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
|
*> \brief \b SGSVTS3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SGSVTS3( M, P, N, A, AF, LDA, B, BF, LDB, U, LDU, V,
* LDV, Q, LDQ, ALPHA, BETA, R, LDR, IWORK, WORK,
* LWORK, RWORK, RESULT )
*
* .. Scalar Arguments ..
* INTEGER LDA, LDB, LDQ, LDR, LDU, LDV, LWORK, M, N, P
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* REAL A( LDA, * ), AF( LDA, * ), ALPHA( * ),
* $ B( LDB, * ), BETA( * ), BF( LDB, * ),
* $ Q( LDQ, * ), R( LDR, * ), RESULT( 6 ),
* $ RWORK( * ), U( LDU, * ), V( LDV, * ),
* $ WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGSVTS3 tests SGGSVD3, which computes the GSVD of an M-by-N matrix A
*> and a P-by-N matrix B:
*> U'*A*Q = D1*R and V'*B*Q = D2*R.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (LDA,M)
*> The M-by-N matrix A.
*> \endverbatim
*>
*> \param[out] AF
*> \verbatim
*> AF is REAL array, dimension (LDA,N)
*> Details of the GSVD of A and B, as returned by SGGSVD3,
*> see SGGSVD3 for further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the arrays A and AF.
*> LDA >= max( 1,M ).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is REAL array, dimension (LDB,P)
*> On entry, the P-by-N matrix B.
*> \endverbatim
*>
*> \param[out] BF
*> \verbatim
*> BF is REAL array, dimension (LDB,N)
*> Details of the GSVD of A and B, as returned by SGGSVD3,
*> see SGGSVD3 for further details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the arrays B and BF.
*> LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is REAL array, dimension(LDU,M)
*> The M by M orthogonal matrix U.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,M).
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is REAL array, dimension(LDV,M)
*> The P by P orthogonal matrix V.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,P).
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is REAL array, dimension(LDQ,N)
*> The N by N orthogonal matrix Q.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is REAL array, dimension (N)
*>
*> The generalized singular value pairs of A and B, the
*> ``diagonal'' matrices D1 and D2 are constructed from
*> ALPHA and BETA, see subroutine SGGSVD3 for details.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is REAL array, dimension(LDQ,N)
*> The upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDR
*> \verbatim
*> LDR is INTEGER
*> The leading dimension of the array R. LDR >= max(1,N).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK,
*> LWORK >= max(M,P,N)*max(M,P,N).
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (max(M,P,N))
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is REAL array, dimension (6)
*> The test ratios:
*> RESULT(1) = norm( U'*A*Q - D1*R ) / ( MAX(M,N)*norm(A)*ULP)
*> RESULT(2) = norm( V'*B*Q - D2*R ) / ( MAX(P,N)*norm(B)*ULP)
*> RESULT(3) = norm( I - U'*U ) / ( M*ULP )
*> RESULT(4) = norm( I - V'*V ) / ( P*ULP )
*> RESULT(5) = norm( I - Q'*Q ) / ( N*ULP )
*> RESULT(6) = 0 if ALPHA is in decreasing order;
*> = ULPINV otherwise.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date August 2015
*
*> \ingroup single_eig
*
* =====================================================================
SUBROUTINE SGSVTS3( M, P, N, A, AF, LDA, B, BF, LDB, U, LDU, V,
$ LDV, Q, LDQ, ALPHA, BETA, R, LDR, IWORK, WORK,
$ LWORK, RWORK, RESULT )
*
* -- LAPACK test routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* August 2015
*
* .. Scalar Arguments ..
INTEGER LDA, LDB, LDQ, LDR, LDU, LDV, LWORK, M, N, P
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL A( LDA, * ), AF( LDA, * ), ALPHA( * ),
$ B( LDB, * ), BETA( * ), BF( LDB, * ),
$ Q( LDQ, * ), R( LDR, * ), RESULT( 6 ),
$ RWORK( * ), U( LDU, * ), V( LDV, * ),
$ WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J, K, L
REAL ANORM, BNORM, RESID, TEMP, ULP, ULPINV, UNFL
* ..
* .. External Functions ..
REAL SLAMCH, SLANGE, SLANSY
EXTERNAL SLAMCH, SLANGE, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGEMM, SGGSVD3, SLACPY, SLASET, SSYRK
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
ULP = SLAMCH( 'Precision' )
ULPINV = ONE / ULP
UNFL = SLAMCH( 'Safe minimum' )
*
* Copy the matrix A to the array AF.
*
CALL SLACPY( 'Full', M, N, A, LDA, AF, LDA )
CALL SLACPY( 'Full', P, N, B, LDB, BF, LDB )
*
ANORM = MAX( SLANGE( '1', M, N, A, LDA, RWORK ), UNFL )
BNORM = MAX( SLANGE( '1', P, N, B, LDB, RWORK ), UNFL )
*
* Factorize the matrices A and B in the arrays AF and BF.
*
CALL SGGSVD3( 'U', 'V', 'Q', M, N, P, K, L, AF, LDA, BF, LDB,
$ ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK,
$ IWORK, INFO )
*
* Copy R
*
DO 20 I = 1, MIN( K+L, M )
DO 10 J = I, K + L
R( I, J ) = AF( I, N-K-L+J )
10 CONTINUE
20 CONTINUE
*
IF( M-K-L.LT.0 ) THEN
DO 40 I = M + 1, K + L
DO 30 J = I, K + L
R( I, J ) = BF( I-K, N-K-L+J )
30 CONTINUE
40 CONTINUE
END IF
*
* Compute A:= U'*A*Q - D1*R
*
CALL SGEMM( 'No transpose', 'No transpose', M, N, N, ONE, A, LDA,
$ Q, LDQ, ZERO, WORK, LDA )
*
CALL SGEMM( 'Transpose', 'No transpose', M, N, M, ONE, U, LDU,
$ WORK, LDA, ZERO, A, LDA )
*
DO 60 I = 1, K
DO 50 J = I, K + L
A( I, N-K-L+J ) = A( I, N-K-L+J ) - R( I, J )
50 CONTINUE
60 CONTINUE
*
DO 80 I = K + 1, MIN( K+L, M )
DO 70 J = I, K + L
A( I, N-K-L+J ) = A( I, N-K-L+J ) - ALPHA( I )*R( I, J )
70 CONTINUE
80 CONTINUE
*
* Compute norm( U'*A*Q - D1*R ) / ( MAX(1,M,N)*norm(A)*ULP ) .
*
RESID = SLANGE( '1', M, N, A, LDA, RWORK )
*
IF( ANORM.GT.ZERO ) THEN
RESULT( 1 ) = ( ( RESID / REAL( MAX( 1, M, N ) ) ) / ANORM ) /
$ ULP
ELSE
RESULT( 1 ) = ZERO
END IF
*
* Compute B := V'*B*Q - D2*R
*
CALL SGEMM( 'No transpose', 'No transpose', P, N, N, ONE, B, LDB,
$ Q, LDQ, ZERO, WORK, LDB )
*
CALL SGEMM( 'Transpose', 'No transpose', P, N, P, ONE, V, LDV,
$ WORK, LDB, ZERO, B, LDB )
*
DO 100 I = 1, L
DO 90 J = I, L
B( I, N-L+J ) = B( I, N-L+J ) - BETA( K+I )*R( K+I, K+J )
90 CONTINUE
100 CONTINUE
*
* Compute norm( V'*B*Q - D2*R ) / ( MAX(P,N)*norm(B)*ULP ) .
*
RESID = SLANGE( '1', P, N, B, LDB, RWORK )
IF( BNORM.GT.ZERO ) THEN
RESULT( 2 ) = ( ( RESID / REAL( MAX( 1, P, N ) ) ) / BNORM ) /
$ ULP
ELSE
RESULT( 2 ) = ZERO
END IF
*
* Compute I - U'*U
*
CALL SLASET( 'Full', M, M, ZERO, ONE, WORK, LDQ )
CALL SSYRK( 'Upper', 'Transpose', M, M, -ONE, U, LDU, ONE, WORK,
$ LDU )
*
* Compute norm( I - U'*U ) / ( M * ULP ) .
*
RESID = SLANSY( '1', 'Upper', M, WORK, LDU, RWORK )
RESULT( 3 ) = ( RESID / REAL( MAX( 1, M ) ) ) / ULP
*
* Compute I - V'*V
*
CALL SLASET( 'Full', P, P, ZERO, ONE, WORK, LDV )
CALL SSYRK( 'Upper', 'Transpose', P, P, -ONE, V, LDV, ONE, WORK,
$ LDV )
*
* Compute norm( I - V'*V ) / ( P * ULP ) .
*
RESID = SLANSY( '1', 'Upper', P, WORK, LDV, RWORK )
RESULT( 4 ) = ( RESID / REAL( MAX( 1, P ) ) ) / ULP
*
* Compute I - Q'*Q
*
CALL SLASET( 'Full', N, N, ZERO, ONE, WORK, LDQ )
CALL SSYRK( 'Upper', 'Transpose', N, N, -ONE, Q, LDQ, ONE, WORK,
$ LDQ )
*
* Compute norm( I - Q'*Q ) / ( N * ULP ) .
*
RESID = SLANSY( '1', 'Upper', N, WORK, LDQ, RWORK )
RESULT( 5 ) = ( RESID / REAL( MAX( 1, N ) ) ) / ULP
*
* Check sorting
*
CALL SCOPY( N, ALPHA, 1, WORK, 1 )
DO 110 I = K + 1, MIN( K+L, M )
J = IWORK( I )
IF( I.NE.J ) THEN
TEMP = WORK( I )
WORK( I ) = WORK( J )
WORK( J ) = TEMP
END IF
110 CONTINUE
*
RESULT( 6 ) = ZERO
DO 120 I = K + 1, MIN( K+L, M ) - 1
IF( WORK( I ).LT.WORK( I+1 ) )
$ RESULT( 6 ) = ULPINV
120 CONTINUE
*
RETURN
*
* End of SGSVTS3
*
END
|
