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# This file is a part of Julia. License is MIT: https://julialang.org/license
module TestHessenberg
using Test, LinearAlgebra, Random
# for tuple tests below
≅(x,y) = all(p -> p[1] ≈ p[2], zip(x,y))
let n = 10
Random.seed!(1234321)
Areal = randn(n,n)/2
Aimg = randn(n,n)/2
b_ = randn(n)
B_ = randn(n,3)
# UpperHessenberg methods not covered by the tests below
@testset "UpperHessenberg" begin
A = Areal
H = UpperHessenberg(A)
AH = triu(A,-1)
@test UpperHessenberg(H) === H
@test parent(H) === A
@test Matrix(H) == Array(H) == H == AH
@test real(H) == real(AH)
@test real(UpperHessenberg{ComplexF64}(A)) == H
@test real(UpperHessenberg{ComplexF64}(H)) == H
sim = similar(H, ComplexF64)
@test sim isa UpperHessenberg{ComplexF64}
@test size(sim) == size(H)
for x in (2,2+3im)
@test x*H == H*x == x*AH
for op in (+,-)
@test op(H,x*I) == op(AH,x*I) == op(op(x*I,H))
@test op(H,x*I)*x == op(AH,x*I)*x == x*op(H,x*I)
end
end
@test [H[i,j] for i=1:size(H,1), j=1:size(H,2)] == triu(A,-1)
H1 = LinearAlgebra.fillstored!(copy(H), 1)
@test H1 == triu(fill(1, n,n), -1)
@test tril(H1.data,-2) == tril(H.data,-2)
A2, H2 = copy(A), copy(H)
A2[1:4,3]=H2[1:4,3]=1:4
H2[5,3]=0
@test H2 == triu(A2,-1)
@test_throws ArgumentError H[5,3]=1
Hc = UpperHessenberg(Areal + im .* Aimg)
AHc = triu(Areal + im .* Aimg,-1)
@test real(Hc) == real(AHc)
@test imag(Hc) == imag(AHc)
@test Array(copy(adjoint(Hc))) == adjoint(Array(Hc))
@test Array(copy(transpose(Hc))) == transpose(Array(Hc))
@test rmul!(copy(Hc), 2.0) == lmul!(2.0, copy(Hc))
H = UpperHessenberg(Areal)
@test Array(Hc + H) == Array(Hc) + Array(H)
@test Array(Hc - H) == Array(Hc) - Array(H)
end
@testset for eltya in (Float32, Float64, ComplexF32, ComplexF64, Int), herm in (false, true)
A_ = eltya == Int ?
rand(1:7, n, n) :
convert(Matrix{eltya}, eltya <: Complex ?
complex.(Areal, Aimg) :
Areal)
A = herm ? Hermitian(A_ + A_') : A_
H = hessenberg(A)
@test Hessenberg(H) === H
eltyh = eltype(H)
@test size(H.Q, 1) == size(A, 1)
@test size(H.Q, 2) == size(A, 2)
@test size(H.Q) == size(A)
@test size(H) == size(A)
@test_throws ErrorException H.Z
@test convert(Array, H) ≈ A
@test (H.Q * H.H) * H.Q' ≈ A ≈ (Matrix(H.Q) * Matrix(H.H)) * Matrix(H.Q)'
@test (H.Q' *A) * H.Q ≈ H.H
#getindex for HessenbergQ
@test H.Q[1,1] ≈ Array(H.Q)[1,1]
# REPL show
hessstring = sprint((t, s) -> show(t, "text/plain", s), H)
qstring = sprint((t, s) -> show(t, "text/plain", s), H.Q)
hstring = sprint((t, s) -> show(t, "text/plain", s), H.H)
@test hessstring == "$(summary(H))\nQ factor:\n$qstring\nH factor:\n$hstring"
#iterate
q,h = H
@test q == H.Q
@test h == H.H
@test convert(Array, 2 * H) ≈ 2 * A ≈ convert(Array, H * 2)
@test convert(Array, H + 2I) ≈ A + 2I ≈ convert(Array, 2I + H)
@test convert(Array, H + (2+4im)I) ≈ A + (2+4im)I ≈ convert(Array, (2+4im)I + H)
@test convert(Array, H - 2I) ≈ A - 2I ≈ -convert(Array, 2I - H)
@test convert(Array, -H) == -convert(Array, H)
@test convert(Array, 2*(H + (2+4im)I)) ≈ 2A + (4+8im)I
b = convert(Vector{eltype(H)}, b_)
B = convert(Matrix{eltype(H)}, B_)
@test H \ b ≈ A \ b ≈ H \ complex(b)
@test H \ B ≈ A \ B ≈ H \ complex(B)
@test (H - I) \ B ≈ (A - I) \ B
@test (H - (3+4im)I) \ B ≈ (A - (3+4im)I) \ B
@test b' / H ≈ b' / A ≈ complex.(b') / H
@test B' / H ≈ B' / A ≈ complex(B') / H
@test B' / (H - I) ≈ B' / (A - I)
@test B' / (H - (3+4im)I) ≈ B' / (A - (3+4im)I)
@test (H - (3+4im)I)' \ B ≈ (A - (3+4im)I)' \ B
@test B' / (H - (3+4im)I)' ≈ B' / (A - (3+4im)I)'
for shift in (0,1,3+4im)
@test det(H + shift*I) ≈ det(A + shift*I)
@test logabsdet(H + shift*I) ≅ logabsdet(A + shift*I)
end
HM = Matrix(h)
@test dot(b, h, b) ≈ dot(h'b, b) ≈ dot(b, HM, b) ≈ dot(HM'b, b)
c = b .+ 1
@test dot(b, h, c) ≈ dot(h'b, c) ≈ dot(b, HM, c) ≈ dot(HM'b, c)
end
end
# check logdet on a matrix that has a positive determinant
let A = [0.5 0.1 0.9 0.4; 0.9 0.7 0.5 0.4; 0.3 0.4 0.9 0.0; 0.4 0.0 0.0 0.5]
@test logdet(hessenberg(A)) ≈ logdet(A) ≈ -3.5065578973199822
end
@testset "Base.propertynames" begin
F = hessenberg([4. 9. 7.; 4. 4. 1.; 4. 3. 2.])
@test Base.propertynames(F) == (:Q, :H, :μ)
@test Base.propertynames(F, true) == (:Q, :H, :μ, :τ, :factors, :uplo)
end
end # module TestHessenberg
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