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diagonal.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
module TestDiagonal
using Test, LinearAlgebra, Random
using LinearAlgebra: BlasFloat, BlasComplex
const BASE_TEST_PATH = joinpath(Sys.BINDIR, "..", "share", "julia", "test")
isdefined(Main, :OffsetArrays) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "OffsetArrays.jl"))
using .Main.OffsetArrays
isdefined(Main, :InfiniteArrays) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "InfiniteArrays.jl"))
using .Main.InfiniteArrays
isdefined(Main, :FillArrays) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "FillArrays.jl"))
using .Main.FillArrays
isdefined(Main, :SizedArrays) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "SizedArrays.jl"))
using .Main.SizedArrays
const n=12 # Size of matrix problem to test
Random.seed!(1)
@testset for relty in (Float32, Float64, BigFloat), elty in (relty, Complex{relty})
dd=convert(Vector{elty}, randn(n))
vv=convert(Vector{elty}, randn(n))
UU=convert(Matrix{elty}, randn(n,n))
if elty <: Complex
dd+=im*convert(Vector{elty}, randn(n))
vv+=im*convert(Vector{elty}, randn(n))
UU+=im*convert(Matrix{elty}, randn(n,n))
end
D = Diagonal(dd)
DM = Matrix(Diagonal(dd))
@testset "constructor" begin
for x in (dd, GenericArray(dd))
@test Diagonal(x)::Diagonal{elty,typeof(x)} == DM
@test Diagonal(x).diag === x
@test Diagonal{elty}(x)::Diagonal{elty,typeof(x)} == DM
@test Diagonal{elty}(x).diag === x
@test Diagonal{elty}(D) === D
end
@test eltype(Diagonal{elty}([1,2,3,4])) == elty
@test isa(Diagonal{elty,Vector{elty}}(GenericArray([1,2,3,4])), Diagonal{elty,Vector{elty}})
@test isa(Diagonal{elty}(rand(Int,n,n)), Diagonal{elty,Vector{elty}})
DI = Diagonal([1,2,3,4])
@test Diagonal(DI) === DI
@test isa(Diagonal{elty}(DI), Diagonal{elty})
# diagonal matrices may be converted to Diagonal
local A = [1 0; 0 2]
local DA = convert(Diagonal{Float32,Vector{Float32}}, A)
@test DA isa Diagonal{Float32,Vector{Float32}}
@test DA == A
# issue #26178
@test_throws MethodError convert(Diagonal, [1,2,3,4])
@test_throws DimensionMismatch convert(Diagonal, [1 2 3 4])
@test_throws InexactError convert(Diagonal, ones(2,2))
# Test reversing
# Test reversing along rows
@test reverse(D, dims=1) == reverse(Matrix(D), dims=1)
# Test reversing along columns
@test reverse(D, dims=2) == reverse(Matrix(D), dims=2)
# Test reversing the entire matrix
@test reverse(D)::Diagonal == reverse(Matrix(D)) == reverse!(copy(D))
end
@testset "Basic properties" begin
@test_throws BoundsError size(D,0)
@test size(D,1) == size(D,2) == length(dd)
@test size(D,3) == 1
@test typeof(convert(Diagonal{ComplexF32},D)) <: Diagonal{ComplexF32}
@test typeof(convert(AbstractMatrix{ComplexF32},D)) <: Diagonal{ComplexF32}
@test Array(real(D)) == real(DM)
@test Array(abs.(D)) == abs.(DM)
@test Array(imag(D)) == imag(DM)
@test parent(D) == dd
@test D[1,1] == dd[1]
@test D[1,2] == 0
@test issymmetric(D)
@test isdiag(D)
@test isdiag(Diagonal([[1 0; 0 1], [1 0; 0 1]]))
@test !isdiag(Diagonal([[1 0; 0 1], [1 0; 1 1]]))
@test istriu(D)
@test istriu(D, -1)
@test !istriu(D, 1)
@test istriu(Diagonal(zero(diag(D))), 1)
@test istril(D)
@test !istril(D, -1)
@test istril(D, 1)
@test istril(Diagonal(zero(diag(D))), -1)
@test Base.isstored(D,1,1)
@test !Base.isstored(D,1,2)
@test_throws BoundsError Base.isstored(D, n + 1, 1)
if elty <: Real
@test ishermitian(D)
end
end
@testset "diag" begin
@test isempty(@inferred diag(D, n+1))
@test isempty(@inferred diag(D, -n-1))
@test (@inferred diag(D))::typeof(dd) == dd
@test (@inferred diag(D, 0))::typeof(dd) == dd
@test (@inferred diag(D, 1))::typeof(dd) == zeros(elty, n-1)
DG = Diagonal(GenericArray(dd))
@test (@inferred diag(DG))::typeof(GenericArray(dd)) == GenericArray(dd)
@test (@inferred diag(DG, 1))::typeof(GenericArray(dd)) == GenericArray(zeros(elty, n-1))
end
@testset "Simple unary functions" begin
for op in (-,)
@test op(D)==op(DM)
end
for func in (det, tr)
@test func(D) ≈ func(DM) atol=n^2*eps(relty)*(1+(elty<:Complex))
end
if eltype(D) <: Real
@test minimum(D) ≈ minimum(DM)
@test maximum(D) ≈ maximum(DM)
end
if relty <: BlasFloat
for func in (exp, cis, sinh, cosh, tanh, sech, csch, coth)
@test func(D) ≈ func(DM) atol=n^3*eps(relty)
end
@test log(Diagonal(abs.(D.diag))) ≈ log(abs.(DM)) atol=n^3*eps(relty)
end
if elty <: BlasComplex
for func in (logdet, sqrt, sin, cos, tan, sec, csc, cot,
asin, acos, atan, asec, acsc, acot,
asinh, acosh, atanh, asech, acsch, acoth)
@test func(D) ≈ func(DM) atol=n^2*eps(relty)*2
end
end
end
@testset "Two-dimensional Euler formula for Diagonal" begin
@test cis(Diagonal([π, π])) ≈ -I
end
@testset "Linear solve" begin
for (v, U) in ((vv, UU), (view(vv, 1:n), view(UU, 1:n, 1:2)))
@test D*v ≈ DM*v atol=n*eps(relty)*(1+(elty<:Complex))
@test D*U ≈ DM*U atol=n^2*eps(relty)*(1+(elty<:Complex))
@test transpose(U)*D ≈ transpose(U)*Array(D)
@test U'*D ≈ U'*Array(D)
if relty != BigFloat
atol_two = 2n^2 * eps(relty) * (1 + (elty <: Complex))
atol_three = 2n^3 * eps(relty) * (1 + (elty <: Complex))
@test D\v ≈ DM\v atol=atol_two
@test D\U ≈ DM\U atol=atol_three
@test ldiv!(D, copy(v)) ≈ DM\v atol=atol_two
@test ldiv!(transpose(D), copy(v)) ≈ DM\v atol=atol_two
@test ldiv!(adjoint(conj(D)), copy(v)) ≈ DM\v atol=atol_two
@test ldiv!(D, copy(U)) ≈ DM\U atol=atol_three
@test ldiv!(transpose(D), copy(U)) ≈ DM\U atol=atol_three
@test ldiv!(adjoint(conj(D)), copy(U)) ≈ DM\U atol=atol_three
# this method tests AbstractMatrix/AbstractVec for second arg
Usym_bad = Symmetric(ones(elty, n+1, n+1))
@test_throws DimensionMismatch ldiv!(D, copy(Usym_bad))
@test ldiv!(zero(v), D, copy(v)) ≈ DM\v atol=atol_two
@test ldiv!(zero(v), transpose(D), copy(v)) ≈ DM\v atol=atol_two
@test ldiv!(zero(v), adjoint(conj(D)), copy(v)) ≈ DM\v atol=atol_two
@test ldiv!(zero(U), D, copy(U)) ≈ DM\U atol=atol_three
@test ldiv!(zero(U), transpose(D), copy(U)) ≈ DM\U atol=atol_three
@test ldiv!(zero(U), adjoint(conj(D)), copy(U)) ≈ DM\U atol=atol_three
Uc = copy(U')
target = rmul!(Uc, Diagonal(inv.(D.diag)))
@test rdiv!(Uc, D) ≈ target atol=atol_three
@test_throws DimensionMismatch rdiv!(Matrix{elty}(I, n-1, n-1), D)
@test_throws SingularException rdiv!(Uc, Diagonal(fill!(similar(D.diag), 0)))
@test rdiv!(Uc, transpose(D)) ≈ target atol=atol_three
@test rdiv!(Uc, adjoint(conj(D))) ≈ target atol=atol_three
@test ldiv!(D, Matrix{eltype(D)}(I, size(D))) ≈ D \ Matrix{eltype(D)}(I, size(D)) atol=atol_three
@test_throws DimensionMismatch ldiv!(D, fill(elty(1), n + 1))
@test_throws SingularException ldiv!(Diagonal(zeros(relty, n)), copy(v))
b = rand(elty, n, n)
@test ldiv!(D, copy(b)) ≈ Array(D)\Array(b)
@test_throws SingularException ldiv!(Diagonal(zeros(elty, n)), copy(b))
b = view(rand(elty, n), Vector(1:n))
b2 = copy(b)
c = ldiv!(D, b)
d = Array(D)\b2
@test c ≈ d
@test_throws SingularException ldiv!(Diagonal(zeros(elty, n)), b)
b = rand(elty, n+1, n+1)
@test_throws DimensionMismatch ldiv!(D, copy(b))
b = view(rand(elty, n+1), Vector(1:n+1))
@test_throws DimensionMismatch ldiv!(D, b)
end
end
end
d = convert(Vector{elty}, randn(n))
D2 = Diagonal(d)
DM2= Matrix(Diagonal(d))
@testset "Binary operations" begin
for op in (+, -, *)
@test Array(op(D, D2)) ≈ op(DM, DM2)
end
@testset "with plain numbers" begin
a = rand()
@test Array(a*D) ≈ a*DM
@test Array(D*a) ≈ DM*a
@test Array(D/a) ≈ DM/a
if elty <: Real
@test Array(abs.(D)^a) ≈ abs.(DM)^a
else
@test Array(D^a) ≈ DM^a
end
@test Diagonal(1:100)^2 == Diagonal((1:100).^2)
p = 3
@test Diagonal(1:100)^p == Diagonal((1:100).^p)
@test Diagonal(1:100)^(-1) == Diagonal(inv.(1:100))
@test Diagonal(1:100)^2.0 == Diagonal((1:100).^2.0)
@test Diagonal(1:100)^(2.0+0im) == Diagonal((1:100).^(2.0+0im))
end
if relty <: BlasFloat
for b in (rand(elty,n,n), rand(elty,n))
@test lmul!(copy(D), copy(b)) ≈ Array(D)*Array(b)
@test lmul!(transpose(copy(D)), copy(b)) ≈ transpose(Array(D))*Array(b)
@test lmul!(adjoint(copy(D)), copy(b)) ≈ Array(D)'*Array(b)
end
end
#a few missing mults
bd = Bidiagonal(D2)
@test D*transpose(D2) ≈ Array(D)*transpose(Array(D2))
@test D2*transpose(D) ≈ Array(D2)*transpose(Array(D))
@test D2*D' ≈ Array(D2)*Array(D)'
#division of two Diagonals
@test D/D2 ≈ Diagonal(D.diag./D2.diag)
@test D\D2 ≈ Diagonal(D2.diag./D.diag)
# QR \ Diagonal
A = rand(elty, n, n)
qrA = qr(A)
@test qrA \ D ≈ A \ D
# HermOrSym
A = rand(elty, n, n)
Asym = Symmetric(A + transpose(A), :U)
Aherm = Hermitian(A + adjoint(A), :U)
for op in (+, -)
@test op(Asym, D) isa Symmetric
@test Array(op(Asym, D)) ≈ Array(Symmetric(op(Array(Asym), Array(D))))
@test op(D, Asym) isa Symmetric
@test Array(op(D, Asym)) ≈ Array(Symmetric(op(Array(D), Array(Asym))))
if !(elty <: Real)
Dr = real(D)
@test op(Aherm, Dr) isa Hermitian
@test Array(op(Aherm, Dr)) ≈ Array(Hermitian(op(Array(Aherm), Array(Dr))))
@test op(Dr, Aherm) isa Hermitian
@test Array(op(Dr, Aherm)) ≈ Array(Hermitian(op(Array(Dr), Array(Aherm))))
end
end
@test Array(D*transpose(Asym)) ≈ Array(D) * Array(transpose(Asym))
@test Array(D*adjoint(Asym)) ≈ Array(D) * Array(adjoint(Asym))
@test Array(D*transpose(Aherm)) ≈ Array(D) * Array(transpose(Aherm))
@test Array(D*adjoint(Aherm)) ≈ Array(D) * Array(adjoint(Aherm))
@test Array(transpose(Asym)*transpose(D)) ≈ Array(transpose(Asym)) * Array(transpose(D))
@test Array(transpose(D)*transpose(Asym)) ≈ Array(transpose(D)) * Array(transpose(Asym))
@test Array(adjoint(Aherm)*adjoint(D)) ≈ Array(adjoint(Aherm)) * Array(adjoint(D))
@test Array(adjoint(D)*adjoint(Aherm)) ≈ Array(adjoint(D)) * Array(adjoint(Aherm))
# Performance specialisations for A*_mul_B!
vvv = similar(vv)
@test (r = Matrix(D) * vv ; mul!(vvv, D, vv) ≈ r ≈ vvv)
@test (r = Matrix(D)' * vv ; mul!(vvv, adjoint(D), vv) ≈ r ≈ vvv)
@test (r = transpose(Matrix(D)) * vv ; mul!(vvv, transpose(D), vv) ≈ r ≈ vvv)
UUU = similar(UU)
for transformA in (identity, adjoint, transpose)
for transformD in (identity, adjoint, transpose)
@test mul!(UUU, transformA(UU), transformD(D)) ≈ transformA(UU) * Matrix(transformD(D))
@test mul!(UUU, transformD(D), transformA(UU)) ≈ Matrix(transformD(D)) * transformA(UU)
end
end
alpha = elty(randn()) # randn(elty) does not work with BigFloat
beta = elty(randn())
@test begin
vvv = similar(vv)
vvv .= randn(size(vvv)) # randn!(vvv) does not work with BigFloat
r = alpha * Matrix(D) * vv + beta * vvv
mul!(vvv, D, vv, alpha, beta) ≈ r ≈ vvv
end
@test begin
vvv = similar(vv)
vvv .= randn(size(vvv)) # randn!(vvv) does not work with BigFloat
r = alpha * Matrix(D)' * vv + beta * vvv
mul!(vvv, adjoint(D), vv, alpha, beta) ≈ r ≈ vvv
end
@test begin
vvv = similar(vv)
vvv .= randn(size(vvv)) # randn!(vvv) does not work with BigFloat
r = alpha * transpose(Matrix(D)) * vv + beta * vvv
mul!(vvv, transpose(D), vv, alpha, beta) ≈ r ≈ vvv
end
@test begin
UUU = similar(UU)
UUU .= randn(size(UUU)) # randn!(UUU) does not work with BigFloat
r = alpha * Matrix(D) * UU + beta * UUU
mul!(UUU, D, UU, alpha, beta) ≈ r ≈ UUU
end
@test begin
UUU = similar(UU)
UUU .= randn(size(UUU)) # randn!(UUU) does not work with BigFloat
r = alpha * Matrix(D)' * UU + beta * UUU
mul!(UUU, adjoint(D), UU, alpha, beta) ≈ r ≈ UUU
end
@test begin
UUU = similar(UU)
UUU .= randn(size(UUU)) # randn!(UUU) does not work with BigFloat
r = alpha * transpose(Matrix(D)) * UU + beta * UUU
mul!(UUU, transpose(D), UU, alpha, beta) ≈ r ≈ UUU
end
# make sure that mul!(A, {Adj|Trans}(B)) works with B as a Diagonal
VV = Array(D)
DD = copy(D)
r = VV * Matrix(D)
@test Array(rmul!(VV, DD)) ≈ r ≈ Array(D)*Array(D)
DD = copy(D)
r = VV * transpose(Array(D))
@test Array(rmul!(VV, transpose(DD))) ≈ r
DD = copy(D)
r = VV * Array(D)'
@test Array(rmul!(VV, adjoint(DD))) ≈ r
# kron
D3 = Diagonal(convert(Vector{elty}, rand(n÷2)))
DM3= Matrix(D3)
@test Matrix(kron(D, D3)) ≈ kron(DM, DM3)
M4 = rand(elty, size(D3,1) + 1, size(D3,2) + 2) # choose a different size from D3
@test kron(D3, M4) ≈ kron(DM3, M4)
@test kron(M4, D3) ≈ kron(M4, DM3)
X = [ones(1,1) for i in 1:2, j in 1:2]
@test kron(I(2), X)[1,3] == zeros(1,1)
X = [ones(2,2) for i in 1:2, j in 1:2]
@test kron(I(2), X)[1,3] == zeros(2,2)
end
@testset "iszero, isone, triu, tril" begin
Dzero = Diagonal(zeros(elty, 10))
Done = Diagonal(ones(elty, 10))
Dmix = Diagonal(zeros(elty, 10))
Dmix[end,end] = one(elty)
@test iszero(Dzero)
@test !isone(Dzero)
@test !iszero(Done)
@test isone(Done)
@test !iszero(Dmix)
@test !isone(Dmix)
@test istriu(D)
@test istril(D)
@test iszero(triu(D,1))
@test triu(D,0) == D
@test triu(D,-1) == D
@test tril(D,1) == D
@test iszero(tril(D,-1))
@test tril(D,0) == D
@test_throws ArgumentError tril(D, -n - 2)
@test_throws ArgumentError tril(D, n)
@test_throws ArgumentError triu(D, -n)
@test_throws ArgumentError triu(D, n + 2)
end
# factorize
@test factorize(D) == D
@testset "Eigensystem" begin
eigD = eigen(D)
@test Diagonal(eigD.values) == D
@test eigD.vectors == Matrix(I, size(D))
eigsortD = eigen(D, sortby=LinearAlgebra.eigsortby)
@test eigsortD.values !== D.diag
@test eigsortD.values == sort(D.diag, by=LinearAlgebra.eigsortby)
@test Matrix(eigsortD) == D
end
@testset "ldiv" begin
v = rand(n + 1)
@test_throws DimensionMismatch D\v
v = rand(n)
@test D\v ≈ DM\v
V = rand(n + 1, n)
@test_throws DimensionMismatch D\V
V = rand(n, n)
@test D\V ≈ DM\V
end
@testset "conj and transpose" begin
@test transpose(D) == D
if elty <: Real
@test transpose(D) === D
@test adjoint(D) === D
elseif elty <: BlasComplex
@test Array(conj(D)) ≈ conj(DM)
@test adjoint(D) == conj(D)
local D2 = copy(D)
local D2adj = adjoint(D2)
D2adj[1,1] = rand(eltype(D2adj))
@test D2[1,1] == adjoint(D2adj[1,1])
@test D2adj' === D2
end
# Translates to Ac/t_mul_B, which is specialized after issue 21286
@test(D' * vv == conj(D) * vv)
@test(transpose(D) * vv == D * vv)
end
# logdet and logabsdet
if relty <: Real
lD = Diagonal(convert(Vector{relty}, rand(n)))
lM = Matrix(lD)
@test logdet(lD) ≈ logdet(lM)
d1, s1 = @inferred logabsdet(lD)
d2, s2 = logabsdet(lM)
@test d1 ≈ d2
@test s1 == s2
@test logdet(Diagonal(relty[-1,-2])) ≈ log(2)
@test_throws DomainError logdet(Diagonal(relty[-1,-2,-3]))
end
@testset "similar" begin
@test isa(similar(D), Diagonal{elty})
@test isa(similar(D, Int), Diagonal{Int})
@test isa(similar(D, (3,2)), Matrix{elty})
@test isa(similar(D, Int, (3,2)), Matrix{Int})
end
# Issue number 10036
# make sure issymmetric/ishermitian work for
# non-real diagonal matrices
@testset "issymmetric/hermitian for complex Diagonal" begin
@test issymmetric(D2)
@test ishermitian(D2)
if elty <: Complex
dc = d .+ elty(1im)
D3 = Diagonal(dc)
@test issymmetric(D3)
@test !ishermitian(D3)
end
end
@testset "svd (#11120/#11247/#1149)" begin
D[1] = 0
U, s, V = svd(D)
@test (U*Diagonal(s))*V' ≈ D
@test svdvals(D) == s
@test svd(D).V == V
end
end
@testset "axes" begin
v = OffsetArray(1:3)
D = Diagonal(v)
@test axes(D) isa NTuple{2,typeof(axes(v,1))}
end
@testset "rdiv! (#40887)" begin
@test rdiv!(Matrix(Diagonal([2.0, 3.0])), Diagonal(2:3)) == Diagonal([1.0, 1.0])
@test rdiv!(fill(3.0, 3, 3), 3.0I(3)) == ones(3,3)
end
@testset "kron (issue #40595)" begin
# custom array type to test that kron on Diagonal matrices preserves types of the parents if possible
struct KronTestArray{T, N, AT} <: AbstractArray{T, N}
data::AT
end
KronTestArray(data::AbstractArray) = KronTestArray{eltype(data), ndims(data), typeof(data)}(data)
Base.size(A::KronTestArray) = size(A.data)
LinearAlgebra.kron(A::KronTestArray, B::KronTestArray) = KronTestArray(kron(A.data, B.data))
Base.getindex(K::KronTestArray{<:Any,N}, i::Vararg{Int,N}) where {N} = K.data[i...]
A = KronTestArray([1, 2, 3]);
@test kron(A, A) isa KronTestArray
Ad = Diagonal(A);
@test kron(Ad, Ad).diag isa KronTestArray
@test kron(Ad, Ad).diag == kron([1, 2, 3], [1, 2, 3])
end
# Define a vector type that does not support `deleteat!`, to ensure that `kron` handles this
struct SimpleVector{T} <: AbstractVector{T}
vec::Vector{T}
end
SimpleVector(x::SimpleVector) = SimpleVector(Vector(x.vec))
SimpleVector{T}(::UndefInitializer, n::Integer) where {T} = SimpleVector(Vector{T}(undef, n))
Base.:(==)(x::SimpleVector, y::SimpleVector) = x == y
Base.axes(x::SimpleVector) = axes(x.vec)
Base.convert(::Type{Vector{T}}, x::SimpleVector) where {T} = convert(Vector{T}, x.vec)
Base.convert(::Type{Vector}, x::SimpleVector{T}) where {T} = convert(Vector{T}, x)
Base.convert(::Type{Array{T}}, x::SimpleVector) where {T} = convert(Vector{T}, x)
Base.convert(::Type{Array}, x::SimpleVector) = convert(Vector, x)
Base.copyto!(x::SimpleVector, y::SimpleVector) = (copyto!(x.vec, y.vec); x)
Base.eltype(::Type{SimpleVector{T}}) where {T} = T
Base.getindex(x::SimpleVector, ind...) = getindex(x.vec, ind...)
Base.kron(x::SimpleVector, y::SimpleVector) = SimpleVector(kron(x.vec, y.vec))
Base.promote_rule(::Type{<:AbstractVector{T}}, ::Type{SimpleVector{U}}) where {T,U} = Vector{promote_type(T, U)}
Base.promote_rule(::Type{SimpleVector{T}}, ::Type{SimpleVector{U}}) where {T,U} = SimpleVector{promote_type(T, U)}
Base.setindex!(x::SimpleVector, val, ind...) = (setindex!(x.vec, val, ind...), x)
Base.similar(x::SimpleVector, ::Type{T}) where {T} = SimpleVector(similar(x.vec, T))
Base.similar(x::SimpleVector, ::Type{T}, dims::Dims{1}) where {T} = SimpleVector(similar(x.vec, T, dims))
Base.size(x::SimpleVector) = size(x.vec)
@testset "kron (issue #46456)" for repr in Any[identity, SimpleVector]
A = Diagonal(repr(randn(10)))
BL = Bidiagonal(repr(randn(10)), repr(randn(9)), :L)
BU = Bidiagonal(repr(randn(10)), repr(randn(9)), :U)
C = SymTridiagonal(repr(randn(10)), repr(randn(9)))
Cl = SymTridiagonal(repr(randn(10)), repr(randn(10)))
D = Tridiagonal(repr(randn(9)), repr(randn(10)), repr(randn(9)))
@test kron(A, BL)::Bidiagonal == kron(Array(A), Array(BL))
@test kron(A, BU)::Bidiagonal == kron(Array(A), Array(BU))
@test kron(A, C)::SymTridiagonal == kron(Array(A), Array(C))
@test kron(A, Cl)::SymTridiagonal == kron(Array(A), Array(Cl))
@test kron(A, D)::Tridiagonal == kron(Array(A), Array(D))
end
@testset "svdvals and eigvals (#11120/#11247)" begin
D = Diagonal(Matrix{Float64}[randn(3,3), randn(2,2)])
@test sort([svdvals(D)...;], rev = true) ≈ svdvals([D.diag[1] zeros(3,2); zeros(2,3) D.diag[2]])
@test sort([eigvals(D)...;], by=LinearAlgebra.eigsortby) ≈ eigvals([D.diag[1] zeros(3,2); zeros(2,3) D.diag[2]])
end
@testset "eigvals should return a copy of the diagonal" begin
D = Diagonal([1, 2, 3])
lam = eigvals(D)
D[3,3] = 4 # should not affect lam
@test lam == [1, 2, 3]
end
@testset "eigmin (#27847)" begin
for _ in 1:100
d = randn(rand(1:10))
D = Diagonal(d)
@test eigmin(D) == minimum(d)
end
end
@testset "isposdef" begin
@test isposdef(Diagonal(1.0 .+ rand(n)))
@test !isposdef(Diagonal(-1.0 * rand(n)))
@test isposdef(Diagonal(complex(1.0, 0.0) .+ rand(n)))
@test !isposdef(Diagonal(complex(1.0, 1.0) .+ rand(n)))
@test isposdef(Diagonal([[1 0; 0 1], [1 0; 0 1]]))
@test !isposdef(Diagonal([[1 0; 0 1], [1 0; 1 1]]))
end
@testset "getindex" begin
d = randn(n)
D = Diagonal(d)
# getindex bounds checking
@test_throws BoundsError D[0, 0]
@test_throws BoundsError D[-1, -2]
@test_throws BoundsError D[n, n + 1]
@test_throws BoundsError D[n + 1, n]
@test_throws BoundsError D[n + 1, n + 1]
# getindex on and off the diagonal
for i in 1:n, j in 1:n
@test D[i, j] == (i == j ? d[i] : 0)
end
end
@testset "setindex!" begin
d = randn(n)
D = Diagonal(d)
# setindex! bounds checking
@test_throws BoundsError D[0, 0] = 0
@test_throws BoundsError D[-1 , -2] = 0
@test_throws BoundsError D[n, n + 1] = 0
@test_throws BoundsError D[n + 1, n] = 0
@test_throws BoundsError D[n + 1, n + 1] = 0
for i in 1:n, j in 1:n
if i == j
# setindex on! the diagonal
@test ((D[i, j] = i) == i; D[i, j] == i)
else
# setindex! off the diagonal
@test ((D[i, j] = 0) == 0; iszero(D[i, j]))
@test_throws ArgumentError D[i, j] = 1
end
end
# setindex should return the destination
@test setindex!(D, 1, 1, 1) === D
end
@testset "Test reverse" begin
D = Diagonal(randn(5))
@test reverse(D, dims=1) == reverse(Matrix(D), dims=1)
@test reverse(D, dims=2) == reverse(Matrix(D), dims=2)
@test reverse(D)::Diagonal == reverse(Matrix(D))
end
@testset "inverse" begin
for d in Any[randn(n), Int[], [1, 2, 3], [1im, 2im, 3im], [1//1, 2//1, 3//1], [1+1im//1, 2//1, 3im//1]]
D = Diagonal(d)
@test inv(D) ≈ inv(Array(D))
end
@test_throws SingularException inv(Diagonal(zeros(n)))
@test_throws SingularException inv(Diagonal([0, 1, 2]))
@test_throws SingularException inv(Diagonal([0im, 1im, 2im]))
end
@testset "pseudoinverse" begin
for d in Any[randn(n), zeros(n), Int[], [0, 2, 0.003], [0im, 1+2im, 0.003im], [0//1, 2//1, 3//100], [0//1, 1//1+2im, 3im//100]]
D = Diagonal(d)
@test pinv(D) ≈ pinv(Array(D))
@test pinv(D, 1.0e-2) ≈ pinv(Array(D), 1.0e-2)
end
end
# allow construct from range
@test all(Diagonal(range(1, stop=3, length=3)) .== Diagonal([1.0,2.0,3.0]))
# Issue 12803
for t in (Float32, Float64, Int, ComplexF64, Rational{Int})
@test Diagonal(Matrix{t}[fill(t(1), 2, 2), fill(t(1), 3, 3)])[2,1] == zeros(t, 3, 2)
end
# Issue 15401
@test Matrix(1.0I, 5, 5) \ Diagonal(fill(1.,5)) == Matrix(I, 5, 5)
@testset "Triangular and Diagonal" begin
function _test_matrix(type)
if type == Int
return rand(1:9, 5, 5)
else
return randn(type, 5, 5)
end
end
types = (Float64, Int, ComplexF64)
for ta in types
D = Diagonal(_test_matrix(ta))
for tb in types
B = _test_matrix(tb)
Tmats = (LowerTriangular(B), UnitLowerTriangular(B), UpperTriangular(B), UnitUpperTriangular(B))
restypes = (LowerTriangular, LowerTriangular, UpperTriangular, UpperTriangular)
for (T, rtype) in zip(Tmats, restypes)
adjtype = (rtype == LowerTriangular) ? UpperTriangular : LowerTriangular
# Triangular * Diagonal
R = T * D
@test R ≈ Array(T) * Array(D)
@test isa(R, rtype)
# Diagonal * Triangular
R = D * T
@test R ≈ Array(D) * Array(T)
@test isa(R, rtype)
# Adjoint of Triangular * Diagonal
R = T' * D
@test R ≈ Array(T)' * Array(D)
@test isa(R, adjtype)
# Diagonal * Adjoint of Triangular
R = D * T'
@test R ≈ Array(D) * Array(T)'
@test isa(R, adjtype)
# Transpose of Triangular * Diagonal
R = transpose(T) * D
@test R ≈ transpose(Array(T)) * Array(D)
@test isa(R, adjtype)
# Diagonal * Transpose of Triangular
R = D * transpose(T)
@test R ≈ Array(D) * transpose(Array(T))
@test isa(R, adjtype)
end
end
end
end
let D1 = Diagonal(rand(5)), D2 = Diagonal(rand(5))
@test LinearAlgebra.rmul!(copy(D1),D2) == D1*D2
@test LinearAlgebra.lmul!(D1,copy(D2)) == D1*D2
@test LinearAlgebra.rmul!(copy(D1),transpose(D2)) == D1*transpose(D2)
@test LinearAlgebra.lmul!(transpose(D1),copy(D2)) == transpose(D1)*D2
@test LinearAlgebra.rmul!(copy(D1),adjoint(D2)) == D1*adjoint(D2)
@test LinearAlgebra.lmul!(adjoint(D1),copy(D2)) == adjoint(D1)*D2
end
@testset "multiplication of a Diagonal with a Matrix" begin
A = collect(reshape(1:8, 4, 2));
B = BigFloat.(A);
DL = Diagonal(collect(axes(A, 1)));
DR = Diagonal(Float16.(collect(axes(A, 2))));
@test DL * A == collect(DL) * A
@test A * DR == A * collect(DR)
@test DL * B == collect(DL) * B
@test B * DR == B * collect(DR)
A = reshape([ones(2,2), ones(2,2)*2, ones(2,2)*3, ones(2,2)*4], 2, 2)
Ac = collect(A)
D = Diagonal([collect(reshape(1:4, 2, 2)), collect(reshape(5:8, 2, 2))])
Dc = collect(D)
@test A * D == Ac * Dc
@test D * A == Dc * Ac
@test D * D == Dc * Dc
AS = similar(A)
mul!(AS, A, D, true, false)
@test AS == A * D
D2 = similar(D)
mul!(D2, D, D)
@test D2 == D * D
copyto!(D2, D)
lmul!(D, D2)
@test D2 == D * D
copyto!(D2, D)
rmul!(D2, D)
@test D2 == D * D
end
@testset "multiplication of 2 Diagonal and a Matrix (#46400)" begin
A = randn(10, 10)
D = Diagonal(randn(10))
D2 = Diagonal(randn(10))
@test D * A * D2 ≈ D * (A * D2)
@test D * A * D2 ≈ (D * A) * D2
@test_throws DimensionMismatch Diagonal(ones(9)) * A * D2
@test_throws DimensionMismatch D * A * Diagonal(ones(9))
end
@testset "multiplication of QR Q-factor and Diagonal (#16615 spot test)" begin
D = Diagonal(randn(5))
Q = qr(randn(5, 5)).Q
@test D * Q' == Array(D) * Q'
Q = qr(randn(5, 5), ColumnNorm()).Q
@test_throws ArgumentError lmul!(Q, D)
end
@testset "block diagonal matrices" begin
D = Diagonal([[1 2; 3 4], [1 2; 3 4]])
Dherm = Diagonal([[1 1+im; 1-im 1], [1 1+im; 1-im 1]])
Dsym = Diagonal([[1 1+im; 1+im 1], [1 1+im; 1+im 1]])
@test adjoint(D) == Diagonal([[1 3; 2 4], [1 3; 2 4]])
@test transpose(D) == Diagonal([[1 3; 2 4], [1 3; 2 4]])
@test adjoint(Dherm) == Dherm
@test transpose(Dherm) == Diagonal([[1 1-im; 1+im 1], [1 1-im; 1+im 1]])
@test adjoint(Dsym) == Diagonal([[1 1-im; 1-im 1], [1 1-im; 1-im 1]])
@test transpose(Dsym) == Dsym
@test diag(D, 0) == diag(D) == [[1 2; 3 4], [1 2; 3 4]]
@test diag(D, 1) == diag(D, -1) == [zeros(Int,2,2)]
@test diag(D, 2) == diag(D, -2) == []
v = [[1, 2], [3, 4]]
@test Dherm' * v == Dherm * v
@test transpose(D) * v == [[7, 10], [15, 22]]
@test issymmetric(D) == false
@test issymmetric(Dherm) == false
@test issymmetric(Dsym) == true
@test ishermitian(D) == false
@test ishermitian(Dherm) == true
@test ishermitian(Dsym) == false
@test exp(D) == Diagonal([exp([1 2; 3 4]), exp([1 2; 3 4])])
@test cis(D) == Diagonal([cis([1 2; 3 4]), cis([1 2; 3 4])])
@test log(D) == Diagonal([log([1 2; 3 4]), log([1 2; 3 4])])
@test sqrt(D) == Diagonal([sqrt([1 2; 3 4]), sqrt([1 2; 3 4])])
@test tr(D) == 10
@test det(D) == 4
M = [1 2; 3 4]
for n in 0:1
D = Diagonal(fill(M, n))
@test D == Matrix{eltype(D)}(D)
end
S = SizedArray{(2,3)}(reshape([1:6;],2,3))
D = Diagonal(fill(S,3))
@test D * fill(S,2,3)' == fill(S * S', 3, 2)
@test fill(S,3,2)' * D == fill(S' * S, 2, 3)
@testset "indexing with non-standard-axes" begin
s = SizedArrays.SizedArray{(2,2)}([1 2; 3 4])
D = Diagonal(fill(s,3))
@test @inferred(D[1,2]) isa typeof(s)
@test all(iszero, D[1,2])
end
@testset "mul!" begin
D1 = Diagonal(fill(ones(2,3), 2))
D2 = Diagonal(fill(ones(3,2), 2))
C = similar(D1, size(D1))
mul!(C, D1, D2)
@test all(x -> size(x) == (2,2), C)
@test C == D1 * D2
D = similar(D1)
mul!(D, D1, D2)
@test all(x -> size(x) == (2,2), D)
@test D == D1 * D2
end
end
@testset "Eigensystem for block diagonal (issue #30681)" begin
I2 = Matrix(I, 2,2)
D = Diagonal([2.0*I2, 3.0*I2])
eigD = eigen(D)
evals = [ 2.0, 2.0, 3.0, 3.0 ]
evecs = [ [[ 1.0, 0.0 ]] [[ 0.0, 1.0 ]] [[ 0.0, 0.0 ]] [[ 0.0, 0.0 ]];
[[ 0.0, 0.0 ]] [[ 0.0, 0.0 ]] [[ 1.0, 0.0 ]] [[ 0.0, 1.0 ]] ]
@test eigD.values == evals
@test eigD.vectors == evecs
@test D * eigD.vectors ≈ eigD.vectors * Diagonal(eigD.values)
I3 = Matrix(I, 3,3)
D = Diagonal([[0.0 -1.0; 1.0 0.0], 2.0*I3])
eigD = eigen(D)
evals = [ -1.0im, 1.0im, 2.0, 2.0, 2.0 ]
evecs = [ [[ 1/sqrt(2)+0im, 1/sqrt(2)*im ]] [[ 1/sqrt(2)+0im, -1/sqrt(2)*im ]] [[ 0.0, 0.0 ]] [[ 0.0, 0.0 ]] [[ 0.0, 0.0]];
[[ 0.0, 0.0, 0.0 ]] [[ 0.0, 0.0, 0.0 ]] [[ 1.0, 0.0, 0.0 ]] [[ 0.0, 1.0, 0.0 ]] [[ 0.0, 0.0, 1.0]] ]
@test eigD.values == evals
@test eigD.vectors ≈ evecs
@test D * eigD.vectors ≈ eigD.vectors * Diagonal(eigD.values)
# test concrete types
D = Diagonal([I2 for _ in 1:4])
@test eigen(D) isa Eigen{Vector{Float64}, Float64, Matrix{Vector{Float64}}, Vector{Float64}}
end
@testset "linear solve for block diagonal matrices" begin
D = Diagonal([rand(2,2) for _ in 1:5])
b = [rand(2,2) for _ in 1:5]
B = [rand(2,2) for _ in 1:5, _ in 1:5]
@test ldiv!(D, copy(b)) ≈ Diagonal(inv.(D.diag)) * b
@test ldiv!(D, copy(B)) ≈ Diagonal(inv.(D.diag)) * B
@test rdiv!(copy(B), D) ≈ B * Diagonal(inv.(D.diag))
end
@testset "multiplication/division with Symmetric/Hermitian" begin
for T in (Float64, ComplexF64)
D = Diagonal(randn(T, n))
A = randn(T, n, n); A = A'A
S = Symmetric(A)
H = Hermitian(A)
for (transform1, transform2) in ((identity, identity),
(identity, adjoint ), (adjoint, identity ), (adjoint, adjoint ),
(identity, transpose), (transpose, identity ), (transpose, transpose) )
@test *(transform1(D), transform2(S)) ≈ *(transform1(Matrix(D)), transform2(Matrix(S)))
@test *(transform1(D), transform2(H)) ≈ *(transform1(Matrix(D)), transform2(Matrix(H)))
@test *(transform1(S), transform2(D)) ≈ *(transform1(Matrix(S)), transform2(Matrix(D)))
@test *(transform1(S), transform2(H)) ≈ *(transform1(Matrix(S)), transform2(Matrix(H)))
@test (transform1(H)/D) * D ≈ transform1(H)
@test (transform1(S)/D) * D ≈ transform1(S)
@test D * (D\transform2(H)) ≈ transform2(H)
@test D * (D\transform2(S)) ≈ transform2(S)
end
end
end
@testset "multiplication of transposes of Diagonal (#22428)" begin
for T in (Float64, ComplexF64)
D = Diagonal(randn(T, 5, 5))
B = Diagonal(randn(T, 5, 5))
DD = Diagonal([randn(T, 2, 2), rand(T, 2, 2)])
BB = Diagonal([randn(T, 2, 2), rand(T, 2, 2)])
fullDD = copyto!(Matrix{Matrix{T}}(undef, 2, 2), DD)
fullBB = copyto!(Matrix{Matrix{T}}(undef, 2, 2), BB)
for (transform1, transform2) in ((identity, identity),
(identity, adjoint ), (adjoint, identity ), (adjoint, adjoint ),
(identity, transpose), (transpose, identity ), (transpose, transpose))
@test *(transform1(D), transform2(B))::typeof(D) ≈ *(transform1(Matrix(D)), transform2(Matrix(B))) atol=2 * eps()
@test *(transform1(DD), transform2(BB))::typeof(DD) == *(transform1(fullDD), transform2(fullBB))
end
M = randn(T, 5, 5)
MM = [randn(T, 2, 2) for _ in 1:2, _ in 1:2]
for transform in (identity, adjoint, transpose)
@test lmul!(transform(D), copy(M)) ≈ *(transform(Matrix(D)), M)
@test rmul!(copy(M), transform(D)) ≈ *(M, transform(Matrix(D)))
@test lmul!(transform(DD), copy(MM)) ≈ *(transform(fullDD), MM)
@test rmul!(copy(MM), transform(DD)) ≈ *(MM, transform(fullDD))
end
end
end
@testset "Diagonal of adjoint/transpose vectors (#23649)" begin
@test Diagonal(adjoint([1, 2, 3])) == Diagonal([1 2 3])
@test Diagonal(transpose([1, 2, 3])) == Diagonal([1 2 3])
end
@testset "Multiplication with adjoint and transpose vectors (#26863)" begin
x = collect(1:2)
xt = transpose(x)
A = reshape([[1 2; 3 4], zeros(Int,2,2), zeros(Int, 2, 2), [5 6; 7 8]], 2, 2)
D = Diagonal(A)
@test x'*D == x'*A == collect(x')*D == collect(x')*A
@test xt*D == xt*A == collect(xt)*D == collect(xt)*A
outadjxD = similar(x'*D); outtrxD = similar(xt*D);
mul!(outadjxD, x', D)
@test outadjxD == x'*D
mul!(outtrxD, xt, D)
@test outtrxD == xt*D
D1 = Diagonal([[1 2; 3 4]])
@test D1 * x' == D1 * collect(x') == collect(D1) * collect(x')
@test D1 * xt == D1 * collect(xt) == collect(D1) * collect(xt)
outD1adjx = similar(D1 * x'); outD1trx = similar(D1 * xt);
mul!(outadjxD, D1, x')
@test outadjxD == D1*x'
mul!(outtrxD, D1, xt)
@test outtrxD == D1*xt
y = [x, x]
yt = transpose(y)
@test y'*D*y == (y'*D)*y == (y'*A)*y
@test yt*D*y == (yt*D)*y == (yt*A)*y
outadjyD = similar(y'*D); outtryD = similar(yt*D);
outadjyD2 = similar(collect(y'*D)); outtryD2 = similar(collect(yt*D));
mul!(outadjyD, y', D)
mul!(outadjyD2, y', D)
@test outadjyD == outadjyD2 == y'*D
mul!(outtryD, yt, D)
mul!(outtryD2, yt, D)
@test outtryD == outtryD2 == yt*D
end
@testset "Multiplication of single element Diagonal (#36746, #40726)" begin
@test_throws DimensionMismatch Diagonal(randn(1)) * randn(5)
@test_throws DimensionMismatch Diagonal(randn(1)) * Diagonal(randn(3, 3))
A = [1 0; 0 2]
v = [3, 4]
@test Diagonal(A) * v == A * v
@test Diagonal(A) * Diagonal(A) == A * A
@test_throws DimensionMismatch [1 0;0 1] * Diagonal([2 3]) # Issue #40726
@test_throws DimensionMismatch lmul!(Diagonal([1]), [1,2,3]) # nearby
end
@testset "Multiplication of a Diagonal with an OffsetArray" begin
# Offset indices should throw
D = Diagonal(1:4)
A = OffsetArray(rand(4,4), 2, 2)
@test_throws ArgumentError D * A
@test_throws ArgumentError A * D
@test_throws ArgumentError mul!(similar(A, size(A)), A, D)
@test_throws ArgumentError mul!(similar(A, size(A)), D, A)
end
@testset "Triangular division by Diagonal #27989" begin
K = 5
for elty in (Float32, Float64, ComplexF32, ComplexF64)
U = UpperTriangular(randn(elty, K, K))
L = LowerTriangular(randn(elty, K, K))
D = Diagonal(randn(elty, K))
@test (U / D)::UpperTriangular{elty} == UpperTriangular(Matrix(U) / Matrix(D))
@test (L / D)::LowerTriangular{elty} == LowerTriangular(Matrix(L) / Matrix(D))
@test (D \ U)::UpperTriangular{elty} == UpperTriangular(Matrix(D) \ Matrix(U))
@test (D \ L)::LowerTriangular{elty} == LowerTriangular(Matrix(D) \ Matrix(L))
end
end
@testset "(Sym)Tridiagonal division by Diagonal" begin
for K in (5, 1), elty in (Float64, ComplexF32), overlength in (1, 0)
S = SymTridiagonal(randn(elty, K), randn(elty, K-overlength))
T = Tridiagonal(randn(elty, K-1), randn(elty, K), randn(elty, K-1))
D = Diagonal(randn(elty, K))
D0 = Diagonal(zeros(elty, K))
@test (D \ S)::Tridiagonal{elty} == Tridiagonal(Matrix(D) \ Matrix(S))
@test (D \ T)::Tridiagonal{elty} == Tridiagonal(Matrix(D) \ Matrix(T))
@test (S / D)::Tridiagonal{elty} == Tridiagonal(Matrix(S) / Matrix(D))
@test (T / D)::Tridiagonal{elty} == Tridiagonal(Matrix(T) / Matrix(D))
@test_throws SingularException D0 \ S
@test_throws SingularException D0 \ T
@test_throws SingularException S / D0
@test_throws SingularException T / D0
end
# 0-length case
S = SymTridiagonal(Float64[], Float64[])
T = Tridiagonal(Float64[], Float64[], Float64[])
D = Diagonal(Float64[])
@test (D \ S)::Tridiagonal{Float64} == T
@test (D \ T)::Tridiagonal{Float64} == T
@test (S / D)::Tridiagonal{Float64} == T
@test (T / D)::Tridiagonal{Float64} == T