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arpack.jl
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arpack.jl
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begin
local n,a,asym,d,v
n = 10
a = rand(n,n)
for elty in (Float32, Float64, Complex64, Complex128)
a = convert(Matrix{elty}, a)
asym = a' + a # symmetric indefinite
apd = a'*a # symmetric positive-definite
(d,v) = eigs(a, nev=3)
Test.@test_approx_eq a*v[:,2] d[2]*v[:,2]
(d,v) = eigs(asym, nev=3)
Test.@test_approx_eq asym*v[:,1] d[1]*v[:,1]
# Test.@test_approx_eq eigs(asym; nev=1, sigma=d[3])[1][1] d[3]
(d,v) = eigs(apd, nev=3)
Test.@test_approx_eq apd*v[:,3] d[3]*v[:,3]
# Test.@test_approx_eq eigs(apd; nev=1, sigma=d[3])[1][1] d[3]
end
end
# Example from Quantum Information Theory
import Base: size, issym, ishermitian
type CPM{T<:Base.LinAlg.BlasFloat}<:AbstractMatrix{T} # completely positive map
kraus::Array{T,3} # kraus operator representation
end
size(Phi::CPM)=(size(Phi.kraus,1)^2,size(Phi.kraus,3)^2)
issym(Phi::CPM)=false
ishermitian(Phi::CPM)=false
function *{T<:Base.LinAlg.BlasFloat}(Phi::CPM{T},rho::Vector{T})
rho=reshape(rho,(size(Phi.kraus,3),size(Phi.kraus,3)))
rho2=zeros(T,(size(Phi.kraus,1),size(Phi.kraus,1)))
for s=1:size(Phi.kraus,2)
As=slice(Phi.kraus,:,s,:)
rho2+=As*rho*As'
end
return reshape(rho2,(size(Phi.kraus,1)^2,))
end
# Generate random isometry
(Q,R)=qr(randn(100,50))
Q=reshape(Q,(50,2,50))
# Construct trace-preserving completely positive map from this
Phi=CPM(Q)
(d,v,numiter,numop,resid) = eigs(Phi,nev=1,which="LM")
# Properties: largest eigenvalue should be 1, largest eigenvector, when reshaped as matrix
# should be a Hermitian positive definite matrix (up to an arbitrary phase)
Test.@test_approx_eq d[1] 1. # largest eigenvalue should be 1.
v=reshape(v,(50,50)) # reshape to matrix
v=v/trace(v) # factor out arbitrary phase
Test.@test_approx_eq normfro(imag(v)) 0. # it should be real
v=real(v)
#Test.@test_approx_eq normfro(v-v') 0. # it shoul be Hermitian
# Comparison to 0. is way to strict for this to work!
v=(v+v')/2
Test.@test isposdef(v)
# Repeat with starting vector
(d2,v2,numiter,numop,resid) = eigs(Phi,nev=1,which="LM",v0=reshape(v,(2500,)))
v2=reshape(v,(50,50))
Test.@test numiter==1
Test.@test_approx_eq v v2