1 RECURSIVE SUBROUTINE DORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS,
2 $ SIGNS, M, P, Q, X11, LDX11, X12,
3 $ LDX12, X21, LDX21, X22, LDX22, THETA,
4 $ U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
5 $ LDV2T, WORK, LWORK, IWORK, INFO )
6 IMPLICIT NONE
7 *
8 * -- LAPACK routine (version 3.3.1) --
9 *
10 * -- Contributed by Brian Sutton of the Randolph-Macon College --
11 * -- November 2010
12 *
13 * -- LAPACK is a software package provided by Univ. of Tennessee, --
14 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
15 *
16 * @precisions normal d -> s
17 *
18 * .. Scalar Arguments ..
19 CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
20 INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
21 $ LDX21, LDX22, LWORK, M, P, Q
22 * ..
23 * .. Array Arguments ..
24 INTEGER IWORK( * )
25 DOUBLE PRECISION THETA( * )
26 DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
27 $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
28 $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
29 $ * )
30 * ..
31 *
32 * Purpose
33 * =======
34 *
35 * DORCSD computes the CS decomposition of an M-by-M partitioned
36 * orthogonal matrix X:
37 *
38 * [ I 0 0 | 0 0 0 ]
39 * [ 0 C 0 | 0 -S 0 ]
40 * [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
41 * X = [-----------] = [---------] [---------------------] [---------] .
42 * [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
43 * [ 0 S 0 | 0 C 0 ]
44 * [ 0 0 I | 0 0 0 ]
45 *
46 * X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
47 * (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
48 * R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
49 * which R = MIN(P,M-P,Q,M-Q).
50 *
51 * Arguments
52 * =========
53 *
54 * JOBU1 (input) CHARACTER
55 * = 'Y': U1 is computed;
56 * otherwise: U1 is not computed.
57 *
58 * JOBU2 (input) CHARACTER
59 * = 'Y': U2 is computed;
60 * otherwise: U2 is not computed.
61 *
62 * JOBV1T (input) CHARACTER
63 * = 'Y': V1T is computed;
64 * otherwise: V1T is not computed.
65 *
66 * JOBV2T (input) CHARACTER
67 * = 'Y': V2T is computed;
68 * otherwise: V2T is not computed.
69 *
70 * TRANS (input) CHARACTER
71 * = 'T': X, U1, U2, V1T, and V2T are stored in row-major
72 * order;
73 * otherwise: X, U1, U2, V1T, and V2T are stored in column-
74 * major order.
75 *
76 * SIGNS (input) CHARACTER
77 * = 'O': The lower-left block is made nonpositive (the
78 * "other" convention);
79 * otherwise: The upper-right block is made nonpositive (the
80 * "default" convention).
81 *
82 * M (input) INTEGER
83 * The number of rows and columns in X.
84 *
85 * P (input) INTEGER
86 * The number of rows in X11 and X12. 0 <= P <= M.
87 *
88 * Q (input) INTEGER
89 * The number of columns in X11 and X21. 0 <= Q <= M.
90 *
91 * X (input/workspace) DOUBLE PRECISION array, dimension (LDX,M)
92 * On entry, the orthogonal matrix whose CSD is desired.
93 *
94 * LDX (input) INTEGER
95 * The leading dimension of X. LDX >= MAX(1,M).
96 *
97 * THETA (output) DOUBLE PRECISION array, dimension (R), in which R =
98 * MIN(P,M-P,Q,M-Q).
99 * C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
100 * S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
101 *
102 * U1 (output) DOUBLE PRECISION array, dimension (P)
103 * If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
104 *
105 * LDU1 (input) INTEGER
106 * The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
107 * MAX(1,P).
108 *
109 * U2 (output) DOUBLE PRECISION array, dimension (M-P)
110 * If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
111 * matrix U2.
112 *
113 * LDU2 (input) INTEGER
114 * The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
115 * MAX(1,M-P).
116 *
117 * V1T (output) DOUBLE PRECISION array, dimension (Q)
118 * If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
119 * matrix V1**T.
120 *
121 * LDV1T (input) INTEGER
122 * The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
123 * MAX(1,Q).
124 *
125 * V2T (output) DOUBLE PRECISION array, dimension (M-Q)
126 * If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
127 * matrix V2**T.
128 *
129 * LDV2T (input) INTEGER
130 * The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
131 * MAX(1,M-Q).
132 *
133 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
134 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
135 * If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
136 * ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
137 * define the matrix in intermediate bidiagonal-block form
138 * remaining after nonconvergence. INFO specifies the number
139 * of nonzero PHI's.
140 *
141 * LWORK (input) INTEGER
142 * The dimension of the array WORK.
143 *
144 * If LWORK = -1, then a workspace query is assumed; the routine
145 * only calculates the optimal size of the WORK array, returns
146 * this value as the first entry of the work array, and no error
147 * message related to LWORK is issued by XERBLA.
148 *
149 * IWORK (workspace) INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
150 *
151 * INFO (output) INTEGER
152 * = 0: successful exit.
153 * < 0: if INFO = -i, the i-th argument had an illegal value.
154 * > 0: DBBCSD did not converge. See the description of WORK
155 * above for details.
156 *
157 * Reference
158 * =========
159 *
160 * [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
161 * Algorithms, 50(1):33-65, 2009.
162 *
163 * ===================================================================
164 *
165 * .. Parameters ..
166 DOUBLE PRECISION REALONE
167 PARAMETER ( REALONE = 1.0D0 )
168 DOUBLE PRECISION NEGONE, ONE, PIOVER2, ZERO
169 PARAMETER ( NEGONE = -1.0D0, ONE = 1.0D0,
170 $ PIOVER2 = 1.57079632679489662D0,
171 $ ZERO = 0.0D0 )
172 * ..
173 * .. Local Scalars ..
174 CHARACTER TRANST, SIGNST
175 INTEGER CHILDINFO, I, IB11D, IB11E, IB12D, IB12E,
176 $ IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB,
177 $ IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1,
178 $ ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN,
179 $ LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN,
180 $ LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN,
181 $ LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN,
182 $ LORGQRWORKOPT, LWORKMIN, LWORKOPT
183 LOGICAL COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2,
184 $ WANTV1T, WANTV2T
185 * ..
186 * .. External Subroutines ..
187 EXTERNAL DBBCSD, DLACPY, DLAPMR, DLAPMT, DLASCL, DLASET,
188 $ DORBDB, DORGLQ, DORGQR, XERBLA
189 * ..
190 * .. External Functions ..
191 LOGICAL LSAME
192 EXTERNAL LSAME
193 * ..
194 * .. Intrinsic Functions
195 INTRINSIC COS, INT, MAX, MIN, SIN
196 * ..
197 * .. Executable Statements ..
198 *
199 * Test input arguments
200 *
201 INFO = 0
202 WANTU1 = LSAME( JOBU1, 'Y' )
203 WANTU2 = LSAME( JOBU2, 'Y' )
204 WANTV1T = LSAME( JOBV1T, 'Y' )
205 WANTV2T = LSAME( JOBV2T, 'Y' )
206 COLMAJOR = .NOT. LSAME( TRANS, 'T' )
207 DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' )
208 LQUERY = LWORK .EQ. -1
209 IF( M .LT. 0 ) THEN
210 INFO = -7
211 ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
212 INFO = -8
213 ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
214 INFO = -9
215 ELSE IF( ( COLMAJOR .AND. LDX11 .LT. MAX(1,P) ) .OR.
216 $ ( .NOT.COLMAJOR .AND. LDX11 .LT. MAX(1,Q) ) ) THEN
217 INFO = -11
218 ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
219 INFO = -14
220 ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
221 INFO = -16
222 ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
223 INFO = -18
224 ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
225 INFO = -20
226 END IF
227 *
228 * Work with transpose if convenient
229 *
230 IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN
231 IF( COLMAJOR ) THEN
232 TRANST = 'T'
233 ELSE
234 TRANST = 'N'
235 END IF
236 IF( DEFAULTSIGNS ) THEN
237 SIGNST = 'O'
238 ELSE
239 SIGNST = 'D'
240 END IF
241 CALL DORCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST, M,
242 $ Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22,
243 $ LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1,
244 $ U2, LDU2, WORK, LWORK, IWORK, INFO )
245 RETURN
246 END IF
247 *
248 * Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
249 * convenient
250 *
251 IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN
252 IF( DEFAULTSIGNS ) THEN
253 SIGNST = 'O'
254 ELSE
255 SIGNST = 'D'
256 END IF
257 CALL DORCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M,
258 $ M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11,
259 $ LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T,
260 $ LDV1T, WORK, LWORK, IWORK, INFO )
261 RETURN
262 END IF
263 *
264 * Compute workspace
265 *
266 IF( INFO .EQ. 0 ) THEN
267 *
268 IPHI = 2
269 ITAUP1 = IPHI + MAX( 1, Q - 1 )
270 ITAUP2 = ITAUP1 + MAX( 1, P )
271 ITAUQ1 = ITAUP2 + MAX( 1, M - P )
272 ITAUQ2 = ITAUQ1 + MAX( 1, Q )
273 IORGQR = ITAUQ2 + MAX( 1, M - Q )
274 CALL DORGQR( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
275 $ CHILDINFO )
276 LORGQRWORKOPT = INT( WORK(1) )
277 LORGQRWORKMIN = MAX( 1, M - Q )
278 IORGLQ = ITAUQ2 + MAX( 1, M - Q )
279 CALL DORGLQ( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
280 $ CHILDINFO )
281 LORGLQWORKOPT = INT( WORK(1) )
282 LORGLQWORKMIN = MAX( 1, M - Q )
283 IORBDB = ITAUQ2 + MAX( 1, M - Q )
284 CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
285 $ X21, LDX21, X22, LDX22, 0, 0, 0, 0, 0, 0, WORK,
286 $ -1, CHILDINFO )
287 LORBDBWORKOPT = INT( WORK(1) )
288 LORBDBWORKMIN = LORBDBWORKOPT
289 IB11D = ITAUQ2 + MAX( 1, M - Q )
290 IB11E = IB11D + MAX( 1, Q )
291 IB12D = IB11E + MAX( 1, Q - 1 )
292 IB12E = IB12D + MAX( 1, Q )
293 IB21D = IB12E + MAX( 1, Q - 1 )
294 IB21E = IB21D + MAX( 1, Q )
295 IB22D = IB21E + MAX( 1, Q - 1 )
296 IB22E = IB22D + MAX( 1, Q )
297 IBBCSD = IB22E + MAX( 1, Q - 1 )
298 CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, 0,
299 $ 0, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, 0,
300 $ 0, 0, 0, 0, 0, 0, 0, WORK, -1, CHILDINFO )
301 LBBCSDWORKOPT = INT( WORK(1) )
302 LBBCSDWORKMIN = LBBCSDWORKOPT
303 LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT,
304 $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKOPT ) - 1
305 LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN,
306 $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKMIN ) - 1
307 WORK(1) = MAX(LWORKOPT,LWORKMIN)
308 *
309 IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
310 INFO = -22
311 ELSE
312 LORGQRWORK = LWORK - IORGQR + 1
313 LORGLQWORK = LWORK - IORGLQ + 1
314 LORBDBWORK = LWORK - IORBDB + 1
315 LBBCSDWORK = LWORK - IBBCSD + 1
316 END IF
317 END IF
318 *
319 * Abort if any illegal arguments
320 *
321 IF( INFO .NE. 0 ) THEN
322 CALL XERBLA( 'DORCSD', -INFO )
323 RETURN
324 ELSE IF( LQUERY ) THEN
325 RETURN
326 END IF
327 *
328 * Transform to bidiagonal block form
329 *
330 CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21,
331 $ LDX21, X22, LDX22, THETA, WORK(IPHI), WORK(ITAUP1),
332 $ WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2),
333 $ WORK(IORBDB), LORBDBWORK, CHILDINFO )
334 *
335 * Accumulate Householder reflectors
336 *
337 IF( COLMAJOR ) THEN
338 IF( WANTU1 .AND. P .GT. 0 ) THEN
339 CALL DLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 )
340 CALL DORGQR( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGQR),
341 $ LORGQRWORK, INFO)
342 END IF
343 IF( WANTU2 .AND. M-P .GT. 0 ) THEN
344 CALL DLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 )
345 CALL DORGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
346 $ WORK(IORGQR), LORGQRWORK, INFO )
347 END IF
348 IF( WANTV1T .AND. Q .GT. 0 ) THEN
349 CALL DLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2),
350 $ LDV1T )
351 V1T(1, 1) = ONE
352 DO J = 2, Q
353 V1T(1,J) = ZERO
354 V1T(J,1) = ZERO
355 END DO
356 CALL DORGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
357 $ WORK(IORGLQ), LORGLQWORK, INFO )
358 END IF
359 IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
360 CALL DLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T )
361 CALL DLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22,
362 $ V2T(P+1,P+1), LDV2T )
363 CALL DORGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
364 $ WORK(IORGLQ), LORGLQWORK, INFO )
365 END IF
366 ELSE
367 IF( WANTU1 .AND. P .GT. 0 ) THEN
368 CALL DLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 )
369 CALL DORGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGLQ),
370 $ LORGLQWORK, INFO)
371 END IF
372 IF( WANTU2 .AND. M-P .GT. 0 ) THEN
373 CALL DLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 )
374 CALL DORGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
375 $ WORK(IORGLQ), LORGLQWORK, INFO )
376 END IF
377 IF( WANTV1T .AND. Q .GT. 0 ) THEN
378 CALL DLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2),
379 $ LDV1T )
380 V1T(1, 1) = ONE
381 DO J = 2, Q
382 V1T(1,J) = ZERO
383 V1T(J,1) = ZERO
384 END DO
385 CALL DORGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
386 $ WORK(IORGQR), LORGQRWORK, INFO )
387 END IF
388 IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
389 CALL DLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T )
390 CALL DLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22,
391 $ V2T(P+1,P+1), LDV2T )
392 CALL DORGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
393 $ WORK(IORGQR), LORGQRWORK, INFO )
394 END IF
395 END IF
396 *
397 * Compute the CSD of the matrix in bidiagonal-block form
398 *
399 CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA,
400 $ WORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
401 $ LDV2T, WORK(IB11D), WORK(IB11E), WORK(IB12D),
402 $ WORK(IB12E), WORK(IB21D), WORK(IB21E), WORK(IB22D),
403 $ WORK(IB22E), WORK(IBBCSD), LBBCSDWORK, INFO )
404 *
405 * Permute rows and columns to place identity submatrices in top-
406 * left corner of (1,1)-block and/or bottom-right corner of (1,2)-
407 * block and/or bottom-right corner of (2,1)-block and/or top-left
408 * corner of (2,2)-block
409 *
410 IF( Q .GT. 0 .AND. WANTU2 ) THEN
411 DO I = 1, Q
412 IWORK(I) = M - P - Q + I
413 END DO
414 DO I = Q + 1, M - P
415 IWORK(I) = I - Q
416 END DO
417 IF( COLMAJOR ) THEN
418 CALL DLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK )
419 ELSE
420 CALL DLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK )
421 END IF
422 END IF
423 IF( M .GT. 0 .AND. WANTV2T ) THEN
424 DO I = 1, P
425 IWORK(I) = M - P - Q + I
426 END DO
427 DO I = P + 1, M - Q
428 IWORK(I) = I - P
429 END DO
430 IF( .NOT. COLMAJOR ) THEN
431 CALL DLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
432 ELSE
433 CALL DLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
434 END IF
435 END IF
436 *
437 RETURN
438 *
439 * End DORCSD
440 *
441 END
442
2 $ SIGNS, M, P, Q, X11, LDX11, X12,
3 $ LDX12, X21, LDX21, X22, LDX22, THETA,
4 $ U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
5 $ LDV2T, WORK, LWORK, IWORK, INFO )
6 IMPLICIT NONE
7 *
8 * -- LAPACK routine (version 3.3.1) --
9 *
10 * -- Contributed by Brian Sutton of the Randolph-Macon College --
11 * -- November 2010
12 *
13 * -- LAPACK is a software package provided by Univ. of Tennessee, --
14 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
15 *
16 * @precisions normal d -> s
17 *
18 * .. Scalar Arguments ..
19 CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
20 INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
21 $ LDX21, LDX22, LWORK, M, P, Q
22 * ..
23 * .. Array Arguments ..
24 INTEGER IWORK( * )
25 DOUBLE PRECISION THETA( * )
26 DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
27 $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
28 $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
29 $ * )
30 * ..
31 *
32 * Purpose
33 * =======
34 *
35 * DORCSD computes the CS decomposition of an M-by-M partitioned
36 * orthogonal matrix X:
37 *
38 * [ I 0 0 | 0 0 0 ]
39 * [ 0 C 0 | 0 -S 0 ]
40 * [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
41 * X = [-----------] = [---------] [---------------------] [---------] .
42 * [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
43 * [ 0 S 0 | 0 C 0 ]
44 * [ 0 0 I | 0 0 0 ]
45 *
46 * X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
47 * (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
48 * R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
49 * which R = MIN(P,M-P,Q,M-Q).
50 *
51 * Arguments
52 * =========
53 *
54 * JOBU1 (input) CHARACTER
55 * = 'Y': U1 is computed;
56 * otherwise: U1 is not computed.
57 *
58 * JOBU2 (input) CHARACTER
59 * = 'Y': U2 is computed;
60 * otherwise: U2 is not computed.
61 *
62 * JOBV1T (input) CHARACTER
63 * = 'Y': V1T is computed;
64 * otherwise: V1T is not computed.
65 *
66 * JOBV2T (input) CHARACTER
67 * = 'Y': V2T is computed;
68 * otherwise: V2T is not computed.
69 *
70 * TRANS (input) CHARACTER
71 * = 'T': X, U1, U2, V1T, and V2T are stored in row-major
72 * order;
73 * otherwise: X, U1, U2, V1T, and V2T are stored in column-
74 * major order.
75 *
76 * SIGNS (input) CHARACTER
77 * = 'O': The lower-left block is made nonpositive (the
78 * "other" convention);
79 * otherwise: The upper-right block is made nonpositive (the
80 * "default" convention).
81 *
82 * M (input) INTEGER
83 * The number of rows and columns in X.
84 *
85 * P (input) INTEGER
86 * The number of rows in X11 and X12. 0 <= P <= M.
87 *
88 * Q (input) INTEGER
89 * The number of columns in X11 and X21. 0 <= Q <= M.
90 *
91 * X (input/workspace) DOUBLE PRECISION array, dimension (LDX,M)
92 * On entry, the orthogonal matrix whose CSD is desired.
93 *
94 * LDX (input) INTEGER
95 * The leading dimension of X. LDX >= MAX(1,M).
96 *
97 * THETA (output) DOUBLE PRECISION array, dimension (R), in which R =
98 * MIN(P,M-P,Q,M-Q).
99 * C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
100 * S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
101 *
102 * U1 (output) DOUBLE PRECISION array, dimension (P)
103 * If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
104 *
105 * LDU1 (input) INTEGER
106 * The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
107 * MAX(1,P).
108 *
109 * U2 (output) DOUBLE PRECISION array, dimension (M-P)
110 * If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
111 * matrix U2.
112 *
113 * LDU2 (input) INTEGER
114 * The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
115 * MAX(1,M-P).
116 *
117 * V1T (output) DOUBLE PRECISION array, dimension (Q)
118 * If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
119 * matrix V1**T.
120 *
121 * LDV1T (input) INTEGER
122 * The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
123 * MAX(1,Q).
124 *
125 * V2T (output) DOUBLE PRECISION array, dimension (M-Q)
126 * If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
127 * matrix V2**T.
128 *
129 * LDV2T (input) INTEGER
130 * The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
131 * MAX(1,M-Q).
132 *
133 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
134 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
135 * If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
136 * ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
137 * define the matrix in intermediate bidiagonal-block form
138 * remaining after nonconvergence. INFO specifies the number
139 * of nonzero PHI's.
140 *
141 * LWORK (input) INTEGER
142 * The dimension of the array WORK.
143 *
144 * If LWORK = -1, then a workspace query is assumed; the routine
145 * only calculates the optimal size of the WORK array, returns
146 * this value as the first entry of the work array, and no error
147 * message related to LWORK is issued by XERBLA.
148 *
149 * IWORK (workspace) INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
150 *
151 * INFO (output) INTEGER
152 * = 0: successful exit.
153 * < 0: if INFO = -i, the i-th argument had an illegal value.
154 * > 0: DBBCSD did not converge. See the description of WORK
155 * above for details.
156 *
157 * Reference
158 * =========
159 *
160 * [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
161 * Algorithms, 50(1):33-65, 2009.
162 *
163 * ===================================================================
164 *
165 * .. Parameters ..
166 DOUBLE PRECISION REALONE
167 PARAMETER ( REALONE = 1.0D0 )
168 DOUBLE PRECISION NEGONE, ONE, PIOVER2, ZERO
169 PARAMETER ( NEGONE = -1.0D0, ONE = 1.0D0,
170 $ PIOVER2 = 1.57079632679489662D0,
171 $ ZERO = 0.0D0 )
172 * ..
173 * .. Local Scalars ..
174 CHARACTER TRANST, SIGNST
175 INTEGER CHILDINFO, I, IB11D, IB11E, IB12D, IB12E,
176 $ IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB,
177 $ IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1,
178 $ ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN,
179 $ LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN,
180 $ LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN,
181 $ LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN,
182 $ LORGQRWORKOPT, LWORKMIN, LWORKOPT
183 LOGICAL COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2,
184 $ WANTV1T, WANTV2T
185 * ..
186 * .. External Subroutines ..
187 EXTERNAL DBBCSD, DLACPY, DLAPMR, DLAPMT, DLASCL, DLASET,
188 $ DORBDB, DORGLQ, DORGQR, XERBLA
189 * ..
190 * .. External Functions ..
191 LOGICAL LSAME
192 EXTERNAL LSAME
193 * ..
194 * .. Intrinsic Functions
195 INTRINSIC COS, INT, MAX, MIN, SIN
196 * ..
197 * .. Executable Statements ..
198 *
199 * Test input arguments
200 *
201 INFO = 0
202 WANTU1 = LSAME( JOBU1, 'Y' )
203 WANTU2 = LSAME( JOBU2, 'Y' )
204 WANTV1T = LSAME( JOBV1T, 'Y' )
205 WANTV2T = LSAME( JOBV2T, 'Y' )
206 COLMAJOR = .NOT. LSAME( TRANS, 'T' )
207 DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' )
208 LQUERY = LWORK .EQ. -1
209 IF( M .LT. 0 ) THEN
210 INFO = -7
211 ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
212 INFO = -8
213 ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
214 INFO = -9
215 ELSE IF( ( COLMAJOR .AND. LDX11 .LT. MAX(1,P) ) .OR.
216 $ ( .NOT.COLMAJOR .AND. LDX11 .LT. MAX(1,Q) ) ) THEN
217 INFO = -11
218 ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
219 INFO = -14
220 ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
221 INFO = -16
222 ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
223 INFO = -18
224 ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
225 INFO = -20
226 END IF
227 *
228 * Work with transpose if convenient
229 *
230 IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN
231 IF( COLMAJOR ) THEN
232 TRANST = 'T'
233 ELSE
234 TRANST = 'N'
235 END IF
236 IF( DEFAULTSIGNS ) THEN
237 SIGNST = 'O'
238 ELSE
239 SIGNST = 'D'
240 END IF
241 CALL DORCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST, M,
242 $ Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22,
243 $ LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1,
244 $ U2, LDU2, WORK, LWORK, IWORK, INFO )
245 RETURN
246 END IF
247 *
248 * Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
249 * convenient
250 *
251 IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN
252 IF( DEFAULTSIGNS ) THEN
253 SIGNST = 'O'
254 ELSE
255 SIGNST = 'D'
256 END IF
257 CALL DORCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M,
258 $ M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11,
259 $ LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T,
260 $ LDV1T, WORK, LWORK, IWORK, INFO )
261 RETURN
262 END IF
263 *
264 * Compute workspace
265 *
266 IF( INFO .EQ. 0 ) THEN
267 *
268 IPHI = 2
269 ITAUP1 = IPHI + MAX( 1, Q - 1 )
270 ITAUP2 = ITAUP1 + MAX( 1, P )
271 ITAUQ1 = ITAUP2 + MAX( 1, M - P )
272 ITAUQ2 = ITAUQ1 + MAX( 1, Q )
273 IORGQR = ITAUQ2 + MAX( 1, M - Q )
274 CALL DORGQR( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
275 $ CHILDINFO )
276 LORGQRWORKOPT = INT( WORK(1) )
277 LORGQRWORKMIN = MAX( 1, M - Q )
278 IORGLQ = ITAUQ2 + MAX( 1, M - Q )
279 CALL DORGLQ( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
280 $ CHILDINFO )
281 LORGLQWORKOPT = INT( WORK(1) )
282 LORGLQWORKMIN = MAX( 1, M - Q )
283 IORBDB = ITAUQ2 + MAX( 1, M - Q )
284 CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
285 $ X21, LDX21, X22, LDX22, 0, 0, 0, 0, 0, 0, WORK,
286 $ -1, CHILDINFO )
287 LORBDBWORKOPT = INT( WORK(1) )
288 LORBDBWORKMIN = LORBDBWORKOPT
289 IB11D = ITAUQ2 + MAX( 1, M - Q )
290 IB11E = IB11D + MAX( 1, Q )
291 IB12D = IB11E + MAX( 1, Q - 1 )
292 IB12E = IB12D + MAX( 1, Q )
293 IB21D = IB12E + MAX( 1, Q - 1 )
294 IB21E = IB21D + MAX( 1, Q )
295 IB22D = IB21E + MAX( 1, Q - 1 )
296 IB22E = IB22D + MAX( 1, Q )
297 IBBCSD = IB22E + MAX( 1, Q - 1 )
298 CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, 0,
299 $ 0, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, 0,
300 $ 0, 0, 0, 0, 0, 0, 0, WORK, -1, CHILDINFO )
301 LBBCSDWORKOPT = INT( WORK(1) )
302 LBBCSDWORKMIN = LBBCSDWORKOPT
303 LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT,
304 $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKOPT ) - 1
305 LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN,
306 $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKMIN ) - 1
307 WORK(1) = MAX(LWORKOPT,LWORKMIN)
308 *
309 IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
310 INFO = -22
311 ELSE
312 LORGQRWORK = LWORK - IORGQR + 1
313 LORGLQWORK = LWORK - IORGLQ + 1
314 LORBDBWORK = LWORK - IORBDB + 1
315 LBBCSDWORK = LWORK - IBBCSD + 1
316 END IF
317 END IF
318 *
319 * Abort if any illegal arguments
320 *
321 IF( INFO .NE. 0 ) THEN
322 CALL XERBLA( 'DORCSD', -INFO )
323 RETURN
324 ELSE IF( LQUERY ) THEN
325 RETURN
326 END IF
327 *
328 * Transform to bidiagonal block form
329 *
330 CALL DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21,
331 $ LDX21, X22, LDX22, THETA, WORK(IPHI), WORK(ITAUP1),
332 $ WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2),
333 $ WORK(IORBDB), LORBDBWORK, CHILDINFO )
334 *
335 * Accumulate Householder reflectors
336 *
337 IF( COLMAJOR ) THEN
338 IF( WANTU1 .AND. P .GT. 0 ) THEN
339 CALL DLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 )
340 CALL DORGQR( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGQR),
341 $ LORGQRWORK, INFO)
342 END IF
343 IF( WANTU2 .AND. M-P .GT. 0 ) THEN
344 CALL DLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 )
345 CALL DORGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
346 $ WORK(IORGQR), LORGQRWORK, INFO )
347 END IF
348 IF( WANTV1T .AND. Q .GT. 0 ) THEN
349 CALL DLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2),
350 $ LDV1T )
351 V1T(1, 1) = ONE
352 DO J = 2, Q
353 V1T(1,J) = ZERO
354 V1T(J,1) = ZERO
355 END DO
356 CALL DORGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
357 $ WORK(IORGLQ), LORGLQWORK, INFO )
358 END IF
359 IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
360 CALL DLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T )
361 CALL DLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22,
362 $ V2T(P+1,P+1), LDV2T )
363 CALL DORGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
364 $ WORK(IORGLQ), LORGLQWORK, INFO )
365 END IF
366 ELSE
367 IF( WANTU1 .AND. P .GT. 0 ) THEN
368 CALL DLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 )
369 CALL DORGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGLQ),
370 $ LORGLQWORK, INFO)
371 END IF
372 IF( WANTU2 .AND. M-P .GT. 0 ) THEN
373 CALL DLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 )
374 CALL DORGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
375 $ WORK(IORGLQ), LORGLQWORK, INFO )
376 END IF
377 IF( WANTV1T .AND. Q .GT. 0 ) THEN
378 CALL DLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2),
379 $ LDV1T )
380 V1T(1, 1) = ONE
381 DO J = 2, Q
382 V1T(1,J) = ZERO
383 V1T(J,1) = ZERO
384 END DO
385 CALL DORGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1),
386 $ WORK(IORGQR), LORGQRWORK, INFO )
387 END IF
388 IF( WANTV2T .AND. M-Q .GT. 0 ) THEN
389 CALL DLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T )
390 CALL DLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22,
391 $ V2T(P+1,P+1), LDV2T )
392 CALL DORGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
393 $ WORK(IORGQR), LORGQRWORK, INFO )
394 END IF
395 END IF
396 *
397 * Compute the CSD of the matrix in bidiagonal-block form
398 *
399 CALL DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA,
400 $ WORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
401 $ LDV2T, WORK(IB11D), WORK(IB11E), WORK(IB12D),
402 $ WORK(IB12E), WORK(IB21D), WORK(IB21E), WORK(IB22D),
403 $ WORK(IB22E), WORK(IBBCSD), LBBCSDWORK, INFO )
404 *
405 * Permute rows and columns to place identity submatrices in top-
406 * left corner of (1,1)-block and/or bottom-right corner of (1,2)-
407 * block and/or bottom-right corner of (2,1)-block and/or top-left
408 * corner of (2,2)-block
409 *
410 IF( Q .GT. 0 .AND. WANTU2 ) THEN
411 DO I = 1, Q
412 IWORK(I) = M - P - Q + I
413 END DO
414 DO I = Q + 1, M - P
415 IWORK(I) = I - Q
416 END DO
417 IF( COLMAJOR ) THEN
418 CALL DLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK )
419 ELSE
420 CALL DLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK )
421 END IF
422 END IF
423 IF( M .GT. 0 .AND. WANTV2T ) THEN
424 DO I = 1, P
425 IWORK(I) = M - P - Q + I
426 END DO
427 DO I = P + 1, M - Q
428 IWORK(I) = I - P
429 END DO
430 IF( .NOT. COLMAJOR ) THEN
431 CALL DLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
432 ELSE
433 CALL DLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
434 END IF
435 END IF
436 *
437 RETURN
438 *
439 * End DORCSD
440 *
441 END
442