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# \hat u_n
function hatun_iteration(e,order,d,v,maxiter)
# gauss legendre weights
weights=gausslegendre(order)
# init V and Eta
(V,V0,Eta,Eta0)=init_veta(weights,d,v)
# init u and rho
u=Array{Array{Complex{Float64}}}(undef,maxiter+1)
u[1]=zeros(Complex{Float64},order)
rho=zeros(Complex{Float64},maxiter+1)
# iterate
for n in 1:maxiter
u[n+1]=A(e,weights,Eta)\b(u[n],e,rho[n],V)
rho[n+1]=rhon(u[n+1],e,weights,V0,Eta0)
end
return (u,rho)
end
# A
function A(e,weights,Eta)
N=length(weights[1])
out=zeros(Complex{Float64},N,N)
for i in 1:N
k=(1-weights[1][i])/(1+weights[1][i])
out[i,i]=k^2+4*e
for j in 1:N
y=(weights[1][j]+1)/2
out[i,j]+=weights[2][j]*(1-y)*Eta[i][j]/(2*(2*pi)^3*y^3)
end
end
return out
end
# b
function b(u,e,rho,V)
out=zeros(Complex{Float64},length(V))
for i in 1:length(V)
out[i]=V[i]+2*e*rho*u[i]^2
end
return out
end
# rho_n
function rhon(u,e,weights,V0,Eta0)
S=V0
for i in 1:length(weights[1])
y=(weights[1][i]+1)/2
S+=-weights[2][i]*(1-y)*u[i]*Eta0[i]/(2*(2*pi)^3*y^3)
end
return 2*e/S
end
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