schur.jl
# This file is a part of Julia. License is MIT: https://julialang.org/license
module TestSchur
using Test, LinearAlgebra, Random
using LinearAlgebra: BlasComplex, BlasFloat, BlasReal, QRPivoted
n = 10
# Split n into 2 parts for tests needing two matrices
n1 = div(n, 2)
n2 = 2*n1
Random.seed!(1234321)
areal = randn(n,n)/2
aimg = randn(n,n)/2
@testset for eltya in (Float32, Float64, ComplexF32, ComplexF64, Int)
a = eltya == Int ? rand(1:7, n, n) : convert(Matrix{eltya}, eltya <: Complex ? complex.(areal, aimg) : areal)
asym = a' + a # symmetric indefinite
apd = a' * a # symmetric positive-definite
for (a, asym, apd) in ((a, asym, apd),
(view(a, 1:n, 1:n),
view(asym, 1:n, 1:n),
view(apd, 1:n, 1:n)))
ε = εa = eps(abs(float(one(eltya))))
d,v = eigen(a)
f = schur(a)
@test f.vectors*f.Schur*f.vectors' ≈ a
@test sort(real(f.values)) ≈ sort(real(d))
@test sort(imag(f.values)) ≈ sort(imag(d))
@test istriu(f.Schur) || eltype(a)<:Real
@test convert(Array, f) ≈ a
@test_throws ErrorException f.A
sch, vecs, vals = schur(UpperTriangular(triu(a)))
@test vecs*sch*vecs' ≈ triu(a)
sch, vecs, vals = schur(UnitUpperTriangular(triu(a)))
@test vecs*sch*vecs' ≈ UnitUpperTriangular(triu(a))
sch, vecs, vals = schur(LowerTriangular(tril(a)))
@test vecs*sch*vecs' ≈ tril(a)
sch, vecs, vals = schur(UnitLowerTriangular(tril(a)))
@test vecs*sch*vecs' ≈ UnitLowerTriangular(tril(a))
sch, vecs, vals = schur(Hermitian(asym))
@test vecs*sch*vecs' ≈ asym
sch, vecs, vals = schur(Symmetric(a + transpose(a)))
@test vecs*sch*vecs' ≈ a + transpose(a)
sch, vecs, vals = schur(Tridiagonal(a + transpose(a)))
@test vecs*sch*vecs' ≈ Tridiagonal(a + transpose(a))
sch, vecs, vals = schur(Bidiagonal(a, :U))
@test vecs*sch*vecs' ≈ Bidiagonal(a, :U)
sch, vecs, vals = schur(Bidiagonal(a, :L))
@test vecs*sch*vecs' ≈ Bidiagonal(a, :L)
tstring = sprint((t, s) -> show(t, "text/plain", s), f.T)
zstring = sprint((t, s) -> show(t, "text/plain", s), f.Z)
vstring = sprint((t, s) -> show(t, "text/plain", s), f.values)
fstring = sprint((t, s) -> show(t, "text/plain", s), f)
@test fstring == "$(summary(f))\nT factor:\n$tstring\nZ factor:\n$(zstring)\neigenvalues:\n$vstring"
@testset "Reorder Schur" begin
# use asym for real schur to enforce tridiag structure
# avoiding partly selection of conj. eigenvalues
ordschura = eltya <: Complex ? a : asym
S = schur(ordschura)
select = bitrand(n)
O = ordschur(S, select)
sum(select) != 0 && @test S.values[findall(select)] ≈ O.values[1:sum(select)]
@test O.vectors*O.Schur*O.vectors' ≈ ordschura
@test_throws ErrorException f.A
Snew = LinearAlgebra.Schur(S.T, S.Z, S.values)
SchurNew = ordschur!(copy(Snew), select)
@test O.vectors ≈ SchurNew.vectors
@test O.Schur ≈ SchurNew.Schur
end
if isa(a, Array)
a1_sf = a[1:n1, 1:n1]
a2_sf = a[n1+1:n2, n1+1:n2]
else
a1_sf = view(a, 1:n1, 1:n1)
a2_sf = view(a, n1+1:n2, n1+1:n2)
end
@testset "Generalized Schur" begin
f = schur(a1_sf, a2_sf)
@test f.Q*f.S*f.Z' ≈ a1_sf
@test f.Q*f.T*f.Z' ≈ a2_sf
@test istriu(f.S) || eltype(a)<:Real
@test istriu(f.T) || eltype(a)<:Real
@test_throws ErrorException f.A
sstring = sprint((t, s) -> show(t, "text/plain", s), f.S)
tstring = sprint((t, s) -> show(t, "text/plain", s), f.T)
qstring = sprint((t, s) -> show(t, "text/plain", s), f.Q)
zstring = sprint((t, s) -> show(t, "text/plain", s), f.Z)
αstring = sprint((t, s) -> show(t, "text/plain", s), f.α)
βstring = sprint((t, s) -> show(t, "text/plain", s), f.β)
fstring = sprint((t, s) -> show(t, "text/plain", s), f)
@test fstring == "$(summary(f))\nS factor:\n$sstring\nT factor:\n$(tstring)\nQ factor:\n$(qstring)\nZ factor:\n$(zstring)\nα:\n$αstring\nβ:\n$βstring"
end
@testset "Reorder Generalized Schur" begin
NS = schur(a1_sf, a2_sf)
# Currently just testing with selecting gen eig values < 1
select = abs2.(NS.values) .< 1
m = sum(select)
S = ordschur(NS, select)
# Make sure that the new factorization still factors matrix
@test S.Q*S.S*S.Z' ≈ a1_sf
@test S.Q*S.T*S.Z' ≈ a2_sf
# Make sure that we have sorted it correctly
@test NS.values[findall(select)] ≈ S.values[1:m]
Snew = LinearAlgebra.GeneralizedSchur(NS.S, NS.T, NS.alpha, NS.beta, NS.Q, NS.Z)
SchurNew = ordschur!(copy(Snew), select)
@test S.Q ≈ SchurNew.Q
@test S.S ≈ SchurNew.S
@test S.T ≈ SchurNew.T
@test S.Z ≈ SchurNew.Z
@test S.alpha ≈ SchurNew.alpha
@test S.beta ≈ SchurNew.beta
sS,sT,sQ,sZ = schur(a1_sf,a2_sf)
@test NS.Q ≈ sQ
@test NS.T ≈ sT
@test NS.S ≈ sS
@test NS.Z ≈ sZ
end
end
@testset "0x0 matrix" for A in (zeros(eltya, 0, 0), view(rand(eltya, 2, 2), 1:0, 1:0))
T, Z, λ = LinearAlgebra.schur(A)
@test T == A
@test Z == A
@test λ == zeros(0)
end
if eltya <: Real
@testset "quasitriangular to triangular" begin
S = schur(a)
SC = Schur{Complex}(S)
@test eltype(SC) == complex(eltype(S))
@test istriu(SC.T)
@test SC.Z*SC.Z' ≈ I
@test SC.Z*SC.T*SC.Z' ≈ a
@test sort(SC.values,by=LinearAlgebra.eigsortby) ≈ sort(S.values,by=LinearAlgebra.eigsortby)
@test Schur{Complex}(SC) === SC === Schur{eltype(SC)}(SC)
@test Schur{eltype(S)}(S) === S
if eltype(S) === Float32
S64 = Schur{Float64}(S)
@test eltype(S64) == Float64
@test S64.Z == S.Z
@test S64.T == S.T
@test S64.values == S.values
end
end
end
@testset "0x0 $eltya matrices" begin
A = zeros(eltya, 0, 0)
B = zeros(eltya, 0, 0)
S = LinearAlgebra.schur(A, B)
@test S.S == A
@test S.T == A
@test S.Q == A
@test S.Z == A
@test S.alpha == zeros(0)
@test S.beta == zeros(0)
end
end
@testset "Generalized Schur convergence" begin
# Check for convergence issues, #40279
problematic_pencils = [
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),
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)]
for (A, B) in problematic_pencils
f = schur(A, B)
@test f.Q*f.S*f.Z' ≈ A
@test f.Q*f.T*f.Z' ≈ B
end
end
@testset "adjoint and transpose for schur (#40941)" begin
A = rand(3, 3)
B = schur(A', A)
C = B.left*B.S*B.right'
D = schur(transpose(A), A)
E = D.left*D.S*D.right'
@test A' ≈ C ≈ E
end
@testset "UpperHessenberg schur" begin
A = UpperHessenberg(rand(ComplexF64, 100, 100))
B = Array(A)
fact1 = schur(A)
fact2 = schur(B)
@test fact1.values ≈ fact2.values
@test fact1.Z * fact1.T * fact1.Z' ≈ B
A = UpperHessenberg(rand(Int32, 50, 50))
B = Array(A)
fact1 = schur(A)
fact2 = schur(B)
@test fact1.values ≈ fact2.values
@test fact1.Z * fact1.T * fact1.Z' ≈ B
end
end # module TestSchur
