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
|
SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
$ IWORK, TAU, WORK, INFO )
*
* -- LAPACK routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* -- April 2011 --
*
* .. Scalar Arguments ..
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
DOUBLE PRECISION TOLA, TOLB
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGGSVP computes orthogonal matrices U, V and Q such that
*
* N-K-L K L
* U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
* L ( 0 0 A23 )
* M-K-L ( 0 0 0 )
*
* N-K-L K L
* = K ( 0 A12 A13 ) if M-K-L < 0;
* M-K ( 0 0 A23 )
*
* N-K-L K L
* V**T*B*Q = L ( 0 0 B13 )
* P-L ( 0 0 0 )
*
* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
* otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
* numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
*
* This decomposition is the preprocessing step for computing the
* Generalized Singular Value Decomposition (GSVD), see subroutine
* DGGSVD.
*
* Arguments
* =========
*
* JOBU (input) CHARACTER*1
* = 'U': Orthogonal matrix U is computed;
* = 'N': U is not computed.
*
* JOBV (input) CHARACTER*1
* = 'V': Orthogonal matrix V is computed;
* = 'N': V is not computed.
*
* JOBQ (input) CHARACTER*1
* = 'Q': Orthogonal matrix Q is computed;
* = 'N': Q is not computed.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. P >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, A contains the triangular (or trapezoidal) matrix
* described in the Purpose section.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
* On entry, the P-by-N matrix B.
* On exit, B contains the triangular matrix described in
* the Purpose section.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* TOLA (input) DOUBLE PRECISION
* TOLB (input) DOUBLE PRECISION
* TOLA and TOLB are the thresholds to determine the effective
* numerical rank of matrix B and a subblock of A. Generally,
* they are set to
* TOLA = MAX(M,N)*norm(A)*MAZHEPS,
* TOLB = MAX(P,N)*norm(B)*MAZHEPS.
* The size of TOLA and TOLB may affect the size of backward
* errors of the decomposition.
*
* K (output) INTEGER
* L (output) INTEGER
* On exit, K and L specify the dimension of the subblocks
* described in Purpose section.
* K + L = effective numerical rank of (A**T,B**T)**T.
*
* U (output) DOUBLE PRECISION array, dimension (LDU,M)
* If JOBU = 'U', U contains the orthogonal matrix U.
* If JOBU = 'N', U is not referenced.
*
* LDU (input) INTEGER
* The leading dimension of the array U. LDU >= max(1,M) if
* JOBU = 'U'; LDU >= 1 otherwise.
*
* V (output) DOUBLE PRECISION array, dimension (LDV,P)
* If JOBV = 'V', V contains the orthogonal matrix V.
* If JOBV = 'N', V is not referenced.
*
* LDV (input) INTEGER
* The leading dimension of the array V. LDV >= max(1,P) if
* JOBV = 'V'; LDV >= 1 otherwise.
*
* Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
* If JOBQ = 'Q', Q contains the orthogonal matrix Q.
* If JOBQ = 'N', Q is not referenced.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= max(1,N) if
* JOBQ = 'Q'; LDQ >= 1 otherwise.
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* TAU (workspace) DOUBLE PRECISION array, dimension (N)
*
* WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P))
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
*
* Further Details
* ===============
*
* The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
* with column pivoting to detect the effective numerical rank of the
* a matrix. It may be replaced by a better rank determination strategy.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL FORWRD, WANTQ, WANTU, WANTV
INTEGER I, J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
$ DORG2R, DORM2R, DORMR2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTU = LSAME( JOBU, 'U' )
WANTV = LSAME( JOBV, 'V' )
WANTQ = LSAME( JOBQ, 'Q' )
FORWRD = .TRUE.
*
INFO = 0
IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 ) THEN
INFO = -5
ELSE IF( N.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -10
ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
INFO = -16
ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
INFO = -18
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -20
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGSVP', -INFO )
RETURN
END IF
*
* QR with column pivoting of B: B*P = V*( S11 S12 )
* ( 0 0 )
*
DO 10 I = 1, N
IWORK( I ) = 0
10 CONTINUE
CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
*
* Update A := A*P
*
CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
*
* Determine the effective rank of matrix B.
*
L = 0
DO 20 I = 1, MIN( P, N )
IF( ABS( B( I, I ) ).GT.TOLB )
$ L = L + 1
20 CONTINUE
*
IF( WANTV ) THEN
*
* Copy the details of V, and form V.
*
CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
IF( P.GT.1 )
$ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
$ LDV )
CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
END IF
*
* Clean up B
*
DO 40 J = 1, L - 1
DO 30 I = J + 1, L
B( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
IF( P.GT.L )
$ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
*
IF( WANTQ ) THEN
*
* Set Q = I and Update Q := Q*P
*
CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
END IF
*
IF( P.GE.L .AND. N.NE.L ) THEN
*
* RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
*
CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
*
* Update A := A*Z**T
*
CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
$ LDA, WORK, INFO )
*
IF( WANTQ ) THEN
*
* Update Q := Q*Z**T
*
CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
$ LDQ, WORK, INFO )
END IF
*
* Clean up B
*
CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
DO 60 J = N - L + 1, N
DO 50 I = J - N + L + 1, L
B( I, J ) = ZERO
50 CONTINUE
60 CONTINUE
*
END IF
*
* Let N-L L
* A = ( A11 A12 ) M,
*
* then the following does the complete QR decomposition of A11:
*
* A11 = U*( 0 T12 )*P1**T
* ( 0 0 )
*
DO 70 I = 1, N - L
IWORK( I ) = 0
70 CONTINUE
CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
*
* Determine the effective rank of A11
*
K = 0
DO 80 I = 1, MIN( M, N-L )
IF( ABS( A( I, I ) ).GT.TOLA )
$ K = K + 1
80 CONTINUE
*
* Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
*
CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
$ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
*
IF( WANTU ) THEN
*
* Copy the details of U, and form U
*
CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
IF( M.GT.1 )
$ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
$ LDU )
CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
END IF
*
IF( WANTQ ) THEN
*
* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
*
CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
END IF
*
* Clean up A: set the strictly lower triangular part of
* A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
*
DO 100 J = 1, K - 1
DO 90 I = J + 1, K
A( I, J ) = ZERO
90 CONTINUE
100 CONTINUE
IF( M.GT.K )
$ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
*
IF( N-L.GT.K ) THEN
*
* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
*
CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
*
IF( WANTQ ) THEN
*
* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
*
CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
$ Q, LDQ, WORK, INFO )
END IF
*
* Clean up A
*
CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
DO 120 J = N - L - K + 1, N - L
DO 110 I = J - N + L + K + 1, K
A( I, J ) = ZERO
110 CONTINUE
120 CONTINUE
*
END IF
*
IF( M.GT.K ) THEN
*
* QR factorization of A( K+1:M,N-L+1:N )
*
CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
*
IF( WANTU ) THEN
*
* Update U(:,K+1:M) := U(:,K+1:M)*U1
*
CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
$ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
$ WORK, INFO )
END IF
*
* Clean up
*
DO 140 J = N - L + 1, N
DO 130 I = J - N + K + L + 1, M
A( I, J ) = ZERO
130 CONTINUE
140 CONTINUE
*
END IF
*
RETURN
*
* End of DGGSVP
*
END
|