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
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
|
# This file is a part of Julia. License is MIT: https://julialang.org/license
module TestTriangular
debug = false
using Test, LinearAlgebra, SparseArrays, Random
using LinearAlgebra: BlasFloat, errorbounds, full!, naivesub!, transpose!,
UnitUpperTriangular, UnitLowerTriangular,
mul!, rdiv!, rmul!, lmul!
debug && println("Triangular matrices")
n = 9
Random.seed!(123)
debug && println("Test basic type functionality")
@test_throws DimensionMismatch LowerTriangular(randn(5, 4))
@test LowerTriangular(randn(3, 3)) |> t -> [size(t, i) for i = 1:3] == [size(Matrix(t), i) for i = 1:3]
# The following test block tries to call all methods in base/linalg/triangular.jl in order for a combination of input element types. Keep the ordering when adding code.
for elty1 in (Float32, Float64, BigFloat, ComplexF32, ComplexF64, Complex{BigFloat}, Int)
# Begin loop for first Triangular matrix
for (t1, uplo1) in ((UpperTriangular, :U),
(UnitUpperTriangular, :U),
(LowerTriangular, :L),
(UnitLowerTriangular, :L))
# Construct test matrix
A1 = t1(elty1 == Int ? rand(1:7, n, n) : convert(Matrix{elty1}, (elty1 <: Complex ? complex.(randn(n, n), randn(n, n)) : randn(n, n)) |> t -> cholesky(t't).U |> t -> uplo1 == :U ? t : copy(t')))
@test t1(A1) === A1
@test t1{elty1}(A1) === A1
# test the ctor works for AbstractMatrix
symm = Symmetric(rand(Int8, n, n))
t1s = t1{elty1}(symm)
@test typeof(t1s) == t1{elty1, Symmetric{elty1, Matrix{elty1}}}
t1t = t1{elty1}(t1(rand(Int8, n, n)))
@test typeof(t1t) == t1{elty1, Matrix{elty1}}
debug && println("elty1: $elty1, A1: $t1")
# Convert
@test convert(AbstractMatrix{elty1}, A1) == A1
@test convert(Matrix, A1) == A1
@test t1{elty1}(convert(AbstractMatrix{elty1}, A1)) == A1
# full!
@test full!(copy(A1)) == A1
# similar
@test isa(similar(A1), t1)
@test eltype(similar(A1)) == elty1
@test isa(similar(A1, Int), t1)
@test eltype(similar(A1, Int)) == Int
@test isa(similar(A1, (3,2)), Matrix{elty1})
@test isa(similar(A1, Int, (3,2)), Matrix{Int})
#copyto!
simA1 = similar(A1)
copyto!(simA1, A1)
@test simA1 == A1
# getindex
let mA1 = Matrix(A1)
# linear indexing
for i in 1:length(A1)
@test A1[i] == mA1[i]
end
# cartesian indexing
for i in 1:size(A1, 1), j in 1:size(A1, 2)
@test A1[i,j] == mA1[i,j]
end
end
@test isa(A1[2:4,1], Vector)
# setindex! (and copy)
A1c = copy(A1)
for i = 1:size(A1, 1)
for j = 1:size(A1, 2)
if uplo1 == :U
if i > j
A1c[i,j] = 0
@test_throws ArgumentError A1c[i,j] = 1
elseif i == j && t1 == UnitUpperTriangular
A1c[i,j] = 1
@test_throws ArgumentError A1c[i,j] = 0
else
A1c[i,j] = 0
@test A1c[i,j] == 0
end
else
if i < j
A1c[i,j] = 0
@test_throws ArgumentError A1c[i,j] = 1
elseif i == j && t1 == UnitLowerTriangular
A1c[i,j] = 1
@test_throws ArgumentError A1c[i,j] = 0
else
A1c[i,j] = 0
@test A1c[i,j] == 0
end
end
end
end
# istril/istriu
if uplo1 == :L
@test istril(A1)
@test !istriu(A1)
@test istriu(A1')
@test istriu(transpose(A1))
@test !istril(A1')
@test !istril(transpose(A1))
else
@test istriu(A1)
@test !istril(A1)
@test istril(A1')
@test istril(transpose(A1))
@test !istriu(A1')
@test !istriu(transpose(A1))
end
#tril/triu
if uplo1 == :L
@test tril(A1,0) == A1
@test tril(A1,-1) == LowerTriangular(tril(Matrix(A1), -1))
@test tril(A1,1) == t1(tril(tril(Matrix(A1), 1)))
@test tril(A1, -n - 2) == zeros(size(A1))
@test tril(A1, n) == A1
@test triu(A1,0) == t1(diagm(0 => diag(A1)))
@test triu(A1,-1) == t1(tril(triu(A1.data,-1)))
@test triu(A1,1) == zeros(size(A1)) # or just @test iszero(triu(A1,1))?
@test triu(A1, -n) == A1
@test triu(A1, n + 2) == zeros(size(A1))
else
@test triu(A1,0) == A1
@test triu(A1,1) == UpperTriangular(triu(Matrix(A1), 1))
@test triu(A1,-1) == t1(triu(triu(Matrix(A1), -1)))
@test triu(A1, -n) == A1
@test triu(A1, n + 2) == zeros(size(A1))
@test tril(A1,0) == t1(diagm(0 => diag(A1)))
@test tril(A1,1) == t1(triu(tril(A1.data,1)))
@test tril(A1,-1) == zeros(size(A1)) # or just @test iszero(tril(A1,-1))?
@test tril(A1, -n - 2) == zeros(size(A1))
@test tril(A1, n) == A1
end
# factorize
@test factorize(A1) == A1
# [c]transpose[!] (test views as well, see issue #14317)
let vrange = 1:n-1, viewA1 = t1(view(A1.data, vrange, vrange))
# transpose
@test copy(transpose(A1)) == transpose(Matrix(A1))
@test copy(transpose(viewA1)) == transpose(Matrix(viewA1))
# adjoint
@test copy(A1') == Matrix(A1)'
@test copy(viewA1') == Matrix(viewA1)'
# transpose!
@test transpose!(copy(A1)) == transpose(A1)
@test transpose!(t1(view(copy(A1).data, vrange, vrange))) == transpose(viewA1)
# adjoint!
@test adjoint!(copy(A1)) == adjoint(A1)
@test adjoint!(t1(view(copy(A1).data, vrange, vrange))) == adjoint(viewA1)
end
# diag
@test diag(A1) == diag(Matrix(A1))
# real
@test real(A1) == real(Matrix(A1))
@test imag(A1) == imag(Matrix(A1))
@test abs.(A1) == abs.(Matrix(A1))
# Unary operations
@test -A1 == -Matrix(A1)
# copy and copyto! (test views as well, see issue #14317)
let vrange = 1:n-1, viewA1 = t1(view(A1.data, vrange, vrange))
# copy
@test copy(A1) == copy(Matrix(A1))
@test copy(viewA1) == copy(Matrix(viewA1))
# copyto!
B = similar(A1)
copyto!(B, A1)
@test B == A1
B = similar(copy(transpose(A1)))
copyto!(B, copy(transpose(A1)))
@test B == copy(transpose(A1))
B = similar(viewA1)
copyto!(B, viewA1)
@test B == viewA1
B = similar(copy(transpose(viewA1)))
copyto!(B, copy(transpose(viewA1)))
@test B == transpose(viewA1)
end
#exp/log
if (elty1 == Float64 || elty1 == ComplexF64) && (t1 == UpperTriangular || t1 == LowerTriangular)
@test exp(Matrix(log(A1))) ≈ A1
end
# scale
if (t1 == UpperTriangular || t1 == LowerTriangular)
unitt = istriu(A1) ? UnitUpperTriangular : UnitLowerTriangular
if elty1 == Int
cr = 2
else
cr = 0.5
end
ci = cr * im
if elty1 <: Real
A1tmp = copy(A1)
rmul!(A1tmp, cr)
@test A1tmp == cr*A1
A1tmp = copy(A1)
lmul!(cr, A1tmp)
@test A1tmp == cr*A1
A1tmp = copy(A1)
A2tmp = unitt(A1)
mul!(A1tmp, A2tmp, cr)
@test A1tmp == cr * A2tmp
A1tmp = copy(A1)
A2tmp = unitt(A1)
mul!(A1tmp, cr, A2tmp)
@test A1tmp == cr * A2tmp
else
A1tmp = copy(A1)
rmul!(A1tmp, ci)
@test A1tmp == ci*A1
A1tmp = copy(A1)
lmul!(ci, A1tmp)
@test A1tmp == ci*A1
A1tmp = copy(A1)
A2tmp = unitt(A1)
mul!(A1tmp, ci, A2tmp)
@test A1tmp == ci * A2tmp
A1tmp = copy(A1)
A2tmp = unitt(A1)
mul!(A1tmp, A2tmp, ci)
@test A1tmp == A2tmp*ci
end
end
# generalized dot
for eltyb in (Float32, Float64, BigFloat, ComplexF32, ComplexF64, Complex{BigFloat})
b1 = convert(Vector{eltyb}, (elty1 <: Complex ? real(A1) : A1)*fill(1., n))
b2 = convert(Vector{eltyb}, (elty1 <: Complex ? real(A1) : A1)*randn(n))
if elty1 in (BigFloat, Complex{BigFloat}) || eltyb in (BigFloat, Complex{BigFloat})
@test dot(b1, A1, b2) ≈ dot(A1'b1, b2) atol=sqrt(max(eps(real(float(one(elty1)))),eps(real(float(one(eltyb))))))*n*n
else
@test dot(b1, A1, b2) ≈ dot(A1'b1, b2) atol=sqrt(max(eps(real(float(one(elty1)))),eps(real(float(one(eltyb))))))*n*n
end
end
# Binary operations
@test A1*0.5 == Matrix(A1)*0.5
@test 0.5*A1 == 0.5*Matrix(A1)
@test A1/0.5 == Matrix(A1)/0.5
@test 0.5\A1 == 0.5\Matrix(A1)
# inversion
@test inv(A1) ≈ inv(lu(Matrix(A1)))
inv(Matrix(A1)) # issue #11298
@test isa(inv(A1), t1)
# make sure the call to LAPACK works right
if elty1 <: BlasFloat
@test LinearAlgebra.inv!(copy(A1)) ≈ inv(lu(Matrix(A1)))
end
# Determinant
@test det(A1) ≈ det(lu(Matrix(A1))) atol=sqrt(eps(real(float(one(elty1)))))*n*n
@test logdet(A1) ≈ logdet(lu(Matrix(A1))) atol=sqrt(eps(real(float(one(elty1)))))*n*n
lada, ladb = logabsdet(A1)
flada, fladb = logabsdet(lu(Matrix(A1)))
@test lada ≈ flada atol=sqrt(eps(real(float(one(elty1)))))*n*n
@test ladb ≈ fladb atol=sqrt(eps(real(float(one(elty1)))))*n*n
# Matrix square root
@test sqrt(A1) |> t -> t*t ≈ A1
# naivesub errors
@test_throws DimensionMismatch naivesub!(A1,Vector{elty1}(undef,n+1))
# eigenproblems
if !(elty1 in (BigFloat, Complex{BigFloat})) # Not handled yet
vals, vecs = eigen(A1)
if (t1 == UpperTriangular || t1 == LowerTriangular) && elty1 != Int # Cannot really handle degenerate eigen space and Int matrices will probably have repeated eigenvalues.
@test vecs*diagm(0 => vals)/vecs ≈ A1 atol=sqrt(eps(float(real(one(vals[1])))))*(opnorm(A1,Inf)*n)^2
end
end
# Condition number tests - can be VERY approximate
if elty1 <:BlasFloat
for p in (1.0, Inf)
@test cond(A1,p) ≈ cond(A1,p) atol=(cond(A1,p)+cond(A1,p))
end
@test cond(A1,2) == cond(Matrix(A1),2)
end
if !(elty1 in (BigFloat, Complex{BigFloat})) # Not implemented yet
svd(A1)
elty1 <: BlasFloat && svd!(copy(A1))
svdvals(A1)
end
# Begin loop for second Triangular matrix
for elty2 in (Float32, Float64, BigFloat, ComplexF32, ComplexF64, Complex{BigFloat}, Int)
for (t2, uplo2) in ((UpperTriangular, :U),
(UnitUpperTriangular, :U),
(LowerTriangular, :L),
(UnitLowerTriangular, :L))
debug && println("elty1: $elty1, A1: $t1, elty2: $elty2")
A2 = t2(elty2 == Int ? rand(1:7, n, n) : convert(Matrix{elty2}, (elty2 <: Complex ? complex.(randn(n, n), randn(n, n)) : randn(n, n)) |> t -> cholesky(t't).U |> t -> uplo2 == :U ? t : copy(t')))
# Convert
if elty1 <: Real && !(elty2 <: Integer)
@test convert(AbstractMatrix{elty2}, A1) == t1(convert(Matrix{elty2}, A1.data))
elseif elty2 <: Real && !(elty1 <: Integer)
@test_throws InexactError convert(AbstractMatrix{elty2}, A1) == t1(convert(Matrix{elty2}, A1.data))
end
# Binary operations
@test A1 + A2 == Matrix(A1) + Matrix(A2)
@test A1 - A2 == Matrix(A1) - Matrix(A2)
# Triangular-Triangualar multiplication and division
@test A1*A2 ≈ Matrix(A1)*Matrix(A2)
@test transpose(A1)*A2 ≈ transpose(Matrix(A1))*Matrix(A2)
@test transpose(A1)*adjoint(A2) ≈ transpose(Matrix(A1))*adjoint(Matrix(A2))
@test adjoint(A1)*transpose(A2) ≈ adjoint(Matrix(A1))*transpose(Matrix(A2))
@test A1'A2 ≈ Matrix(A1)'Matrix(A2)
@test A1*transpose(A2) ≈ Matrix(A1)*transpose(Matrix(A2))
@test A1*A2' ≈ Matrix(A1)*Matrix(A2)'
@test transpose(A1)*transpose(A2) ≈ transpose(Matrix(A1))*transpose(Matrix(A2))
@test A1'A2' ≈ Matrix(A1)'Matrix(A2)'
@test A1/A2 ≈ Matrix(A1)/Matrix(A2)
@test A1\A2 ≈ Matrix(A1)\Matrix(A2)
offsizeA = Matrix{Float64}(I, n+1, n+1)
@test_throws DimensionMismatch offsizeA / A2
@test_throws DimensionMismatch offsizeA / transpose(A2)
@test_throws DimensionMismatch offsizeA / A2'
@test_throws DimensionMismatch offsizeA * A2
@test_throws DimensionMismatch offsizeA * transpose(A2)
@test_throws DimensionMismatch offsizeA * A2'
@test_throws DimensionMismatch transpose(A2) * offsizeA
@test_throws DimensionMismatch A2' * offsizeA
@test_throws DimensionMismatch A2 * offsizeA
if (uplo1 == uplo2 && elty1 == elty2 != Int && t1 != UnitLowerTriangular && t1 != UnitUpperTriangular)
@test rdiv!(copy(A1), copy(A2)) ≈ A1/A2 ≈ Matrix(A1)/Matrix(A2)
end
if (uplo1 != uplo2 && elty1 == elty2 != Int && t2 != UnitLowerTriangular && t2 != UnitUpperTriangular)
@test lmul!(adjoint(copy(A1)), copy(A2)) ≈ A1'*A2 ≈ Matrix(A1)'*Matrix(A2)
@test lmul!(transpose(copy(A1)), copy(A2)) ≈ transpose(A1)*A2 ≈ transpose(Matrix(A1))*Matrix(A2)
@test ldiv!(adjoint(copy(A1)), copy(A2)) ≈ A1'\A2 ≈ Matrix(A1)'\Matrix(A2)
@test ldiv!(transpose(copy(A1)), copy(A2)) ≈ transpose(A1)\A2 ≈ transpose(Matrix(A1))\Matrix(A2)
end
if (uplo1 != uplo2 && elty1 == elty2 != Int && t1 != UnitLowerTriangular && t1 != UnitUpperTriangular)
@test rmul!(copy(A1), adjoint(copy(A2))) ≈ A1*A2' ≈ Matrix(A1)*Matrix(A2)'
@test rmul!(copy(A1), transpose(copy(A2))) ≈ A1*transpose(A2) ≈ Matrix(A1)*transpose(Matrix(A2))
@test rdiv!(copy(A1), adjoint(copy(A2))) ≈ A1/A2' ≈ Matrix(A1)/Matrix(A2)'
@test rdiv!(copy(A1), transpose(copy(A2))) ≈ A1/transpose(A2) ≈ Matrix(A1)/transpose(Matrix(A2))
end
end
end
for eltyB in (Float32, Float64, BigFloat, ComplexF32, ComplexF64, Complex{BigFloat})
B = convert(Matrix{eltyB}, (elty1 <: Complex ? real(A1) : A1)*fill(1., n, n))
debug && println("elty1: $elty1, A1: $t1, B: $eltyB")
if !(eltyB in (BigFloat, Complex{BigFloat})) # rand does not support BigFloat and Complex{BigFloat} as of Dec 2015
Tri = Tridiagonal(rand(eltyB,n-1),rand(eltyB,n),rand(eltyB,n-1))
@test lmul!(Tri,copy(A1)) ≈ Tri*Matrix(A1)
Tri = Tridiagonal(rand(eltyB,n-1),rand(eltyB,n),rand(eltyB,n-1))
C = Matrix{promote_type(elty1,eltyB)}(undef, n, n)
mul!(C, Tri, copy(A1))
@test C ≈ Tri*Matrix(A1)
Tri = Tridiagonal(rand(eltyB,n-1),rand(eltyB,n),rand(eltyB,n-1))
mul!(C, copy(A1), Tri)
@test C ≈ Matrix(A1)*Tri
end
# Triangular-dense Matrix/vector multiplication
@test A1*B[:,1] ≈ Matrix(A1)*B[:,1]
@test A1*B ≈ Matrix(A1)*B
@test transpose(A1)*B[:,1] ≈ transpose(Matrix(A1))*B[:,1]
@test A1'B[:,1] ≈ Matrix(A1)'B[:,1]
@test transpose(A1)*B ≈ transpose(Matrix(A1))*B
@test A1'B ≈ Matrix(A1)'B
@test A1*transpose(B) ≈ Matrix(A1)*transpose(B)
@test adjoint(A1)*transpose(B) ≈ Matrix(A1)'*transpose(B)
@test transpose(A1)*adjoint(B) ≈ transpose(Matrix(A1))*adjoint(B)
@test A1*B' ≈ Matrix(A1)*B'
@test B*A1 ≈ B*Matrix(A1)
@test transpose(B[:,1])*A1 ≈ transpose(B[:,1])*Matrix(A1)
@test B[:,1]'A1 ≈ B[:,1]'Matrix(A1)
@test transpose(B)*A1 ≈ transpose(B)*Matrix(A1)
@test transpose(B)*adjoint(A1) ≈ transpose(B)*Matrix(A1)'
@test adjoint(B)*transpose(A1) ≈ adjoint(B)*transpose(Matrix(A1))
@test B'A1 ≈ B'Matrix(A1)
@test B*transpose(A1) ≈ B*transpose(Matrix(A1))
@test B*A1' ≈ B*Matrix(A1)'
@test transpose(B[:,1])*transpose(A1) ≈ transpose(B[:,1])*transpose(Matrix(A1))
@test B[:,1]'A1' ≈ B[:,1]'Matrix(A1)'
@test transpose(B)*transpose(A1) ≈ transpose(B)*transpose(Matrix(A1))
@test B'A1' ≈ B'Matrix(A1)'
if eltyB == elty1
@test mul!(similar(B), A1, B) ≈ Matrix(A1)*B
@test mul!(similar(B), A1, adjoint(B)) ≈ Matrix(A1)*B'
@test mul!(similar(B), A1, transpose(B)) ≈ Matrix(A1)*transpose(B)
@test mul!(similar(B), adjoint(A1), adjoint(B)) ≈ Matrix(A1)'*B'
@test mul!(similar(B), transpose(A1), transpose(B)) ≈ transpose(Matrix(A1))*transpose(B)
@test mul!(similar(B), transpose(A1), adjoint(B)) ≈ transpose(Matrix(A1))*B'
@test mul!(similar(B), adjoint(A1), transpose(B)) ≈ Matrix(A1)'*transpose(B)
@test mul!(similar(B), adjoint(A1), B) ≈ Matrix(A1)'*B
@test mul!(similar(B), transpose(A1), B) ≈ transpose(Matrix(A1))*B
# test also vector methods
B1 = vec(B[1,:])
@test mul!(similar(B1), A1, B1) ≈ Matrix(A1)*B1
@test mul!(similar(B1), adjoint(A1), B1) ≈ Matrix(A1)'*B1
@test mul!(similar(B1), transpose(A1), B1) ≈ transpose(Matrix(A1))*B1
end
#error handling
Ann, Bmm, bm = A1, Matrix{eltyB}(undef, n+1, n+1), Vector{eltyB}(undef, n+1)
@test_throws DimensionMismatch lmul!(Ann, bm)
@test_throws DimensionMismatch rmul!(Bmm, Ann)
@test_throws DimensionMismatch lmul!(transpose(Ann), bm)
@test_throws DimensionMismatch lmul!(adjoint(Ann), bm)
@test_throws DimensionMismatch rmul!(Bmm, adjoint(Ann))
@test_throws DimensionMismatch rmul!(Bmm, transpose(Ann))
# ... and division
@test A1\B[:,1] ≈ Matrix(A1)\B[:,1]
@test A1\B ≈ Matrix(A1)\B
@test transpose(A1)\B[:,1] ≈ transpose(Matrix(A1))\B[:,1]
@test A1'\B[:,1] ≈ Matrix(A1)'\B[:,1]
@test transpose(A1)\B ≈ transpose(Matrix(A1))\B
@test A1'\B ≈ Matrix(A1)'\B
@test A1\transpose(B) ≈ Matrix(A1)\transpose(B)
@test A1\B' ≈ Matrix(A1)\B'
@test transpose(A1)\transpose(B) ≈ transpose(Matrix(A1))\transpose(B)
@test A1'\B' ≈ Matrix(A1)'\B'
Ann, bm = A1, Vector{elty1}(undef,n+1)
@test_throws DimensionMismatch Ann\bm
@test_throws DimensionMismatch Ann'\bm
@test_throws DimensionMismatch transpose(Ann)\bm
if t1 == UpperTriangular || t1 == LowerTriangular
@test_throws LinearAlgebra.SingularException naivesub!(t1(zeros(elty1,n,n)),fill(eltyB(1),n))
end
@test B/A1 ≈ B/Matrix(A1)
@test B/transpose(A1) ≈ B/transpose(Matrix(A1))
@test B/A1' ≈ B/Matrix(A1)'
@test transpose(B)/A1 ≈ transpose(B)/Matrix(A1)
@test B'/A1 ≈ B'/Matrix(A1)
@test transpose(B)/transpose(A1) ≈ transpose(B)/transpose(Matrix(A1))
@test B'/A1' ≈ B'/Matrix(A1)'
# Error bounds
!(elty1 in (BigFloat, Complex{BigFloat})) && !(eltyB in (BigFloat, Complex{BigFloat})) && errorbounds(A1, A1\B, B)
end
end
end
# Matrix square root
Atn = UpperTriangular([-1 1 2; 0 -2 2; 0 0 -3])
Atp = UpperTriangular([1 1 2; 0 2 2; 0 0 3])
@test sqrt(Atn) |> t->t*t ≈ Atn
@test typeof(sqrt(Atn)[1,1]) <: Complex
@test sqrt(Atp) |> t->t*t ≈ Atp
@test typeof(sqrt(Atp)[1,1]) <: Real
Areal = randn(n, n)/2
Aimg = randn(n, n)/2
A2real = randn(n, n)/2
A2img = randn(n, n)/2
for eltya in (Float32, Float64, ComplexF32, ComplexF64, BigFloat, Int)
A = eltya == Int ? rand(1:7, n, n) : convert(Matrix{eltya}, eltya <: Complex ? complex.(Areal, Aimg) : Areal)
# a2 = eltya == Int ? rand(1:7, n, n) : convert(Matrix{eltya}, eltya <: Complex ? complex.(a2real, a2img) : a2real)
εa = eps(abs(float(one(eltya))))
for eltyb in (Float32, Float64, ComplexF32, ComplexF64)
εb = eps(abs(float(one(eltyb))))
ε = max(εa,εb)
debug && println("\ntype of A: ", eltya, " type of b: ", eltyb, "\n")
debug && println("Solve upper triangular system")
Atri = UpperTriangular(lu(A).U) |> t -> eltya <: Complex && eltyb <: Real ? real(t) : t # Here the triangular matrix can't be too badly conditioned
b = convert(Matrix{eltyb}, Matrix(Atri)*fill(1., n, 2))
x = Matrix(Atri) \ b
debug && println("Test error estimates")
if eltya != BigFloat && eltyb != BigFloat
for i = 1:2
@test norm(x[:,1] .- 1) <= errorbounds(UpperTriangular(A), x, b)[1][i]
end
end
debug && println("Test forward error [JIN 5705] if this is not a BigFloat")
x = Atri \ b
γ = n*ε/(1 - n*ε)
if eltya != BigFloat
bigA = big.(Atri)
x̂ = fill(1., n, 2)
for i = 1:size(b, 2)
@test norm(x̂[:,i] - x[:,i], Inf)/norm(x̂[:,i], Inf) <= condskeel(bigA, x̂[:,i])*γ/(1 - condskeel(bigA)*γ)
end
end
debug && println("Test backward error [JIN 5705]")
for i = 1:size(b, 2)
@test norm(abs.(b[:,i] - Atri*x[:,i]), Inf) <= γ * norm(Atri, Inf) * norm(x[:,i], Inf)
end
debug && println("Solve lower triangular system")
Atri = UpperTriangular(lu(A).U) |> t -> eltya <: Complex && eltyb <: Real ? real(t) : t # Here the triangular matrix can't be too badly conditioned
b = convert(Matrix{eltyb}, Matrix(Atri)*fill(1., n, 2))
x = Matrix(Atri)\b
debug && println("Test error estimates")
if eltya != BigFloat && eltyb != BigFloat
for i = 1:2
@test norm(x[:,1] .- 1) <= errorbounds(UpperTriangular(A), x, b)[1][i]
end
end
debug && println("Test forward error [JIN 5705] if this is not a BigFloat")
b = (b0 = Atri*fill(1, n, 2); convert(Matrix{eltyb}, eltyb == Int ? trunc.(b0) : b0))
x = Atri \ b
γ = n*ε/(1 - n*ε)
if eltya != BigFloat
bigA = big.(Atri)
x̂ = fill(1., n, 2)
for i = 1:size(b, 2)
@test norm(x̂[:,i] - x[:,i], Inf)/norm(x̂[:,i], Inf) <= condskeel(bigA, x̂[:,i])*γ/(1 - condskeel(bigA)*γ)
end
end
debug && println("Test backward error [JIN 5705]")
for i = 1:size(b, 2)
@test norm(abs.(b[:,i] - Atri*x[:,i]), Inf) <= γ * norm(Atri, Inf) * norm(x[:,i], Inf)
end
end
end
# Issue 10742 and similar
@test istril(UpperTriangular(diagm(0 => [1,2,3,4])))
@test istriu(LowerTriangular(diagm(0 => [1,2,3,4])))
@test isdiag(UpperTriangular(diagm(0 => [1,2,3,4])))
@test isdiag(LowerTriangular(diagm(0 => [1,2,3,4])))
@test !isdiag(UpperTriangular(rand(4, 4)))
@test !isdiag(LowerTriangular(rand(4, 4)))
# Test throwing in fallbacks for non BlasFloat/BlasComplex in A_rdiv_Bx!
let n = 5
A = rand(Float16, n, n)
B = rand(Float16, n-1, n-1)
@test_throws DimensionMismatch rdiv!(A, LowerTriangular(B))
@test_throws DimensionMismatch rdiv!(A, UpperTriangular(B))
@test_throws DimensionMismatch rdiv!(A, UnitLowerTriangular(B))
@test_throws DimensionMismatch rdiv!(A, UnitUpperTriangular(B))
@test_throws DimensionMismatch rdiv!(A, adjoint(LowerTriangular(B)))
@test_throws DimensionMismatch rdiv!(A, adjoint(UpperTriangular(B)))
@test_throws DimensionMismatch rdiv!(A, adjoint(UnitLowerTriangular(B)))
@test_throws DimensionMismatch rdiv!(A, adjoint(UnitUpperTriangular(B)))
@test_throws DimensionMismatch rdiv!(A, transpose(LowerTriangular(B)))
@test_throws DimensionMismatch rdiv!(A, transpose(UpperTriangular(B)))
@test_throws DimensionMismatch rdiv!(A, transpose(UnitLowerTriangular(B)))
@test_throws DimensionMismatch rdiv!(A, transpose(UnitUpperTriangular(B)))
end
@test isdiag(LowerTriangular(UpperTriangular(randn(3,3))))
@test isdiag(UpperTriangular(LowerTriangular(randn(3,3))))
# Issue 16196
@test UpperTriangular(Matrix(1.0I, 3, 3)) \ view(fill(1., 3), [1,2,3]) == fill(1., 3)
# dimensional correctness:
const BASE_TEST_PATH = joinpath(Sys.BINDIR, "..", "share", "julia", "test")
isdefined(Main, :Furlongs) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "Furlongs.jl"))
using .Main.Furlongs
LinearAlgebra.sylvester(a::Furlong,b::Furlong,c::Furlong) = -c / (a + b)
let A = UpperTriangular([Furlong(1) Furlong(4); Furlong(0) Furlong(1)])
@test sqrt(A) == Furlong{1//2}.(UpperTriangular([1 2; 0 1]))
end
@testset "similar should preserve underlying storage type" begin
local m, n = 4, 3
sparsemat = sprand(m, m, 0.5)
for TriType in (UpperTriangular, LowerTriangular, UnitUpperTriangular, UnitLowerTriangular)
trisparsemat = TriType(sparsemat)
@test isa(similar(trisparsemat), typeof(trisparsemat))
@test isa(similar(trisparsemat, Float32), TriType{Float32,<:SparseMatrixCSC{Float32}})
@test isa(similar(trisparsemat, (n, n)), typeof(sparsemat))
@test isa(similar(trisparsemat, Float32, (n, n)), SparseMatrixCSC{Float32})
end
end
@testset "special printing of Lower/UpperTriangular" begin
@test occursin(r"3×3 (LinearAlgebra\.)?LowerTriangular{Int64,Array{Int64,2}}:\n 2 ⋅ ⋅\n 2 2 ⋅\n 2 2 2",
sprint(show, MIME"text/plain"(), LowerTriangular(2ones(Int64,3,3))))
@test occursin(r"3×3 (LinearAlgebra\.)?UnitLowerTriangular{Int64,Array{Int64,2}}:\n 1 ⋅ ⋅\n 2 1 ⋅\n 2 2 1",
sprint(show, MIME"text/plain"(), UnitLowerTriangular(2ones(Int64,3,3))))
@test occursin(r"3×3 (LinearAlgebra\.)?UpperTriangular{Int64,Array{Int64,2}}:\n 2 2 2\n ⋅ 2 2\n ⋅ ⋅ 2",
sprint(show, MIME"text/plain"(), UpperTriangular(2ones(Int64,3,3))))
@test occursin(r"3×3 (LinearAlgebra\.)?UnitUpperTriangular{Int64,Array{Int64,2}}:\n 1 2 2\n ⋅ 1 2\n ⋅ ⋅ 1",
sprint(show, MIME"text/plain"(), UnitUpperTriangular(2ones(Int64,3,3))))
end
@testset "adjoint/transpose triangular/vector multiplication" begin
for elty in (Float64, ComplexF64), trity in (UpperTriangular, LowerTriangular)
A1 = trity(rand(elty, 1, 1))
b1 = rand(elty, 1)
A4 = trity(rand(elty, 4, 4))
b4 = rand(elty, 4)
@test A1 * b1' ≈ Matrix(A1) * b1'
@test_throws DimensionMismatch A4 * b4'
@test A1 * transpose(b1) ≈ Matrix(A1) * transpose(b1)
@test_throws DimensionMismatch A4 * transpose(b4)
@test A1' * b1' ≈ Matrix(A1') * b1'
@test_throws DimensionMismatch A4' * b4'
@test A1' * transpose(b1) ≈ Matrix(A1') * transpose(b1)
@test_throws DimensionMismatch A4' * transpose(b4)
@test transpose(A1) * transpose(b1) ≈ Matrix(transpose(A1)) * transpose(b1)
@test_throws DimensionMismatch transpose(A4) * transpose(b4)
@test transpose(A1) * b1' ≈ Matrix(transpose(A1)) * b1'
@test_throws DimensionMismatch transpose(A4) * b4'
@test b1' * transpose(A1) ≈ b1' * Matrix(transpose(A1))
@test b4' * transpose(A4) ≈ b4' * Matrix(transpose(A4))
@test transpose(b1) * A1' ≈ transpose(b1) * Matrix(A1')
@test transpose(b4) * A4' ≈ transpose(b4) * Matrix(A4')
end
end
@testset "Error condition for powm" begin
A = UpperTriangular(rand(ComplexF64, 10, 10))
@test_throws ArgumentError LinearAlgebra.powm!(A, 2.2)
A = LowerTriangular(rand(ComplexF64, 10, 10))
At = copy(transpose(A))
p = rand()
@test LinearAlgebra.powm(A, p) == transpose(LinearAlgebra.powm!(At, p))
@test_throws ArgumentError LinearAlgebra.powm(A, 2.2)
end
# Issue 35058
let A = [0.9999999999999998 4.649058915617843e-16 -1.3149405273715513e-16 9.9959579317056e-17; -8.326672684688674e-16 1.0000000000000004 2.9280733590254494e-16 -2.9993900031619594e-16; 9.43689570931383e-16 -1.339206523454095e-15 1.0000000000000007 -8.550505126287743e-16; -6.245004513516506e-16 -2.0122792321330962e-16 1.183061278035052e-16 1.0000000000000002],
B = [0.09648289218436859 0.023497875751503007 0.0 0.0; 0.023497875751503007 0.045787575150300804 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0]
@test sqrt(A*B*A')^2 ≈ A*B*A'
end
end # module TestTriangular
|
