1: *> \brief \b DGGSVP3
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGGSVP3 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvp3.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvp3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvp3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
22: * TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
23: * IWORK, TAU, WORK, LWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBQ, JOBU, JOBV
27: * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK
28: * DOUBLE PRECISION TOLA, TOLB
29: * ..
30: * .. Array Arguments ..
31: * INTEGER IWORK( * )
32: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
33: * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> DGGSVP3 computes orthogonal matrices U, V and Q such that
43: *>
44: *> N-K-L K L
45: *> U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
46: *> L ( 0 0 A23 )
47: *> M-K-L ( 0 0 0 )
48: *>
49: *> N-K-L K L
50: *> = K ( 0 A12 A13 ) if M-K-L < 0;
51: *> M-K ( 0 0 A23 )
52: *>
53: *> N-K-L K L
54: *> V**T*B*Q = L ( 0 0 B13 )
55: *> P-L ( 0 0 0 )
56: *>
57: *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
58: *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
59: *> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
60: *> numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
61: *>
62: *> This decomposition is the preprocessing step for computing the
63: *> Generalized Singular Value Decomposition (GSVD), see subroutine
64: *> DGGSVD3.
65: *> \endverbatim
66: *
67: * Arguments:
68: * ==========
69: *
70: *> \param[in] JOBU
71: *> \verbatim
72: *> JOBU is CHARACTER*1
73: *> = 'U': Orthogonal matrix U is computed;
74: *> = 'N': U is not computed.
75: *> \endverbatim
76: *>
77: *> \param[in] JOBV
78: *> \verbatim
79: *> JOBV is CHARACTER*1
80: *> = 'V': Orthogonal matrix V is computed;
81: *> = 'N': V is not computed.
82: *> \endverbatim
83: *>
84: *> \param[in] JOBQ
85: *> \verbatim
86: *> JOBQ is CHARACTER*1
87: *> = 'Q': Orthogonal matrix Q is computed;
88: *> = 'N': Q is not computed.
89: *> \endverbatim
90: *>
91: *> \param[in] M
92: *> \verbatim
93: *> M is INTEGER
94: *> The number of rows of the matrix A. M >= 0.
95: *> \endverbatim
96: *>
97: *> \param[in] P
98: *> \verbatim
99: *> P is INTEGER
100: *> The number of rows of the matrix B. P >= 0.
101: *> \endverbatim
102: *>
103: *> \param[in] N
104: *> \verbatim
105: *> N is INTEGER
106: *> The number of columns of the matrices A and B. N >= 0.
107: *> \endverbatim
108: *>
109: *> \param[in,out] A
110: *> \verbatim
111: *> A is DOUBLE PRECISION array, dimension (LDA,N)
112: *> On entry, the M-by-N matrix A.
113: *> On exit, A contains the triangular (or trapezoidal) matrix
114: *> described in the Purpose section.
115: *> \endverbatim
116: *>
117: *> \param[in] LDA
118: *> \verbatim
119: *> LDA is INTEGER
120: *> The leading dimension of the array A. LDA >= max(1,M).
121: *> \endverbatim
122: *>
123: *> \param[in,out] B
124: *> \verbatim
125: *> B is DOUBLE PRECISION array, dimension (LDB,N)
126: *> On entry, the P-by-N matrix B.
127: *> On exit, B contains the triangular matrix described in
128: *> the Purpose section.
129: *> \endverbatim
130: *>
131: *> \param[in] LDB
132: *> \verbatim
133: *> LDB is INTEGER
134: *> The leading dimension of the array B. LDB >= max(1,P).
135: *> \endverbatim
136: *>
137: *> \param[in] TOLA
138: *> \verbatim
139: *> TOLA is DOUBLE PRECISION
140: *> \endverbatim
141: *>
142: *> \param[in] TOLB
143: *> \verbatim
144: *> TOLB is DOUBLE PRECISION
145: *>
146: *> TOLA and TOLB are the thresholds to determine the effective
147: *> numerical rank of matrix B and a subblock of A. Generally,
148: *> they are set to
149: *> TOLA = MAX(M,N)*norm(A)*MACHEPS,
150: *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
151: *> The size of TOLA and TOLB may affect the size of backward
152: *> errors of the decomposition.
153: *> \endverbatim
154: *>
155: *> \param[out] K
156: *> \verbatim
157: *> K is INTEGER
158: *> \endverbatim
159: *>
160: *> \param[out] L
161: *> \verbatim
162: *> L is INTEGER
163: *>
164: *> On exit, K and L specify the dimension of the subblocks
165: *> described in Purpose section.
166: *> K + L = effective numerical rank of (A**T,B**T)**T.
167: *> \endverbatim
168: *>
169: *> \param[out] U
170: *> \verbatim
171: *> U is DOUBLE PRECISION array, dimension (LDU,M)
172: *> If JOBU = 'U', U contains the orthogonal matrix U.
173: *> If JOBU = 'N', U is not referenced.
174: *> \endverbatim
175: *>
176: *> \param[in] LDU
177: *> \verbatim
178: *> LDU is INTEGER
179: *> The leading dimension of the array U. LDU >= max(1,M) if
180: *> JOBU = 'U'; LDU >= 1 otherwise.
181: *> \endverbatim
182: *>
183: *> \param[out] V
184: *> \verbatim
185: *> V is DOUBLE PRECISION array, dimension (LDV,P)
186: *> If JOBV = 'V', V contains the orthogonal matrix V.
187: *> If JOBV = 'N', V is not referenced.
188: *> \endverbatim
189: *>
190: *> \param[in] LDV
191: *> \verbatim
192: *> LDV is INTEGER
193: *> The leading dimension of the array V. LDV >= max(1,P) if
194: *> JOBV = 'V'; LDV >= 1 otherwise.
195: *> \endverbatim
196: *>
197: *> \param[out] Q
198: *> \verbatim
199: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
200: *> If JOBQ = 'Q', Q contains the orthogonal matrix Q.
201: *> If JOBQ = 'N', Q is not referenced.
202: *> \endverbatim
203: *>
204: *> \param[in] LDQ
205: *> \verbatim
206: *> LDQ is INTEGER
207: *> The leading dimension of the array Q. LDQ >= max(1,N) if
208: *> JOBQ = 'Q'; LDQ >= 1 otherwise.
209: *> \endverbatim
210: *>
211: *> \param[out] IWORK
212: *> \verbatim
213: *> IWORK is INTEGER array, dimension (N)
214: *> \endverbatim
215: *>
216: *> \param[out] TAU
217: *> \verbatim
218: *> TAU is DOUBLE PRECISION array, dimension (N)
219: *> \endverbatim
220: *>
221: *> \param[out] WORK
222: *> \verbatim
223: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
224: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
225: *> \endverbatim
226: *>
227: *> \param[in] LWORK
228: *> \verbatim
229: *> LWORK is INTEGER
230: *> The dimension of the array WORK.
231: *>
232: *> If LWORK = -1, then a workspace query is assumed; the routine
233: *> only calculates the optimal size of the WORK array, returns
234: *> this value as the first entry of the WORK array, and no error
235: *> message related to LWORK is issued by XERBLA.
236: *> \endverbatim
237: *>
238: *> \param[out] INFO
239: *> \verbatim
240: *> INFO is INTEGER
241: *> = 0: successful exit
242: *> < 0: if INFO = -i, the i-th argument had an illegal value.
243: *> \endverbatim
244: *
245: * Authors:
246: * ========
247: *
248: *> \author Univ. of Tennessee
249: *> \author Univ. of California Berkeley
250: *> \author Univ. of Colorado Denver
251: *> \author NAG Ltd.
252: *
253: *> \date August 2015
254: *
255: *> \ingroup doubleOTHERcomputational
256: *
257: *> \par Further Details:
258: * =====================
259: *>
260: *> \verbatim
261: *>
262: *> The subroutine uses LAPACK subroutine DGEQP3 for the QR factorization
263: *> with column pivoting to detect the effective numerical rank of the
264: *> a matrix. It may be replaced by a better rank determination strategy.
265: *>
266: *> DGGSVP3 replaces the deprecated subroutine DGGSVP.
267: *>
268: *> \endverbatim
269: *>
270: * =====================================================================
271: SUBROUTINE DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
272: $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
273: $ IWORK, TAU, WORK, LWORK, INFO )
274: *
275: * -- LAPACK computational routine (version 3.7.0) --
276: * -- LAPACK is a software package provided by Univ. of Tennessee, --
277: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
278: * August 2015
279: *
280: IMPLICIT NONE
281: *
282: * .. Scalar Arguments ..
283: CHARACTER JOBQ, JOBU, JOBV
284: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
285: $ LWORK
286: DOUBLE PRECISION TOLA, TOLB
287: * ..
288: * .. Array Arguments ..
289: INTEGER IWORK( * )
290: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
291: $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
292: * ..
293: *
294: * =====================================================================
295: *
296: * .. Parameters ..
297: DOUBLE PRECISION ZERO, ONE
298: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
299: * ..
300: * .. Local Scalars ..
301: LOGICAL FORWRD, WANTQ, WANTU, WANTV, LQUERY
302: INTEGER I, J, LWKOPT
303: * ..
304: * .. External Functions ..
305: LOGICAL LSAME
306: EXTERNAL LSAME
307: * ..
308: * .. External Subroutines ..
309: EXTERNAL DGEQP3, DGEQR2, DGERQ2, DLACPY, DLAPMT,
310: $ DLASET, DORG2R, DORM2R, DORMR2, XERBLA
311: * ..
312: * .. Intrinsic Functions ..
313: INTRINSIC ABS, MAX, MIN
314: * ..
315: * .. Executable Statements ..
316: *
317: * Test the input parameters
318: *
319: WANTU = LSAME( JOBU, 'U' )
320: WANTV = LSAME( JOBV, 'V' )
321: WANTQ = LSAME( JOBQ, 'Q' )
322: FORWRD = .TRUE.
323: LQUERY = ( LWORK.EQ.-1 )
324: LWKOPT = 1
325: *
326: * Test the input arguments
327: *
328: INFO = 0
329: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
330: INFO = -1
331: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
332: INFO = -2
333: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
334: INFO = -3
335: ELSE IF( M.LT.0 ) THEN
336: INFO = -4
337: ELSE IF( P.LT.0 ) THEN
338: INFO = -5
339: ELSE IF( N.LT.0 ) THEN
340: INFO = -6
341: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
342: INFO = -8
343: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
344: INFO = -10
345: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
346: INFO = -16
347: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
348: INFO = -18
349: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
350: INFO = -20
351: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
352: INFO = -24
353: END IF
354: *
355: * Compute workspace
356: *
357: IF( INFO.EQ.0 ) THEN
358: CALL DGEQP3( P, N, B, LDB, IWORK, TAU, WORK, -1, INFO )
359: LWKOPT = INT( WORK ( 1 ) )
360: IF( WANTV ) THEN
361: LWKOPT = MAX( LWKOPT, P )
362: END IF
363: LWKOPT = MAX( LWKOPT, MIN( N, P ) )
364: LWKOPT = MAX( LWKOPT, M )
365: IF( WANTQ ) THEN
366: LWKOPT = MAX( LWKOPT, N )
367: END IF
368: CALL DGEQP3( M, N, A, LDA, IWORK, TAU, WORK, -1, INFO )
369: LWKOPT = MAX( LWKOPT, INT( WORK ( 1 ) ) )
370: LWKOPT = MAX( 1, LWKOPT )
371: WORK( 1 ) = DBLE( LWKOPT )
372: END IF
373: *
374: IF( INFO.NE.0 ) THEN
375: CALL XERBLA( 'DGGSVP3', -INFO )
376: RETURN
377: END IF
378: IF( LQUERY ) THEN
379: RETURN
380: ENDIF
381: *
382: * QR with column pivoting of B: B*P = V*( S11 S12 )
383: * ( 0 0 )
384: *
385: DO 10 I = 1, N
386: IWORK( I ) = 0
387: 10 CONTINUE
388: CALL DGEQP3( P, N, B, LDB, IWORK, TAU, WORK, LWORK, INFO )
389: *
390: * Update A := A*P
391: *
392: CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
393: *
394: * Determine the effective rank of matrix B.
395: *
396: L = 0
397: DO 20 I = 1, MIN( P, N )
398: IF( ABS( B( I, I ) ).GT.TOLB )
399: $ L = L + 1
400: 20 CONTINUE
401: *
402: IF( WANTV ) THEN
403: *
404: * Copy the details of V, and form V.
405: *
406: CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
407: IF( P.GT.1 )
408: $ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
409: $ LDV )
410: CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
411: END IF
412: *
413: * Clean up B
414: *
415: DO 40 J = 1, L - 1
416: DO 30 I = J + 1, L
417: B( I, J ) = ZERO
418: 30 CONTINUE
419: 40 CONTINUE
420: IF( P.GT.L )
421: $ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
422: *
423: IF( WANTQ ) THEN
424: *
425: * Set Q = I and Update Q := Q*P
426: *
427: CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
428: CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
429: END IF
430: *
431: IF( P.GE.L .AND. N.NE.L ) THEN
432: *
433: * RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
434: *
435: CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
436: *
437: * Update A := A*Z**T
438: *
439: CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
440: $ LDA, WORK, INFO )
441: *
442: IF( WANTQ ) THEN
443: *
444: * Update Q := Q*Z**T
445: *
446: CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
447: $ LDQ, WORK, INFO )
448: END IF
449: *
450: * Clean up B
451: *
452: CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
453: DO 60 J = N - L + 1, N
454: DO 50 I = J - N + L + 1, L
455: B( I, J ) = ZERO
456: 50 CONTINUE
457: 60 CONTINUE
458: *
459: END IF
460: *
461: * Let N-L L
462: * A = ( A11 A12 ) M,
463: *
464: * then the following does the complete QR decomposition of A11:
465: *
466: * A11 = U*( 0 T12 )*P1**T
467: * ( 0 0 )
468: *
469: DO 70 I = 1, N - L
470: IWORK( I ) = 0
471: 70 CONTINUE
472: CALL DGEQP3( M, N-L, A, LDA, IWORK, TAU, WORK, LWORK, INFO )
473: *
474: * Determine the effective rank of A11
475: *
476: K = 0
477: DO 80 I = 1, MIN( M, N-L )
478: IF( ABS( A( I, I ) ).GT.TOLA )
479: $ K = K + 1
480: 80 CONTINUE
481: *
482: * Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
483: *
484: CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
485: $ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
486: *
487: IF( WANTU ) THEN
488: *
489: * Copy the details of U, and form U
490: *
491: CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
492: IF( M.GT.1 )
493: $ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
494: $ LDU )
495: CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
496: END IF
497: *
498: IF( WANTQ ) THEN
499: *
500: * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
501: *
502: CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
503: END IF
504: *
505: * Clean up A: set the strictly lower triangular part of
506: * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
507: *
508: DO 100 J = 1, K - 1
509: DO 90 I = J + 1, K
510: A( I, J ) = ZERO
511: 90 CONTINUE
512: 100 CONTINUE
513: IF( M.GT.K )
514: $ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
515: *
516: IF( N-L.GT.K ) THEN
517: *
518: * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
519: *
520: CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
521: *
522: IF( WANTQ ) THEN
523: *
524: * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
525: *
526: CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
527: $ Q, LDQ, WORK, INFO )
528: END IF
529: *
530: * Clean up A
531: *
532: CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
533: DO 120 J = N - L - K + 1, N - L
534: DO 110 I = J - N + L + K + 1, K
535: A( I, J ) = ZERO
536: 110 CONTINUE
537: 120 CONTINUE
538: *
539: END IF
540: *
541: IF( M.GT.K ) THEN
542: *
543: * QR factorization of A( K+1:M,N-L+1:N )
544: *
545: CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
546: *
547: IF( WANTU ) THEN
548: *
549: * Update U(:,K+1:M) := U(:,K+1:M)*U1
550: *
551: CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
552: $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
553: $ WORK, INFO )
554: END IF
555: *
556: * Clean up
557: *
558: DO 140 J = N - L + 1, N
559: DO 130 I = J - N + K + L + 1, M
560: A( I, J ) = ZERO
561: 130 CONTINUE
562: 140 CONTINUE
563: *
564: END IF
565: *
566: WORK( 1 ) = DBLE( LWKOPT )
567: RETURN
568: *
569: * End of DGGSVP3
570: *
571: END
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