1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
|
# This file is a part of Julia. License is MIT: https://julialang.org/license
module TestSVD
using Test, LinearAlgebra, Random
using LinearAlgebra: BlasComplex, BlasFloat, BlasReal, QRPivoted
@testset "Simple svdvals / svd tests" begin
≊(x,y) = isapprox(x,y,rtol=1e-15)
m1 = [2 0; 0 0]
m2 = [2 -2; 1 1]/sqrt(2)
m2c = Complex.([2 -2; 1 1]/sqrt(2))
@test @inferred(svdvals(m1)) ≊ [2, 0]
@test @inferred(svdvals(m2)) ≊ [2, 1]
@test @inferred(svdvals(m2c)) ≊ [2, 1]
sf1 = svd(m1)
sf2 = svd(m2)
@test sf1.S ≊ [2, 0]
@test sf2.S ≊ [2, 1]
# U & Vt are unitary
I22 = Matrix(I, 2, 2)
@test sf1.U*sf1.U' ≊ I22
@test sf1.Vt*sf1.Vt' ≊ I22
@test sf2.U*sf2.U' ≊ I22
@test sf2.Vt*sf2.Vt' ≊ I22
# SVD not uniquely determined, so just test we can reconstruct the
# matrices from the factorization as expected.
@test sf1.U*Diagonal(sf1.S)*sf1.Vt' ≊ m1
@test sf2.U*Diagonal(sf2.S)*sf2.Vt' ≊ m2
@test ldiv!([0., 0.], svd(Matrix(I, 2, 2)), [1., 1.]) ≊ [1., 1.]
@test inv(svd(Matrix(I, 2, 2))) ≈ I
@test inv(svd([1 2; 3 4])) ≈ [-2.0 1.0; 1.5 -0.5]
@test inv(svd([1 0 1; 0 1 0])) ≈ [0.5 0.0; 0.0 1.0; 0.5 0.0]
@test_throws SingularException inv(svd([0 0; 0 0]))
@test inv(svd([1+2im 3+4im; 5+6im 7+8im])) ≈ [-0.5 + 0.4375im 0.25 - 0.1875im; 0.375 - 0.3125im -0.125 + 0.0625im]
end
n = 10
Random.seed!(1234321)
areal = randn(n,n)/2
aimg = randn(n,n)/2
@testset for eltya in (Float32, Float64, ComplexF32, ComplexF64, Int)
aa = eltya == Int ? rand(1:7, n, n) : convert(Matrix{eltya}, eltya <: Complex ? complex.(areal, aimg) : areal)
asym = aa' + aa # symmetric indefinite
for a in (aa, view(aa, 1:n, 1:n))
usv = svd(a)
@testset "singular value decomposition" begin
@test usv.S === svdvals(usv)
@test usv.U * (Diagonal(usv.S) * usv.Vt) ≈ a
@test convert(Array, usv) ≈ a
@test usv.Vt' ≈ usv.V
@test_throws ErrorException usv.Z
b = rand(eltya,n)
@test usv\b ≈ a\b
@test Base.propertynames(usv) == (:U, :S, :V, :Vt)
@test size(usv) == size(a)
if eltya <: BlasFloat
svdz = svd!(Matrix{eltya}(undef,0,0))
@test svdz.U ≈ Matrix{eltya}(I, 0, 0)
@test svdz.S ≈ real(zeros(eltya,0))
@test svdz.Vt ≈ Matrix{eltya}(I, 0, 0)
end
end
@testset "singular value decomposition of adjoint/transpose" begin
for transform in (adjoint, transpose)
usv = svd(transform(a))
@test usv.S === svdvals(usv)
@test usv.U * (Diagonal(usv.S) * usv.Vt) ≈ transform(a)
@test convert(Array, usv) ≈ transform(a)
@test usv.Vt' ≈ usv.V
@test_throws ErrorException usv.Z
b = rand(eltya,n)
@test usv\b ≈ transform(a)\b
end
end
@testset "Generalized svd" begin
a_svd = a[1:div(n, 2), :]
gsvd = svd(a,a_svd)
@test gsvd.U*gsvd.D1*gsvd.R*gsvd.Q' ≈ a
@test gsvd.V*gsvd.D2*gsvd.R*gsvd.Q' ≈ a_svd
@test usv.Vt' ≈ usv.V
@test_throws ErrorException usv.Z
@test_throws ErrorException gsvd.Z
@test gsvd.vals ≈ svdvals(a,a_svd)
α = eltya == Int ? -1 : rand(eltya)
β = svd(α)
@test β.S == [abs(α)]
@test svdvals(α) == abs(α)
u,v,q,d1,d2,r0 = svd(a,a_svd)
@test u ≈ gsvd.U
@test v ≈ gsvd.V
@test d1 ≈ gsvd.D1
@test d2 ≈ gsvd.D2
@test q ≈ gsvd.Q
@test gsvd.a.^2 + gsvd.b.^2 ≈ fill(1, length(gsvd.a))
@test gsvd.alpha.^2 + gsvd.beta.^2 ≈ ones(eltya, length(gsvd.a))
#testing the other layout for D1 & D2
b = rand(eltya,n,2*n)
c = rand(eltya,n,2*n)
gsvd = svd(b,c)
@test gsvd.U*gsvd.D1*gsvd.R*gsvd.Q' ≈ b
@test gsvd.V*gsvd.D2*gsvd.R*gsvd.Q' ≈ c
end
end
@testset "singular value decomposition of Hermitian/real-Symmetric" begin
for T in (eltya <: Real ? (Symmetric, Hermitian) : (Hermitian,))
usv = svd(T(asym))
@test usv.S === svdvals(usv)
@test usv.U * (Diagonal(usv.S) * usv.Vt) ≈ T(asym)
@test convert(Array, usv) ≈ T(asym)
@test usv.Vt' ≈ usv.V
@test_throws ErrorException usv.Z
b = rand(eltya,n)
@test usv\b ≈ T(asym)\b
end
end
if eltya <: LinearAlgebra.BlasReal
@testset "Number input" begin
x, y = randn(eltya, 2)
@test svd(x) == svd(fill(x, 1, 1))
@test svdvals(x) == first(svdvals(fill(x, 1, 1)))
@test svd(x, y) == svd(fill(x, 1, 1), fill(y, 1, 1))
@test svdvals(x, y) ≈ first(svdvals(fill(x, 1, 1), fill(y, 1, 1)))
end
end
if eltya != Int
@testset "isequal, ==, and hash" begin
x, y = rand(eltya), convert(eltya, NaN)
Fx, Fy = svd(x), svd(y)
@test Fx == Fx
@test !(Fy == Fy)
@test isequal(Fy, Fy)
@test hash(Fx) == hash(Fx)
@test hash(Fx, UInt(1)) == hash(Fx, UInt(1))
@test hash(Fy) == hash(Fy)
@test hash(Fy, UInt(1)) == hash(Fy, UInt(1))
end
end
end
@testset "SVD Algorithms" begin
≊(x,y) = isapprox(x,y,rtol=1e-15)
x = [0.1 0.2; 0.3 0.4]
for alg in [LinearAlgebra.QRIteration(), LinearAlgebra.DivideAndConquer()]
sx1 = svd(x, alg = alg)
@test sx1.U * Diagonal(sx1.S) * sx1.Vt ≊ x
@test sx1.V * sx1.Vt ≊ I
@test sx1.U * sx1.U' ≊ I
@test all(sx1.S .≥ 0)
sx2 = svd!(copy(x), alg = alg)
@test sx2.U * Diagonal(sx2.S) * sx2.Vt ≊ x
@test sx2.V * sx2.Vt ≊ I
@test sx2.U * sx2.U' ≊ I
@test all(sx2.S .≥ 0)
end
end
@testset "REPL printing of SVD" begin
svdd = svd(randn(3, 3))
svdstring = sprint((t, s) -> show(t, "text/plain", s), svdd)
ustring = sprint((t, s) -> show(t, "text/plain", s), svdd.U)
sstring = sprint((t, s) -> show(t, "text/plain", s), svdd.S)
vtstring = sprint((t, s) -> show(t, "text/plain", s), svdd.Vt)
@test svdstring == "$(summary(svdd))\nU factor:\n$ustring\nsingular values:\n$sstring\nVt factor:\n$vtstring"
end
@testset "REPL printing of Generalized SVD" begin
a = randn(3, 3)
b = randn(3, 3)
svdd = svd(a, b)
svdstring = sprint((t, s) -> show(t, "text/plain", s), svdd)
ustring = sprint((t, s) -> show(t, "text/plain", s), svdd.U)
qstring = sprint((t, s) -> show(t, "text/plain", s), svdd.Q)
vstring = sprint((t, s) -> show(t, "text/plain", s), svdd.V)
d1string = sprint((t, s) -> show(t, "text/plain", s), svdd.D1)
d2string = sprint((t, s) -> show(t, "text/plain", s), svdd.D2)
r0string = sprint((t, s) -> show(t, "text/plain", s), svdd.R0)
@test svdstring == "$(summary(svdd))\nU factor:\n$ustring\nV factor:\n$vstring\nQ factor:\n$qstring\nD1 factor:\n$d1string\nD2 factor:\n$d2string\nR0 factor:\n$r0string"
end
end # module TestSVD
|
