493,0.
|
|
||||||
8,1,3,3,2,2,0,0,2,2.202162451867089,2.3467827533568713,0.
|
|
||||||
4,1,7,1,0,2,0,3,2,1.1507774013957701,1.7553231930689552,0.
|
|
||||||
3,3,7,1,7,0,3,3,3,1.7207018027964498,0.5156168080476098,0.
|
|
||||||
10,3,2,3,2,2,8,1,2,1.8402526798628767,2.8981293881514993,0.
|
|
||||||
4,3,8,1,0,3,0,3,3,2.471579266664868,0.9515272650259292,0.
|
|
||||||
9,2,8,3,10,3,1,2,1,2.1675261170354094,0.38546777353112605,0.
|
|
||||||
2,2,7,3,4,3,7,3,3,2.76860332907484,0.7916486882071214,0.
|
|
||||||
2,3,3,3,5,1,1,0,1,1.6146747022554235,2.8377955838430013,0.
|
|
||||||
4,3,5,0,7,3,2,3,3,0.9198951336314427,0.9950080709108984,0.
|
|
||||||
7,3,4,2,2,2,8,0,2,0.2364795732399818,1.839193911487496,0.
|
|
||||||
7,3,3,3,5,1,6,2,3,0.29514953289458434,0.3780481641738338,0.
|
|
||||||
5,2,4,1,9,3,1,0,3,2.8086957499249543,2.1260743936378903,0.
|
|
||||||
4,1,6,1,1,0,10,0,0,2.6628402589565523,2.425950546585856,0.
|
|
||||||
8,3,1,0,6,3,0,3,3,2.1328726647005265,0.9431361454676961,0.
|
|
||||||
4,3,5,0,5,1,0,3,3,2.576495013458395,0.37293785437117544,0.
|
|
||||||
10,3,4,0,1,2,2,3,3,2.0030278179077214,1.1187270744696427,0.
|
|
||||||
9,1,8,2,1,1,5,3,2,2.789813153068671,2.6763038014457665,0.
|
|
||||||
5,0,6,1,5,3,9,3,1,1.9319197502299517,0.6216330166798851,0.
|
|
||||||
8,0,6,2,7,3,8,2,2,1.8084358691372344,2.0181750785519634,0.
|
|
||||||
3,1,5,1,9,0,7,1,1,1.599191638971793,1.552252944957802,0.
|
|
||||||
1,0,7,2,6,3,4,2,2,2.4217191310288078,1.2381047980553532,0.
|
|
||||||
9,3,2,0,1,2,4,2,3,1.261927284387386,0.39708064622835604,0.
|
|
||||||
0,1,5,1,5,0,0,2,2,0.407989632316895,2.4021451804173477,0.
|
|
||||||
3,1,6,3,9,2,2,1,2,1.2053920704239278,0.25375827636999704,0.
|
|
||||||
9,2,7,0,5,2,5,3,2,2.539812963473447,0.7382167735661622,0.
|
|
||||||
10,2,7,3,4,3,0,0,3,0.8427632747449358,0.7622120453149561,0.
|
|
||||||
5,2,0,2,7,3,5,0,3,0.460843656858851,0.583650962079461,0.
|
|
||||||
8,1,8,0,6,2,0,2,1,2.056601089239093,0.22262414697020771,0.
|
|
||||||
4,2,4,2,6,1,0,3,3,1.4808637079563942,1.358717819227813,0.
|
|
||||||
10,3,4,3,4,0,5,2,2,1.5284998408295385,2.925365977036159,0.
|
|
||||||
10,1,1,1,5,1,10,1,0,1.1846342029623314,1.186468360104492,0.
|
|
||||||
9,0,5,2,1,2,2,3,2,1.733114516642603,2.8283467464705456,0.
|
|
||||||
8,3,8,1,10,3,4,3,3,0.8918790824078111,0.9129770894549143,0.
|
|
||||||
10,0,9,3,5,3,8,3,3,0.6417078137732939,1.965954991753998,0.
|
|
||||||
10,2,8,2,8,2,8,3,4,0.6600917142427227,0.987969186819146,0.
|
|
||||||
8,2,4,1,3,2,9,1,3,0.8935573241720247,2.1162172596314806,0.
|
|
||||||
4,2,10,3,7,2,7,2,2,1.5052737158592535,2.7171501665361504,0.
|
|
||||||
0,1,3,1,5,2,5,0,2,1.577080591236359,0.34164973402053134,0.
|
|
||||||
0,2,10,3,6,3,6,0,3,2.7925439641102736,1.831313118698632,0.
|
|
||||||
1,0,9,3,9,0,0,3,3,2.3331462089933934,1.6945470525813482,0.
|
|
||||||
8,2,7,3,7,2,3,1,1,1.4988059394628408,1.3530552540483156,0.
|
|
||||||
9,2,8,1,8,3,6,1,2,0.6337531145346973,1.5268564417495032,0.
|
|
||||||
10,1,0,0,7,2,7,2,1,1.3790583258395,1.5669277734464737,0.
|
|
||||||
3,0,0,2,6,0,4,2,2,2.4730048216795737,2.70759381187844,0.
|
|
||||||
8,1,3,1,5,3,8,2,2,0.8399302875607697,0.32436518041750384,0.
|
|
||||||
4,1,1,2,4,3,10,1,3,0.8437127682376069,1.7940889253407324,0.
|
|
||||||
9,0,3,1,4,3,2,2,1,0.8755496194878809,0.9675651711989977,0.
|
|
||||||
9,0,2,2,3,1,8,2,2,1.4950146011492311,2.8152388609955157,0.
|
|
||||||
7,0,8,3,2,3,1,3,3,2.2587603015484863,2.0666085413750848,0.
|
|
||||||
10,3,9,1,4,3,0,2,4,1.5489892079797567,1.7151221413658782,0.
|
|
||||||
3,0,1,2,2,3,4,1,2,0.25514014111567285,2.06281752109991,0.
|
|
||||||
5,1,1,0,9,1,3,2,1,1.3329695370285548,0.5298097905264418,0.
|
|
||||||
8,1,1,0,5,0,9,1,1,2.95254173854617,2.1687131876307753,0.
|
|
||||||
6,1,8,0,5,2,1,2,1,1.5801597077900533,2.257296177275017,0.
|
|
||||||
10,0,9,1,9,3,4,3,1,2.5352867964035397,1.7981404960338239,0.
|
|
||||||
10,2,1,2,0,0,9,3,3,2.06463437201601,1.4295850747781111,0.
|
|
||||||
1,0,1,3,6,3,6,3,3,0.8872777463742612,1.9002353924129451,0.
|
|
||||||
6,3,10,0,2,3,2,1,3,1.2410196118280048,1.8568476669071874,0.
|
|
||||||
8,2,1,2,2,3,6,3,2,0.5302021471052729,1.2216709678909177,0.
|
|
||||||
1,2,8,1,0,3,3,0,3,2.8876975946711045,2.717322643102454,0.
|
|
||||||
4,2,4,1,0,1,7,0,1,2.0026155982539864,0.7640140889966269,0.
|
|
||||||
1,0,1,1,4,2,3,2,1,1.016582775534538,2.3841546013418045,0.
|
|
||||||
2,0,3,1,10,1,1,1,1,1.6404532954231827,2.1972778830632356,0.
|
|
||||||
7,3,1,3,2,2,4,3,4,2.648817905371118,2.798529790102262,0.
|
|
||||||
9,0,4,1,6,0,2,1,1,2.3138662599243407,1.9450229178773055,0.
|
|
||||||
0,1,8,1,1,1,1,0,1,0.6524221878214105,2.9034613327001972,0.
|
|
||||||
6,1,6,3,0,1,6,2,3,1.831532710689828,2.890598375121809,0.
|
|
||||||
4,1,0,0,0,2,7,3,1,0.626504505885948,2.6340691505629463,0.
|
|
||||||
10,3,2,3,10,3,1,1,2,1.6591641837977482,2.452206213376977,0.
|
|
||||||
3,1,6,2,2,2,7,1,3,1.1999874026225852,2.795586000355012,3.3573329331683617
|
|
||||||
8,3,1,2,1,3,5,2,5,0.6830778416708307,1.1113967340568016,0.
|
|
||||||
7,2,3,1,5,3,8,2,3,1.714181314975816,1.3518067184764888,0.
|
|
||||||
0,2,4,2,0,1,4,1,2,1.060932248034836,0.8036155418186053,-1.1371377045290691
|
|
||||||
4,3,9,3,10,3,10,3,5,0.42490400067847744,1.9191837942593741,0.
|
|
||||||
0,0,6,1,2,2,7,2,1,1.4606282497051195,2.390102981429126,0.
|
|
||||||
8,0,0,1,9,0,10,1,1,2.9779138627844803,1.482942802025069,0.
|
|
||||||
1,2,2,3,3,3,8,3,4,1.540918746101882,0.9621051195430765,0.
|
|
||||||
2,1,1,3,2,0,1,3,3,2.0584800719627525,1.7796724319652748,0.
|
|
||||||
10,2,5,2,4,0,0,2,2,1.2759898818923578,0.6080747387544254,0.
|
|
||||||
1,0,9,3,2,2,10,2,3,2.835152414163625,1.2470542223023808,0.
|
|
||||||
1,2,2,0,6,0,7,2,2,1.8642054302207116,0.48490065380939873,0.
|
|
||||||
5,1,10,2,5,0,10,3,3,0.6428303158043476,0.5501263433952537,2.105493662515726
|
|
||||||
1,1,1,1,4,2,0,1,1,2.91586289712962,0.2030446107349677,0.
|
|
||||||
4,2,0,2,1,3,1,0,3,0.31239590050870536,2.3530015462264577,0.
|
|
||||||
10,0,0,3,0,1,4,3,3,1.4066735483371904,0.3492189542356794,0.
|
|
||||||
3,1,9,3,6,2,4,2,4,2.039761320004641,2.0376049586912632,0.
|
|
||||||
4,1,3,3,2,2,8,2,2,2.485347427066465,2.2425974670112305,0.
|
|
||||||
8,0,10,3,7,3,3,0,3,1.145305638226675,0.7430070058758362,0.
|
|
||||||
1,0,4,3,1,3,3,2,3,2.236729779510484,2.518871850291589,0.
|
|
||||||
9,3,10,2,0,0,9,3,3,2.2231614378407896,2.934256387523143,0.
|
|
||||||
1,3,7,1,6,2,7,1,3,1.739591417445606,2.852892536289553,0.
|
|
||||||
9,0,0,2,7,1,0,2,2,2.628613403491599,1.948716181798507,0.
|
|
||||||
2,0,5,3,3,2,4,3,3,0.7244831235767495,0.307815884940033,0.
|
|
||||||
9,2,2,3,4,0,9,2,2,1.4712965684171477,0.7152116770614816,0.
|
|
||||||
10,0,7,1,7,2,2,2,1,2.0944053463871253,0.7877074060643912,0.
|
|
||||||
10,2,8,2,4,1,4,3,3,1.2400259243830605,1.533481685252715,0.
|
|
||||||
7,2,2,0,6,3,2,1,2,0.7570369287424406,1.5727691945398687,2.4154000428935993
|
|
||||||
9,0,9,2,4,2,5,0,2,2.9903195228568853,1.7342728120955062,0.
|
|
||||||
0,2,2,1,2,2,2,2,3,2.871118456330721,1.8840932443901108,0.
|
|
||||||
1,3,4,1,9,2,8,1,3,2.2820457592186356,2.8820550176739523,0.
|
|
||||||
4,3,9,1,5,3,7,1,2,0.5157719636123437,2.2186070481213065,0.
|
|
||||||
3,3,9,3,7,2,1,1,1,1.806277095079202,2.303389106635813,0.
|
|
||||||
4,2,0,0,8,0,10,2,2,0.4100439145977339,1.0520725633546872,0.
|
|
||||||
10,1,6,0,4,2,2,3,1,1.126129220884609,1.5550003535454513,0.
|
|
||||||
3,2,10,1,6,1,10,1,1,0.2658540542443206,0.608227371386918,0.
|
|
||||||
6,3,5,1,4,1,0,3,3,2.705056186004474,0.42463741187699755,0.
|
|
||||||
10,0,5,2,5,1,1,2,2,0.8756297484714799,0.7584762754047034,0.
|
|
||||||
7,1,7,3,5,1,0,3,4,2.914828464578812,2.261191790401713,0.
|
|
||||||
10,0,3,1,10,2,4,3,1,2.50303873714987,1.5051451493521428,0.
|
|
||||||
1,1,10,0,1,3,0,3,1,1.3072742851724795,1.9423953478998177,0.
|
|
||||||
1,1,3,2,3,2,0,3,2,2.3584710757323544,1.1305990899916512,0.
|
|
||||||
10,3,2,3,3,1,6,1,0,1.4654067038271226,1.985658804098854,0.
|
|
||||||
7,2,0,2,9,2,0,1,1,1.3873373238347515,0.7703862068685314,0.
|
|
||||||
8,3,7,0,8,2,5,1,3,0.8253295318767577,2.899984116328765,0.
|
|
||||||
1,1,0,0,0,1,10,2,1,2.3641832402927276,2.1083504299969444,0.
|
|
||||||
8,2,7,3,10,1,0,2,2,1.3652741505836983,2.0949097737438462,0.
|
|
||||||
8,3,10,0,4,2,5,1,3,0.24020243315960244,1.9022194572611664,0.
|
|
||||||
9,1,9,2,0,0,4,1,1,2.5376357730342507,1.8526159863451914,0.
|
|
||||||
7,3,9,3,3,3,7,0,3,1.6807661985105629,2.687649747358564,0.
|
|
||||||
7,2,2,0,9,2,8,1,2,2.4565884018951802,0.8982169296897196,0.
|
|
||||||
9,2,0,3,10,3,5,1,2,0.6353524665165655,1.0371641317201128,0.
|
|
||||||
3,0,7,3,2,2,1,1,3,2.7958304824728186,2.4767642915078376,0.
|
|
||||||
9,0,7,1,10,1,4,0,1,0.25616825512317254,1.308344122398832,0.
|
|
||||||
1,2,1,3,10,1,1,1,2,2.224763269329724,2.231499665082513,0.
|
|
||||||
8,0,4,3,0,2,5,3,3,0.770816371156068,2.3322630361929546,0.
|
|
||||||
5,1,1,2,2,2,3,1,2,0.8022984805467428,1.4426168425194534,0.
|
|
||||||
7,0,5,0,0,2,10,2,0,1.9511722131189702,0.4950560622323814,0.
|
|
||||||
9,3,7,1,7,0,10,2,2,2.775167770563364,0.45822632372936667,0.
|
|
||||||
7,2,7,3,9,0,6,1,1,2.8779651825740737,2.4436285639972706,0.
|
|
||||||
1,1,6,3,0,0,3,3,3,0.5442622594206528,2.9853682012412737,0.
|
|
||||||
10,2,8,1,4,2,7,3,2,1.329367888136959,0.9413429526552286,0.
|
|
||||||
7,3,0,1,4,1,9,1,2,1.3457614299418887,1.5531678736115984,0.
|
|
||||||
4,2,8,2,10,1,3,3,3,0.8248650313839705,0.6616884801498313,0.
|
|
||||||
0,0,8,3,3,1,10,2,3,2.0673228373578825,1.4463899838479248,0.
|
|
||||||
0,3,0,2,9,3,0,3,5,0.906810420580336,2.811065359991593,0.
|
|
||||||
2,1,2,0,4,3,8,2,1,1.4719829696395097,1.5045170591445123,0.
|
|
||||||
8,3,4,2,7,0,6,2,2,2.934501293001735,2.475691074854602,0.
|
|
||||||
4,1,7,3,9,1,2,2,3,1.5229908742702083,2.7910336359576577,0.
|
|
||||||
3,0,4,3,9,2,8,2,3,0.7395750631044415,2.7437950070306316,0.
|
|
||||||
0,3,10,0,8,3,5,0,3,2.6218495656463876,0.917324005589486,0.
|
|
||||||
0,1,10,3,3,3,7,3,4,1.2601693603785153,2.8311848091672465,0.
|
|
||||||
5,3,3,1,0,2,0,1,2,1.5322630072331016,2.4612811983034675,0.
|
|
||||||
2,0,10,3,4,3,8,2,3,2.581504744416619,2.3352397148922988,0.
|
|
||||||
8,2,10,1,8,2,10,1,1,1.1752269484482167,1.7646051998689414,0.
|
|
||||||
0,2,4,1,10,2,7,0,2,1.1544888515337508,2.637958865288499,0.
|
|
||||||
4,0,6,3,6,1,7,3,3,1.4304562327889254,1.498204529248202,0.
|
|
||||||
8,2,6,1,3,3,6,0,3,2.3823736912605336,2.0477368833060705,0.
|
|
||||||
4,2,2,3,10,1,3,1,1,2.5205545321391254,0.7220948301952252,0.
|
|
||||||
10,0,7,1,8,3,9,2,1,1.2260538418022682,2.451257871532359,0.
|
|
||||||
0,0,0,3,10,1,10,3,3,2.006553045177734,1.173388489802477,0.
|
|
||||||
6,3,3,3,1,3,1,2,5,2.9612859268853233,1.4353422269551883,0.
|
|
||||||
2,1,2,0,2,3,1,2,1,1.3130842056605396,1.1311416250752515,0.
|
|
||||||
3,3,9,3,5,1,3,3,3,0.4024708536057622,2.0642879809097643,0.
|
|
||||||
8,2,7,2,3,2,7,2,0,1.2953268563208105,0.5693758931801445,0.
|
|
||||||
0,3,9,1,8,1,8,1,2,2.152069679767771,1.41039953394532,0.
|
|
||||||
10,3,10,1,2,1,2,3,3,1.922953978768665,1.3543149100065444,0.
|
|
||||||
4,2,5,1,10,3,8,0,3,0.6098612038176467,0.5440300294800884,0.
|
|
||||||
8,2,7,1,3,1,8,2,2,0.608144934570964,2.548777086726803,0.
|
|
||||||
9,2,3,0,6,0,10,2,2,1.7250046113327357,1.415947697499866,0.
|
|
||||||
4,1,3,2,7,0,3,3,3,1.5376308517695847,2.8988547754211718,0.
|
|
||||||
5,3,4,0,6,3,4,2,3,0.7636088061341804,0.7149031039865208,0.
|
|
||||||
4,3,9,3,2,0,6,2,2,1.8376914709492826,1.479579605670744,0.
|
|
||||||
2,3,10,2,5,2,10,0,2,0.8658425814507988,1.1218947490443987,0.
|
|
||||||
6,0,9,1,2,3,10,2,1,2.7269511334702985,1.2873996903274714,0.
|
|
||||||
5,3,1,1,4,1,2,2,3,2.0472755894627443,1.4404427131274424,0.
|
|
||||||
7,3,5,1,5,3,1,0,3,2.1329904508839563,1.7676149035568383,0.
|
|
||||||
0,1,6,1,3,2,1,3,1,1.1352856084425453,2.0153824999161527,0.
|
|
||||||
9,1,4,2,9,2,1,0,2,0.5735267931557,0.883812410190215,0.
|
|
||||||
10,2,8,0,2,2,0,0,2,2.286451077255511,2.3140961306516,0.
|
|
||||||
3,3,3,3,2,2,0,2,2,0.7388292443690174,1.6936545691041545,0.
|
|
||||||
10,2,2,1,10,0,10,2,2,1.4443453550869583,1.112093007841184,0.
|
|
||||||
7,3,6,1,10,3,2,2,4,1.8267825436432625,1.0428273092046476,0.
|
|
||||||
0,0,1,3,9,2,9,1,3,2.4360668218156167,1.0158379194857559,0.
|
|
||||||
9,0,6,3,8,1,5,2,3,2.4459546670789223,2.5273734233477834,0.
|
|
||||||
8,2,0,2,9,3,2,2,1,0.9849892765217048,0.31610510276469483,0.
|
|
||||||
2,0,6,0,3,1,9,1,0,1.646375524767966,0.39139222602227974,0.
|
|
||||||
2,3,0,2,6,0,10,1,1,0.3040006591059585,1.784871103068122,0.
|
|
||||||
7,2,2,2,4,3,6,0,3,1.5667275423919191,1.7334318105948183,0.
|
|
||||||
5,3,2,1,1,2,4,0,2,0.4724440845832545,2.0191544118568805,0.
|
|
||||||
2,1,3,2,6,1,8,3,2,1.6569706450379993,1.22551530354268,0.
|
|
||||||
4,0,8,2,10,3,6,2,2,1.5517249658736345,1.763047858243108,0.
|
|
||||||
6,2,4,2,0,3,3,0,3,2.056076795383781,1.693356584470254,0.
|
|
||||||
1,3,6,1,4,1,3,3,3,1.9168275535161614,2.2496624307522994,0.
|
|
||||||
6,1,1,1,8,1,3,0,1,0.7505340273835843,1.1586206830861343,0.
|
|
||||||
1,2,9,2,8,2,1,0,2,2.5465024812722747,0.40185310698150234,0.
|
|
||||||
6,2,1,3,8,3,5,3,2,1.0806918086518902,2.785171921529357,0.
|
|
||||||
2,3,9,1,0,1,7,3,2,2.649104146816554,0.22330923947941184,0.
|
|
||||||
3,3,2,2,8,2,7,1,1,1.4153027357986057,0.7592362643035888,0.
|
|
||||||
1,2,4,1,2,0,8,1,1,0.3716613874614527,2.741327801112848,0.
|
|
||||||
8,2,4,0,7,3,8,2,2,2.772853450624477,0.7349370680695975,0.
|
|
||||||
4,2,10,1,9,1,10,2,2,2.3205630100773416,0.7942049271730163,0.
|
|
||||||
1,3,6,3,6,1,4,3,2,1.3281998811443967,1.0363866114967322,0.
|
|
||||||
3,2,7,3,3,1,3,3,4,0.8164145918403629,1.064457528831932,0.
|
|
||||||
5,1,6,0,6,1,5,0,1,2.7770950000390737,1.3710296853982933,0.
|
|
||||||
4,3,3,3,3,1,2,2,3,2.714056705000763,2.4602516368227603,0.
|
|
||||||
4,1,5,2,3,2,7,3,1,0.35146521178282075,0.3571498607132635,0.
|
|
||||||
9,1,8,2,9,1,6,1,1,1.2274240317357688,1.6715623227186507,0.
|
|
||||||
1,1,3,3,8,3,7,0,3,1.9317099576956442,1.552216114675943,0.
|
|
||||||
9,0,3,1,8,0,2,1,1,2.9043448611385188,2.7338758740571345,0.
|
|
||||||
6,0,8,2,2,2,4,1,2,1.675816848063465,0.8679109639840443,0.
|
|
||||||
9,1,9,1,0,1,1,0,1,0.596382672070606,1.7119233696062492,0.
|
|
||||||
7,2,3,1,5,1,0,3,3,1.374644910756329,2.8427817664580024,0.
|
|
||||||
6,3,10,3,7,1,10,1,2,2.6167816000584327,0.8461412388948104,0.
|
|
||||||
5,2,7,1,6,3,2,3,2,1.372612635683653,2.0961606258142194,0.
|
|
||||||
8,1,7,2,8,3,1,2,3,0.9231592095990986,0.4730174634532238,0.
|
|
||||||
3,1,1,1,10,1,10,3,2,0.3461817829038991,1.5964106156876117,0.
|
|
||||||
8,0,7,3,0,2,4,1,3,2.096022436900264,1.9588093081620253,0.
|
|
||||||
4,3,9,3,5,1,9,1,0,2.449961912097761,0.4920648568697943,0.
|
|
||||||
4,0,8,1,6,2,1,3,1,2.7092656850999868,1.2113036872224647,0.
|
|
||||||
2,2,4,3,2,2,1,0,2,0.7822138935161549,0.8243720955516407,0.
|
|
||||||
5,3,9,3,8,0,7,3,3,1.245926765783301,2.995825714452799,0.
|
|
||||||
9,1,9,0,5,3,1,3,1,0.9447956603373777,1.9688135363735153,0.
|
|
||||||
5,2,2,2,3,3,7,1,2,2.2346399253675413,0.7417391778003699,0.
|
|
||||||
0,1,8,3,0,2,2,3,2,1.0232322455743281,1.8925470263446593,0.
|
|
||||||
5,3,2,3,8,2,9,0,2,2.9817835096507697,2.111249932049714,0.
|
|
||||||
9,2,8,2,1,1,4,3,4,0.4083625562135289,0.40305961637541055,0.
|
|
||||||
4,3,9,2,8,2,10,1,3,2.7964161333721353,2.78601941028056,0.
|
|
||||||
1,1,10,0,10,0,10,1,1,2.8918303445485822,2.7282191833543576,0.
|
|
||||||
3,1,1,3,0,1,9,2,3,0.5883986968362152,1.1131593470756491,0.
|
|
||||||
3,1,5,3,6,3,10,0,3,1.29122181890919,2.1687719665184835,0.
|
|
||||||
5,1,1,0,10,1,9,1,1,2.7757481174210463,2.3835512920725517,0.
|
|
||||||
3,3,1,3,10,0,6,1,1,1.127749467977892,1.3592596107053452,0.
|
|
||||||
0,1,10,2,10,2,4,0,2,1.403190963591391,1.7426120757596841,0.
|
|
||||||
8,2,0,3,2,3,4,2,3,2.21376917027845,1.2373565371204496,0.
|
|
||||||
6,3,2,2,0,2,6,2,4,2.0909192232076848,2.9595518129553477,0.
|
|
||||||
2,0,4,3,0,3,2,0,3,2.9389053651691066,0.2571903449038695,0.
|
|
||||||
3,0,8,2,2,2,8,3,2,1.1292930350844212,2.662958886705143,0.
|
|
||||||
4,2,9,3,3,1,8,2,2,1.2069732981501304,0.6603448818737543,0.
|
|
||||||
8,1,1,2,9,2,2,3,2,1.176331991745739,2.9348246139214273,0.
|
|
||||||
8,2,9,1,3,2,6,0,2,1.2579465199621849,2.8708297284639936,0.
|
|
||||||
2,1,0,0,4,2,0,2,1,0.5068840995027317,1.4926314796061328,0.
|
|
||||||
8,0,5,2,0,1,6,1,2,2.083459207268281,2.044350098956075,0.
|
|
||||||
8,1,3,1,8,3,3,1,2,0.27182349339562206,0.25278077977555524,0.
|
|
||||||
9,0,9,3,8,2,7,1,3,2.0842851912303697,1.4617238592678117,0.
|
|
||||||
6,0,7,3,4,2,5,3,3,0.5438422711246416,1.1960642527951837,0.
|
|
||||||
3,0,0,3,1,3,5,2,3,2.142935683266975,0.3991912324294198,0.
|
|
||||||
5,2,1,1,6,3,6,2,2,1.7422616583207757,1.1999207722763399,0.
|
|
||||||
1,2,1,0,8,2,9,0,2,0.32666758600821844,2.6968272363747605,0.
|
|
||||||
3,2,0,0,10,1,6,1,2,1.6221941314416704,1.1434410020543084,0.
|
|
||||||
0,3,1,2,10,1,3,1,2,0.9726067645442806,0.5905895434956969,0.
|
|
||||||
0,1,0,0,1,0,7,1,1,2.900837043628563,2.949017267115262,0.
|
|
||||||
8,3,8,1,2,0,0,2,2,1.5234948494086824,0.3368477864264441,0.
|
|
||||||
7,2,4,2,8,1,6,1,1,0.650243384315508,0.5631520583276046,0.
|
|
||||||
3,3,9,3,2,0,9,0,0,2.3807738065036146,0.579866727058429,0.
|
|
||||||
4,3,7,0,10,2,6,3,3,1.3173017138545768,0.5962764653299519,0.
|
|
||||||
8,3,0,3,0,3,2,1,4,0.6053424778386112,0.8254011764061571,0.
|
|
||||||
9,1,3,1,1,0,2,2,2,0.9352316673114225,2.648521223427407,0.
|
|
||||||
4,3,6,2,1,3,8,3,3,1.1811263754353778,0.27949826874059935,0.
|
|
||||||
6,2,9,1,0,3,8,3,2,1.7891686343587905,2.5475273097794897,0.
|
|
||||||
3,3,6,3,1,1,4,2,1,0.8652651806579512,2.144133642324886,0.
|
|
||||||
8,2,5,1,10,3,3,1,3,1.3153722390010687,2.370685736109543,0.
|
|
||||||
9,1,5,2,7,3,2,1,2,1.4002015422875607,1.5704174280943155,0.
|
|
||||||
4,3,8,3,9,1,4,2,1,1.1067444816569871,2.7633308770019767,0.
|
|
||||||
8,1,5,1,7,0,7,1,1,1.3033564335609653,1.8205597872081913,0.
|
|
||||||
6,2,10,2,0,2,9,0,2,1.417347861506408,0.8228017043867935,0.
|
|
||||||
3,3,5,2,8,1,5,1,1,2.3178061623921504,2.446585257893128,0.
|
|
||||||
0,2,5,2,5,1,0,0,1,0.417745684372842,0.962045539464802,0.
|
|
||||||
8,1,8,1,8,1,5,0,1,2.241624712766236,2.117102285343411,0.
|
|
||||||
5,2,1,1,2,0,2,3,3,1.452790633495452,2.3926515680383504,0.
|
|
||||||
7,1,0,3,9,1,9,1,2,1.320602559814661,0.5022153836743799,0.
|
|
||||||
10,2,8,3,1,1,4,3,4,0.8992253656072662,2.3674150876262807,0.
|
|
||||||
2,3,9,0,10,1,9,2,3,1.0158467671907352,1.376506498457696,0.
|
|
||||||
1,3,1,1,4,1,4,1,2,1.8583872135299577,2.4330946543470144,0.
|
|
||||||
8,0,9,1,3,0,7,1,1,2.9549925009110574,0.7408650898965572,0.
|
|
||||||
5,1,3,1,2,2,0,0,2,0.7685503537061233,1.8557635823419316,0.
|
|
||||||
10,1,7,0,3,1,7,1,1,2.91588044339084,0.4955239475168276,0.
|
|
||||||
6,2,10,0,10,3,1,1,2,1.2165093404056426,0.48447018353308113,0.
|
|
||||||
8,3,0,2,9,0,9,3,3,1.2458573416728456,1.8984850604019927,0.
|
|
||||||
5,1,6,1,5,2,2,3,2,1.6678698181298168,0.27271548312207194,0.
|
|
||||||
1,0,5,1,6,3,7,3,1,2.4353366447686176,2.251622116348118,0.
|
|
||||||
4,1,6,2,10,3,8,1,3,0.9785359746307756,2.2569876645542513,0.
|
|
||||||
7,1,8,0,3,0,8,1,1,2.6254607678979696,1.4870987629995316,0.
|
|
||||||
7,2,4,0,4,1,6,2,2,2.4860925929709445,2.078702116156186,0.
|
|
||||||
4,3,10,0,2,3,4,1,3,2.7833993206816183,2.4091684481860574,0.
|
|
||||||
10,2,3,2,10,0,3,2,2,2.600869455586518,2.7023313585291886,0.
|
|
||||||
10,0,3,2,1,1,5,1,2,0.6308140012941346,2.192072324433969,0.
|
|
||||||
0,2,1,3,8,3,9,2,5,0.47211349059914465,1.2835269519982067,0.
|
|
||||||
1,3,5,3,9,0,8,1,1,0.39051019613873894,0.6191713045924181,0.
|
|
||||||
8,2,1,2,5,2,10,1,1,2.8083327076207043,2.7889141797722985,0.
|
|
||||||
0,1,6,2,4,3,1,3,3,2.576880910721891,2.8940089415951142,0.
|
|
||||||
10,2,9,2,3,1,7,1,2,0.4344721828159619,2.6268647157878835,0.
|
|
||||||
4,1,0,3,8,0,10,3,3,2.3791812481389023,0.41047900744269716,0.
|
|
||||||
9,0,0,3,9,2,2,1,3,0.3591915419041567,2.665840985111095,0.
|
|
||||||
9,3,6,2,5,1,5,2,3,2.1661396893720006,1.1610985541712369,0.
|
|
||||||
7,3,2,3,2,1,5,2,2,1.2178868457103484,1.906937613481415,0.
|
|
||||||
9,1,4,2,1,2,5,2,1,1.76351959987508,0.310172698567285,0.
|
|
||||||
10,1,4,2,5,1,5,1,2,2.9057790676388597,2.246935216027948,0.
|
|
||||||
9,3,6,1,7,0,5,2,2,1.8228283848861624,0.5136268936144317,0.
|
|
||||||
4,1,2,1,10,2,1,2,2,2.4146521536083805,0.4173254666656194,0.
|
|
||||||
9,1,2,1,3,2,8,0,2,2.091075002425989,1.6190364383142475,0.
|
|
||||||
5,2,7,1,2,2,7,1,2,1.9881551763498084,0.9065836043584428,0.
|
|
||||||
5,3,6,2,4,1,7,2,1,0.2437619648775846,0.4633502649952983,0.
|
|
||||||
6,1,8,1,10,2,5,1,1,1.979325271197399,1.1385489265368367,0.
|
|
||||||
3,2,6,2,4,0,8,3,3,2.761222760478871,2.281187219578926,0.
|
|
||||||
7,3,2,2,3,0,4,1,1,1.387675975897055,2.978542623739007,0.
|
|
||||||
0,2,8,0,0,1,2,2,2,0.7224763006498796,1.564980597384034,0.
|
|
||||||
3,3,7,1,10,2,10,2,3,2.9525286612356716,0.8408929108171121,0.
|
|
||||||
0,1,7,2,2,1,9,0,1,0.9126033819378172,2.8275866134399896,0.
|
|
||||||
9,2,1,0,8,0,0,2,2,1.0314013900322792,0.6732978424388585,0.
|
|
||||||
8,2,9,1,7,1,9,0,1,1.3865069980063334,2.540865404640523,0.
|
|
||||||
6,2,0,2,2,1,9,2,3,0.4235665273207312,1.6846937203861359,0.
|
|
||||||
5,2,5,2,5,0,3,1,1,2.7659439645673256,1.0115836172242267,0.
|
|
||||||
0,1,9,3,6,3,2,1,3,1.2065541331332832,1.0405234817364937,0.
|
|
||||||
8,1,6,1,4,0,1,1,1,0.7227473981225692,1.155456043230863,0.
|
|
||||||
2,3,8,0,1,0,3,3,3,1.9299731710816523,0.6103164598521023,0.
|
|
||||||
4,0,2,3,0,2,1,2,3,2.8621863382346,2.9310316681902524,0.
|
|
||||||
8,3,5,1,8,2,6,3,4,0.45896263960829176,1.9911804294619255,0.
|
|
||||||
1,0,3,2,6,2,4,0,2,2.335589089980126,1.0652452461608548,0.
|
|
||||||
3,1,1,3,1,2,10,3,3,1.0154942249838212,1.8603629108124498,0.
|
|
||||||
2,1,6,0,3,2,2,2,1,0.6444259802836747,1.4787144338375233,0.
|
|
||||||
7,1,10,2,2,1,3,2,3,0.7147755411709733,0.741536139217259,0.
|
|
||||||
0,2,1,3,2,1,8,1,1,1.8857744159033487,2.4763969783476716,0.
|
|
||||||
2,3,8,0,10,3,4,0,3,1.9960210966818677,2.9863811890951855,0.
|
|
||||||
2,3,2,3,2,1,5,2,2,1.4578550110452344,2.0655616115569337,0.
|
|
||||||
8,3,7,2,0,0,0,2,2,2.1690558898141488,1.6547046613975827,0.
|
|
||||||
7,2,3,1,6,3,3,0,3,2.553795269242924,2.98657012675977,-5.8584894789029045
|
|
||||||
5,2,2,2,8,0,7,0,0,2.111885434964618,1.9937513659932842,0.
|
|
||||||
8,3,8,0,8,2,0,3,3,2.8705609121506805,1.3331637714091094,0.
|
|
||||||
3,2,6,3,1,2,0,1,1,2.34251648574082,0.5435767229545014,0.
|
|
||||||
4,0,6,2,6,0,9,2,2,1.20338522636726,2.788327210207373,0.
|
|
||||||
7,1,1,0,10,1,0,0,1,2.3923670262306302,2.4029578275837586,0.
|
|
||||||
0,1,8,3,6,2,0,3,2,1.203543583831212,2.036143250730075,0.
|
|
||||||
5,3,3,1,10,3,9,1,2,2.7314904241739955,0.21670230017016534,0.
|
|
||||||
7,3,0,3,4,0,9,3,3,0.7221556673410454,2.504574860198268,0.
|
|
||||||
0,1,9,0,7,2,2,1,1,0.6129040077484231,0.9377073179769542,0.
|
|
||||||
8,3,8,3,8,0,6,1,1,0.2916837472356941,1.6706587000955677,0.
|
|
||||||
5,0,8,1,2,1,9,1,1,1.1071388188914164,2.907305251123775,0.
|
|
||||||
4,2,3,3,2,1,9,0,1,2.6049414020919137,1.1423144552963707,0.
|
|
||||||
0,2,3,2,6,0,3,1,1,0.6589475462254843,0.3458167300233934,0.
|
|
||||||
5,1,7,2,1,1,9,1,2,0.37152940436379245,2.257037185691721,0.
|
|
||||||
5,2,5,2,9,3,3,2,4,1.1411180980151068,1.360756047176233,0.
|
|
||||||
7,2,5,3,4,3,6,3,5,2.885010245830152,0.6362691751312788,0.
|
|
||||||
5,2,9,3,10,1,0,1,2,0.37940369466801593,1.2334735865845734,0.
|
|
||||||
0,0,7,2,1,2,7,1,2,1.5479759698453543,1.9801733480094859,0.
|
|
||||||
2,2,4,0,10,0,10,2,2,1.3293709201965624,1.2689927819499371,0.
|
|
||||||
9,0,3,1,0,2,8,2,1,2.164068098414817,1.2453603926890962,0.
|
|
||||||
0,2,9,2,3,1,4,0,1,0.6696321137903438,2.6084879932864853,0.
|
|
||||||
8,1,3,3,4,1,6,1,2,2.042466846990008,1.8892178256161216,0.
|
|
||||||
2,3,2,1,10,3,10,2,3,1.4323483837153708,1.0105488574564596,0.
|
|
||||||
5,0,10,3,7,3,10,2,3,2.726039671844071,2.8549137956975468,0.
|
|
||||||
2,3,1,2,3,3,10,1,3,2.351151357868736,0.36675214929021926,0.
|
|
||||||
2,0,7,0,9,1,3,1,0,0.7394282604433373,0.2998036281738994,0.
|
|
||||||
4,3,0,3,6,2,9,2,4,1.4189068677052719,2.9473449090470467,0.
|
|
||||||
9,2,1,3,9,2,7,0,2,2.2125653578966116,0.6449925605947713,0.
|
|
||||||
3,3,9,3,7,0,2,0,0,2.6527765476432785,0.7017535912522956,0.
|
|
||||||
3,2,1,3,10,3,10,2,3,1.3545764993952725,1.4463176767351857,0.
|
|
||||||
6,0,6,3,7,3,10,1,3,0.5231215491635299,1.5951002259931988,0.
|
|
||||||
10,1,0,3,0,0,3,2,2,0.932146690038874,0.6396108542280041,0.
|
|
||||||
1,3,4,3,3,1,5,1,2,2.6568428894223652,1.0041895351349086,0.
|
|
||||||
2,0,6,3,7,2,5,2,3,2.1388490384614993,1.084038962927921,0.
|
|
||||||
1,0,4,3,3,3,6,1,3,1.1954487430499983,1.3480169360596266,0.
|
|
||||||
4,2,10,3,7,0,8,3,3,0.9989698852671967,0.31492500891608577,0.
|
|
||||||
0,1,5,1,4,1,7,1,2,0.2915533255281222,0.8488169939496006,0.
|
|
||||||
5,3,8,2,6,3,10,0,3,1.3273506939181288,1.9576844972216119,0.
|
|
||||||
4,1,1,1,8,1,3,1,0,2.5512768516580167,1.1776908814863933,0.
|
|
||||||
0,1,7,2,4,3,5,2,3,2.55835426097922,0.7999378464474947,0.
|
|
||||||
7,1,3,2,4,3,4,3,3,2.9469964384408645,1.0130439824706348,0.
|
|
||||||
10,1,10,1,4,0,9,2,2,2.29282320065841,1.5895923699679324,0.
|
|
||||||
3,3,8,2,9,2,8,3,5,1.4288368272508416,0.32914872565251896,0.
|
|
||||||
5,2,10,0,7,1,4,2,2,2.0963238426388164,2.5184909838350205,0.
|
|
||||||
7,3,4,0,1,2,10,3,3,0.7462285283562706,1.9678766341114837,0.
|
|
||||||
5,1,6,0,6,1,1,0,1,2.742103144450633,1.0100914398280834,0.
|
|
||||||
6,2,8,2,1,0,8,0,0,2.9686558639239102,2.571359181752765,0.
|
|
||||||
6,2,1,0,8,3,0,2,2,2.8725089493368277,2.890444927773988,0.
|
|
||||||
3,2,6,3,9,1,2,2,1,0.9140224473432359,1.7614216539656447,0.
|
|
||||||
9,1,3,1,2,3,8,2,2,1.5224977087087135,1.7130595192975742,0.
|
|
||||||
2,1,5,1,1,1,2,1,0,0.620215788594543,2.125090314126055,0.
|
|
||||||
3,3,2,2,9,3,10,3,5,2.1571418399260214,0.2352162581649102,0.
|
|
||||||
4,1,3,3,8,2,1,0,2,2.5710875717760064,1.8947511401060053,0.
|
|
||||||
8,2,5,2,6,1,8,1,0,1.5775238404669114,2.9176906027237717,0.
|
|
||||||
10,2,1,2,2,3,3,3,2,2.9869659703459304,0.6494176627258623,0.
|
|
||||||
9,1,7,1,7,0,5,0,0,0.35843660067087235,1.1434109912448518,0.
|
|
||||||
8,0,0,3,0,1,7,3,3,2.5846819654664843,0.30192303129821685,0.
|
|
||||||
9,0,10,1,3,0,4,1,1,2.2400726780346583,2.76156913953512,0.
|
|
||||||
0,2,6,1,3,1,3,0,1,1.2200817282296477,0.8137895784475973,0.
|
|
||||||
4,3,0,1,10,1,3,2,3,1.1171419500565256,2.8469536184972846,0.
|
|
||||||
6,1,3,2,2,3,3,0,3,0.8843975271072293,2.0058645383535723,0.
|
|
||||||
0,2,4,3,3,1,1,2,1,2.612997115865329,0.34104461891724425,0.
|
|
||||||
3,0,1,3,6,3,10,2,3,2.716076032829566,1.791883854821867,0.
|
|
||||||
0,3,1,3,1,2,7,2,2,2.989190167535284,0.6804444348264811,0.
|
|
||||||
7,0,7,0,6,1,8,1,0,1.4624896265248113,1.7583815597493881,0.
|
|
||||||
6,3,6,3,1,1,9,2,3,2.4493465867634194,1.3172740005089132,0.
|
|
||||||
3,2,1,2,5,3,8,1,2,2.9637052407119135,1.386678142676876,0.
|
|
||||||
|
|
|
@ -1,43 +0,0 @@
|
||||||
using QuadGK
|
|
||||||
include("../math.jl")
|
|
||||||
|
|
||||||
# Testing ∫dp p u' u where u' and u' may have different l
|
|
||||||
|
|
||||||
function explicit_integral(np, lp, n, l, μω)
|
|
||||||
integrand(p) = p * (-1)^(n + np) * 1/sqrt(μω) * ho_basis(l, n, 1/sqrt(μω) * p) * ho_basis(lp, np, 1/sqrt(μω) * p)
|
|
||||||
(integral, _) = quadgk(integrand, 0, Inf; atol=1e-10)
|
|
||||||
return integral
|
|
||||||
end
|
|
||||||
|
|
||||||
n_list = 0 : 10
|
|
||||||
l_list = 0 : 5
|
|
||||||
μω_list = [1, 2.66]
|
|
||||||
|
|
||||||
for np in n_list
|
|
||||||
for n in n_list
|
|
||||||
for lp in l_list
|
|
||||||
for l in l_list
|
|
||||||
for μω in μω_list
|
|
||||||
println("Checking n=$n, np=$np, l=$l, lp=$lp, μω=$μω")
|
|
||||||
expl = explicit_integral(np, lp, n, l, μω)
|
|
||||||
anal = integral(np, lp, n, l, μω)
|
|
||||||
@assert isapprox(expl, anal; atol=1e-8) "Explicit=$(expl) and analytical=$(anal)"
|
|
||||||
end
|
|
||||||
end
|
|
||||||
end
|
|
||||||
end
|
|
||||||
end
|
|
||||||
|
|
||||||
# Testing the full Racah's reduction formula
|
|
||||||
|
|
||||||
test_data = DelimitedFiles.readdlm("test/racah_test_data.csv", ',') # format = n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ, μ1ω1, μ2ω2, racah_matrix_element
|
|
||||||
test_data_ints = Int.(test_data[:, 1:(end - 3)])
|
|
||||||
test_data_floats = test_data[:, (end - 2):end]
|
|
||||||
|
|
||||||
for i in 1 : size(test_data)[1]
|
|
||||||
n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ = test_data_ints[i, :]
|
|
||||||
μ1ω1, μ2ω2, racah_matrix_element = test_data_floats[i, :]
|
|
||||||
println("Checking n1'=$n1p, l1'=$l1p, n2'=$n2p, l2'=$l2p, n1=$n1, l1=$l1, n2=$n2, l2=$l2, Λ=$Λ, μ1ω1=$μ1ω1, μ2ω2=$μ2ω2")
|
|
||||||
val = racahs_reduction_formula(n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ, μ1ω1, μ2ω2)
|
|
||||||
@assert isapprox(val, racah_matrix_element; atol=1e-8) "Formula=$(val) and test reference=$racah_matrix_element"
|
|
||||||
end
|
|
||||||
|
|
@ -1,30 +0,0 @@
|
||||||
using Plots
|
|
||||||
include("../common.jl")
|
|
||||||
include("../p_space.jl")
|
|
||||||
|
|
||||||
vertices = [0, 0.5 - 0.3im, 1, 6]
|
|
||||||
subdivisions = [64, 64, 64]
|
|
||||||
p, w = get_mesh(vertices, subdivisions)
|
|
||||||
|
|
||||||
# resonance example from my thesis
|
|
||||||
V_basis(p, q) = 2*g0(4, p, q) - 3*g0(2, p, q)
|
|
||||||
|
|
||||||
basis_eig = eigen(get_H_matrix(V_basis, p, w))
|
|
||||||
basis = basis_eig.vectors .* w
|
|
||||||
|
|
||||||
basis_evals = basis_eig.values
|
|
||||||
E_target = 0.7
|
|
||||||
E = nearest(basis_evals, E_target)
|
|
||||||
|
|
||||||
print("pole E = $E")
|
|
||||||
|
|
||||||
# ResonanceEC: Eq. (20)
|
|
||||||
V_system(c) = (p, q) -> c*(-5*g0(sqrt(3), p, q) + 2*g0(sqrt(10), p, q))
|
|
||||||
|
|
||||||
H = get_H_matrix(V_system(0.45), p, w)
|
|
||||||
H_berggren = transpose(basis) * H * basis
|
|
||||||
|
|
||||||
evals = eigvals(H)
|
|
||||||
scatter(real.(evals), imag.(evals), label="E")
|
|
||||||
scatter!(real.(basis_evals), imag.(basis_evals), label="basis", markershape=:x)
|
|
||||||
xlims!((0, 1))
|
|
||||||
Loading…
Reference in New Issue