1: *> \brief <b> DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGGSVD3 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvd3.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvd3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvd3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
22: * LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
23: * LWORK, IWORK, 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: * ..
29: * .. Array Arguments ..
30: * INTEGER IWORK( * )
31: * DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
32: * $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
33: * $ V( LDV, * ), WORK( * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> DGGSVD3 computes the generalized singular value decomposition (GSVD)
43: *> of an M-by-N real matrix A and P-by-N real matrix B:
44: *>
45: *> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
46: *>
47: *> where U, V and Q are orthogonal matrices.
48: *> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
49: *> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
50: *> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
51: *> following structures, respectively:
52: *>
53: *> If M-K-L >= 0,
54: *>
55: *> K L
56: *> D1 = K ( I 0 )
57: *> L ( 0 C )
58: *> M-K-L ( 0 0 )
59: *>
60: *> K L
61: *> D2 = L ( 0 S )
62: *> P-L ( 0 0 )
63: *>
64: *> N-K-L K L
65: *> ( 0 R ) = K ( 0 R11 R12 )
66: *> L ( 0 0 R22 )
67: *>
68: *> where
69: *>
70: *> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
71: *> S = diag( BETA(K+1), ... , BETA(K+L) ),
72: *> C**2 + S**2 = I.
73: *>
74: *> R is stored in A(1:K+L,N-K-L+1:N) on exit.
75: *>
76: *> If M-K-L < 0,
77: *>
78: *> K M-K K+L-M
79: *> D1 = K ( I 0 0 )
80: *> M-K ( 0 C 0 )
81: *>
82: *> K M-K K+L-M
83: *> D2 = M-K ( 0 S 0 )
84: *> K+L-M ( 0 0 I )
85: *> P-L ( 0 0 0 )
86: *>
87: *> N-K-L K M-K K+L-M
88: *> ( 0 R ) = K ( 0 R11 R12 R13 )
89: *> M-K ( 0 0 R22 R23 )
90: *> K+L-M ( 0 0 0 R33 )
91: *>
92: *> where
93: *>
94: *> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
95: *> S = diag( BETA(K+1), ... , BETA(M) ),
96: *> C**2 + S**2 = I.
97: *>
98: *> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
99: *> ( 0 R22 R23 )
100: *> in B(M-K+1:L,N+M-K-L+1:N) on exit.
101: *>
102: *> The routine computes C, S, R, and optionally the orthogonal
103: *> transformation matrices U, V and Q.
104: *>
105: *> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
106: *> A and B implicitly gives the SVD of A*inv(B):
107: *> A*inv(B) = U*(D1*inv(D2))*V**T.
108: *> If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is
109: *> also equal to the CS decomposition of A and B. Furthermore, the GSVD
110: *> can be used to derive the solution of the eigenvalue problem:
111: *> A**T*A x = lambda* B**T*B x.
112: *> In some literature, the GSVD of A and B is presented in the form
113: *> U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
114: *> where U and V are orthogonal and X is nonsingular, D1 and D2 are
115: *> ``diagonal''. The former GSVD form can be converted to the latter
116: *> form by taking the nonsingular matrix X as
117: *>
118: *> X = Q*( I 0 )
119: *> ( 0 inv(R) ).
120: *> \endverbatim
121: *
122: * Arguments:
123: * ==========
124: *
125: *> \param[in] JOBU
126: *> \verbatim
127: *> JOBU is CHARACTER*1
128: *> = 'U': Orthogonal matrix U is computed;
129: *> = 'N': U is not computed.
130: *> \endverbatim
131: *>
132: *> \param[in] JOBV
133: *> \verbatim
134: *> JOBV is CHARACTER*1
135: *> = 'V': Orthogonal matrix V is computed;
136: *> = 'N': V is not computed.
137: *> \endverbatim
138: *>
139: *> \param[in] JOBQ
140: *> \verbatim
141: *> JOBQ is CHARACTER*1
142: *> = 'Q': Orthogonal matrix Q is computed;
143: *> = 'N': Q is not computed.
144: *> \endverbatim
145: *>
146: *> \param[in] M
147: *> \verbatim
148: *> M is INTEGER
149: *> The number of rows of the matrix A. M >= 0.
150: *> \endverbatim
151: *>
152: *> \param[in] N
153: *> \verbatim
154: *> N is INTEGER
155: *> The number of columns of the matrices A and B. N >= 0.
156: *> \endverbatim
157: *>
158: *> \param[in] P
159: *> \verbatim
160: *> P is INTEGER
161: *> The number of rows of the matrix B. P >= 0.
162: *> \endverbatim
163: *>
164: *> \param[out] K
165: *> \verbatim
166: *> K is INTEGER
167: *> \endverbatim
168: *>
169: *> \param[out] L
170: *> \verbatim
171: *> L is INTEGER
172: *>
173: *> On exit, K and L specify the dimension of the subblocks
174: *> described in Purpose.
175: *> K + L = effective numerical rank of (A**T,B**T)**T.
176: *> \endverbatim
177: *>
178: *> \param[in,out] A
179: *> \verbatim
180: *> A is DOUBLE PRECISION array, dimension (LDA,N)
181: *> On entry, the M-by-N matrix A.
182: *> On exit, A contains the triangular matrix R, or part of R.
183: *> See Purpose for details.
184: *> \endverbatim
185: *>
186: *> \param[in] LDA
187: *> \verbatim
188: *> LDA is INTEGER
189: *> The leading dimension of the array A. LDA >= max(1,M).
190: *> \endverbatim
191: *>
192: *> \param[in,out] B
193: *> \verbatim
194: *> B is DOUBLE PRECISION array, dimension (LDB,N)
195: *> On entry, the P-by-N matrix B.
196: *> On exit, B contains the triangular matrix R if M-K-L < 0.
197: *> See Purpose for details.
198: *> \endverbatim
199: *>
200: *> \param[in] LDB
201: *> \verbatim
202: *> LDB is INTEGER
203: *> The leading dimension of the array B. LDB >= max(1,P).
204: *> \endverbatim
205: *>
206: *> \param[out] ALPHA
207: *> \verbatim
208: *> ALPHA is DOUBLE PRECISION array, dimension (N)
209: *> \endverbatim
210: *>
211: *> \param[out] BETA
212: *> \verbatim
213: *> BETA is DOUBLE PRECISION array, dimension (N)
214: *>
215: *> On exit, ALPHA and BETA contain the generalized singular
216: *> value pairs of A and B;
217: *> ALPHA(1:K) = 1,
218: *> BETA(1:K) = 0,
219: *> and if M-K-L >= 0,
220: *> ALPHA(K+1:K+L) = C,
221: *> BETA(K+1:K+L) = S,
222: *> or if M-K-L < 0,
223: *> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
224: *> BETA(K+1:M) =S, BETA(M+1:K+L) =1
225: *> and
226: *> ALPHA(K+L+1:N) = 0
227: *> BETA(K+L+1:N) = 0
228: *> \endverbatim
229: *>
230: *> \param[out] U
231: *> \verbatim
232: *> U is DOUBLE PRECISION array, dimension (LDU,M)
233: *> If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
234: *> If JOBU = 'N', U is not referenced.
235: *> \endverbatim
236: *>
237: *> \param[in] LDU
238: *> \verbatim
239: *> LDU is INTEGER
240: *> The leading dimension of the array U. LDU >= max(1,M) if
241: *> JOBU = 'U'; LDU >= 1 otherwise.
242: *> \endverbatim
243: *>
244: *> \param[out] V
245: *> \verbatim
246: *> V is DOUBLE PRECISION array, dimension (LDV,P)
247: *> If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
248: *> If JOBV = 'N', V is not referenced.
249: *> \endverbatim
250: *>
251: *> \param[in] LDV
252: *> \verbatim
253: *> LDV is INTEGER
254: *> The leading dimension of the array V. LDV >= max(1,P) if
255: *> JOBV = 'V'; LDV >= 1 otherwise.
256: *> \endverbatim
257: *>
258: *> \param[out] Q
259: *> \verbatim
260: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
261: *> If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
262: *> If JOBQ = 'N', Q is not referenced.
263: *> \endverbatim
264: *>
265: *> \param[in] LDQ
266: *> \verbatim
267: *> LDQ is INTEGER
268: *> The leading dimension of the array Q. LDQ >= max(1,N) if
269: *> JOBQ = 'Q'; LDQ >= 1 otherwise.
270: *> \endverbatim
271: *>
272: *> \param[out] WORK
273: *> \verbatim
274: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
275: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
276: *> \endverbatim
277: *>
278: *> \param[in] LWORK
279: *> \verbatim
280: *> LWORK is INTEGER
281: *> The dimension of the array WORK.
282: *>
283: *> If LWORK = -1, then a workspace query is assumed; the routine
284: *> only calculates the optimal size of the WORK array, returns
285: *> this value as the first entry of the WORK array, and no error
286: *> message related to LWORK is issued by XERBLA.
287: *> \endverbatim
288: *>
289: *> \param[out] IWORK
290: *> \verbatim
291: *> IWORK is INTEGER array, dimension (N)
292: *> On exit, IWORK stores the sorting information. More
293: *> precisely, the following loop will sort ALPHA
294: *> for I = K+1, min(M,K+L)
295: *> swap ALPHA(I) and ALPHA(IWORK(I))
296: *> endfor
297: *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
298: *> \endverbatim
299: *>
300: *> \param[out] INFO
301: *> \verbatim
302: *> INFO is INTEGER
303: *> = 0: successful exit.
304: *> < 0: if INFO = -i, the i-th argument had an illegal value.
305: *> > 0: if INFO = 1, the Jacobi-type procedure failed to
306: *> converge. For further details, see subroutine DTGSJA.
307: *> \endverbatim
308: *
309: *> \par Internal Parameters:
310: * =========================
311: *>
312: *> \verbatim
313: *> TOLA DOUBLE PRECISION
314: *> TOLB DOUBLE PRECISION
315: *> TOLA and TOLB are the thresholds to determine the effective
316: *> rank of (A**T,B**T)**T. Generally, they are set to
317: *> TOLA = MAX(M,N)*norm(A)*MACHEPS,
318: *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
319: *> The size of TOLA and TOLB may affect the size of backward
320: *> errors of the decomposition.
321: *> \endverbatim
322: *
323: * Authors:
324: * ========
325: *
326: *> \author Univ. of Tennessee
327: *> \author Univ. of California Berkeley
328: *> \author Univ. of Colorado Denver
329: *> \author NAG Ltd.
330: *
331: *> \ingroup doubleGEsing
332: *
333: *> \par Contributors:
334: * ==================
335: *>
336: *> Ming Gu and Huan Ren, Computer Science Division, University of
337: *> California at Berkeley, USA
338: *>
339: *
340: *> \par Further Details:
341: * =====================
342: *>
343: *> DGGSVD3 replaces the deprecated subroutine DGGSVD.
344: *>
345: * =====================================================================
346: SUBROUTINE DGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
347: $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
348: $ WORK, LWORK, IWORK, INFO )
349: *
350: * -- LAPACK driver routine --
351: * -- LAPACK is a software package provided by Univ. of Tennessee, --
352: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
353: *
354: * .. Scalar Arguments ..
355: CHARACTER JOBQ, JOBU, JOBV
356: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
357: $ LWORK
358: * ..
359: * .. Array Arguments ..
360: INTEGER IWORK( * )
361: DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
362: $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
363: $ V( LDV, * ), WORK( * )
364: * ..
365: *
366: * =====================================================================
367: *
368: * .. Local Scalars ..
369: LOGICAL WANTQ, WANTU, WANTV, LQUERY
370: INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
371: DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
372: * ..
373: * .. External Functions ..
374: LOGICAL LSAME
375: DOUBLE PRECISION DLAMCH, DLANGE
376: EXTERNAL LSAME, DLAMCH, DLANGE
377: * ..
378: * .. External Subroutines ..
379: EXTERNAL DCOPY, DGGSVP3, DTGSJA, XERBLA
380: * ..
381: * .. Intrinsic Functions ..
382: INTRINSIC MAX, MIN
383: * ..
384: * .. Executable Statements ..
385: *
386: * Decode and test the input parameters
387: *
388: WANTU = LSAME( JOBU, 'U' )
389: WANTV = LSAME( JOBV, 'V' )
390: WANTQ = LSAME( JOBQ, 'Q' )
391: LQUERY = ( LWORK.EQ.-1 )
392: LWKOPT = 1
393: *
394: * Test the input arguments
395: *
396: INFO = 0
397: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
398: INFO = -1
399: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
400: INFO = -2
401: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
402: INFO = -3
403: ELSE IF( M.LT.0 ) THEN
404: INFO = -4
405: ELSE IF( N.LT.0 ) THEN
406: INFO = -5
407: ELSE IF( P.LT.0 ) THEN
408: INFO = -6
409: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
410: INFO = -10
411: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
412: INFO = -12
413: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
414: INFO = -16
415: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
416: INFO = -18
417: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
418: INFO = -20
419: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
420: INFO = -24
421: END IF
422: *
423: * Compute workspace
424: *
425: IF( INFO.EQ.0 ) THEN
426: CALL DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
427: $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
428: $ WORK, -1, INFO )
429: LWKOPT = N + INT( WORK( 1 ) )
430: LWKOPT = MAX( 2*N, LWKOPT )
431: LWKOPT = MAX( 1, LWKOPT )
432: WORK( 1 ) = DBLE( LWKOPT )
433: END IF
434: *
435: IF( INFO.NE.0 ) THEN
436: CALL XERBLA( 'DGGSVD3', -INFO )
437: RETURN
438: END IF
439: IF( LQUERY ) THEN
440: RETURN
441: ENDIF
442: *
443: * Compute the Frobenius norm of matrices A and B
444: *
445: ANORM = DLANGE( '1', M, N, A, LDA, WORK )
446: BNORM = DLANGE( '1', P, N, B, LDB, WORK )
447: *
448: * Get machine precision and set up threshold for determining
449: * the effective numerical rank of the matrices A and B.
450: *
451: ULP = DLAMCH( 'Precision' )
452: UNFL = DLAMCH( 'Safe Minimum' )
453: TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
454: TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
455: *
456: * Preprocessing
457: *
458: CALL DGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
459: $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
460: $ WORK( N+1 ), LWORK-N, INFO )
461: *
462: * Compute the GSVD of two upper "triangular" matrices
463: *
464: CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
465: $ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
466: $ WORK, NCYCLE, INFO )
467: *
468: * Sort the singular values and store the pivot indices in IWORK
469: * Copy ALPHA to WORK, then sort ALPHA in WORK
470: *
471: CALL DCOPY( N, ALPHA, 1, WORK, 1 )
472: IBND = MIN( L, M-K )
473: DO 20 I = 1, IBND
474: *
475: * Scan for largest ALPHA(K+I)
476: *
477: ISUB = I
478: SMAX = WORK( K+I )
479: DO 10 J = I + 1, IBND
480: TEMP = WORK( K+J )
481: IF( TEMP.GT.SMAX ) THEN
482: ISUB = J
483: SMAX = TEMP
484: END IF
485: 10 CONTINUE
486: IF( ISUB.NE.I ) THEN
487: WORK( K+ISUB ) = WORK( K+I )
488: WORK( K+I ) = SMAX
489: IWORK( K+I ) = K + ISUB
490: ELSE
491: IWORK( K+I ) = K + I
492: END IF
493: 20 CONTINUE
494: *
495: WORK( 1 ) = DBLE( LWKOPT )
496: RETURN
497: *
498: * End of DGGSVD3
499: *
500: END
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