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
|
*> \brief \b SORBDB3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SORBDB3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sorbdb3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sorbdb3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sorbdb3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SORBDB3( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
* TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
* ..
* .. Array Arguments ..
* REAL PHI(*), THETA(*)
* REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
* $ X11(LDX11,*), X21(LDX21,*)
* ..
*
*
*> \par Purpose:
* =============
*>
*>\verbatim
*>
*> SORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
*> matrix X with orthonomal columns:
*>
*> [ B11 ]
*> [ X11 ] [ P1 | ] [ 0 ]
*> [-----] = [---------] [-----] Q1**T .
*> [ X21 ] [ | P2 ] [ B21 ]
*> [ 0 ]
*>
*> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
*> Q, or M-Q. Routines SORBDB1, SORBDB2, and SORBDB4 handle cases in
*> which M-P is not the minimum dimension.
*>
*> The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
*> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
*> Householder vectors.
*>
*> B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
*> implicitly by angles THETA, PHI.
*>
*>\endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows X11 plus the number of rows in X21.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is INTEGER
*> The number of columns in X11 and X21. 0 <= Q <= M.
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*> X11 is REAL array, dimension (LDX11,Q)
*> On entry, the top block of the matrix X to be reduced. On
*> exit, the columns of tril(X11) specify reflectors for P1 and
*> the rows of triu(X11,1) specify reflectors for Q1.
*> \endverbatim
*>
*> \param[in] LDX11
*> \verbatim
*> LDX11 is INTEGER
*> The leading dimension of X11. LDX11 >= P.
*> \endverbatim
*>
*> \param[in,out] X21
*> \verbatim
*> X21 is REAL array, dimension (LDX21,Q)
*> On entry, the bottom block of the matrix X to be reduced. On
*> exit, the columns of tril(X21) specify reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX21
*> \verbatim
*> LDX21 is INTEGER
*> The leading dimension of X21. LDX21 >= M-P.
*> \endverbatim
*>
*> \param[out] THETA
*> \verbatim
*> THETA is REAL array, dimension (Q)
*> The entries of the bidiagonal blocks B11, B21 are defined by
*> THETA and PHI. See Further Details.
*> \endverbatim
*>
*> \param[out] PHI
*> \verbatim
*> PHI is REAL array, dimension (Q-1)
*> The entries of the bidiagonal blocks B11, B21 are defined by
*> THETA and PHI. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP1
*> \verbatim
*> TAUP1 is REAL array, dimension (P)
*> The scalar factors of the elementary reflectors that define
*> P1.
*> \endverbatim
*>
*> \param[out] TAUP2
*> \verbatim
*> TAUP2 is REAL array, dimension (M-P)
*> The scalar factors of the elementary reflectors that define
*> P2.
*> \endverbatim
*>
*> \param[out] TAUQ1
*> \verbatim
*> TAUQ1 is REAL array, dimension (Q)
*> The scalar factors of the elementary reflectors that define
*> Q1.
*> \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 >= M-Q.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*>
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date July 2012
*
*> \ingroup realOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The upper-bidiagonal blocks B11, B21 are represented implicitly by
*> angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
*> in each bidiagonal band is a product of a sine or cosine of a THETA
*> with a sine or cosine of a PHI. See [1] or SORCSD for details.
*>
*> P1, P2, and Q1 are represented as products of elementary reflectors.
*> See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
*> and SORGLQ.
*> \endverbatim
*
*> \par References:
* ================
*>
*> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*> Algorithms, 50(1):33-65, 2009.
*>
* =====================================================================
SUBROUTINE SORBDB3( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
$ TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* July 2012
*
* .. Scalar Arguments ..
INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
* ..
* .. Array Arguments ..
REAL PHI(*), THETA(*)
REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
$ X11(LDX11,*), X21(LDX21,*)
* ..
*
* ====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E0 )
* ..
* .. Local Scalars ..
REAL C, S
INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
$ LWORKMIN, LWORKOPT
LOGICAL LQUERY
* ..
* .. External Subroutines ..
EXTERNAL SLARF, SLARFGP, SORBDB5, SROT, XERBLA
* ..
* .. External Functions ..
REAL SNRM2
EXTERNAL SNRM2
* ..
* .. Intrinsic Function ..
INTRINSIC ATAN2, COS, MAX, SIN, SQRT
* ..
* .. Executable Statements ..
*
* Test input arguments
*
INFO = 0
LQUERY = LWORK .EQ. -1
*
IF( M .LT. 0 ) THEN
INFO = -1
ELSE IF( 2*P .LT. M .OR. P .GT. M ) THEN
INFO = -2
ELSE IF( Q .LT. M-P .OR. M-Q .LT. M-P ) THEN
INFO = -3
ELSE IF( LDX11 .LT. MAX( 1, P ) ) THEN
INFO = -5
ELSE IF( LDX21 .LT. MAX( 1, M-P ) ) THEN
INFO = -7
END IF
*
* Compute workspace
*
IF( INFO .EQ. 0 ) THEN
ILARF = 2
LLARF = MAX( P, M-P-1, Q-1 )
IORBDB5 = 2
LORBDB5 = Q-1
LWORKOPT = MAX( ILARF+LLARF-1, IORBDB5+LORBDB5-1 )
LWORKMIN = LWORKOPT
WORK(1) = LWORKOPT
IF( LWORK .LT. LWORKMIN .AND. .NOT.LQUERY ) THEN
INFO = -14
END IF
END IF
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'SORBDB3', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Reduce rows 1, ..., M-P of X11 and X21
*
DO I = 1, M-P
*
IF( I .GT. 1 ) THEN
CALL SROT( Q-I+1, X11(I-1,I), LDX11, X21(I,I), LDX11, C, S )
END IF
*
CALL SLARFGP( Q-I+1, X21(I,I), X21(I,I+1), LDX21, TAUQ1(I) )
S = X21(I,I)
X21(I,I) = ONE
CALL SLARF( 'R', P-I+1, Q-I+1, X21(I,I), LDX21, TAUQ1(I),
$ X11(I,I), LDX11, WORK(ILARF) )
CALL SLARF( 'R', M-P-I, Q-I+1, X21(I,I), LDX21, TAUQ1(I),
$ X21(I+1,I), LDX21, WORK(ILARF) )
C = SQRT( SNRM2( P-I+1, X11(I,I), 1 )**2
$ + SNRM2( M-P-I, X21(I+1,I), 1 )**2 )
THETA(I) = ATAN2( S, C )
*
CALL SORBDB5( P-I+1, M-P-I, Q-I, X11(I,I), 1, X21(I+1,I), 1,
$ X11(I,I+1), LDX11, X21(I+1,I+1), LDX21,
$ WORK(IORBDB5), LORBDB5, CHILDINFO )
CALL SLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
IF( I .LT. M-P ) THEN
CALL SLARFGP( M-P-I, X21(I+1,I), X21(I+2,I), 1, TAUP2(I) )
PHI(I) = ATAN2( X21(I+1,I), X11(I,I) )
C = COS( PHI(I) )
S = SIN( PHI(I) )
X21(I+1,I) = ONE
CALL SLARF( 'L', M-P-I, Q-I, X21(I+1,I), 1, TAUP2(I),
$ X21(I+1,I+1), LDX21, WORK(ILARF) )
END IF
X11(I,I) = ONE
CALL SLARF( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I), X11(I,I+1),
$ LDX11, WORK(ILARF) )
*
END DO
*
* Reduce the bottom-right portion of X11 to the identity matrix
*
DO I = M-P + 1, Q
CALL SLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
X11(I,I) = ONE
CALL SLARF( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I), X11(I,I+1),
$ LDX11, WORK(ILARF) )
END DO
*
RETURN
*
* End of SORBDB3
*
END
|
