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 This routine is deprecated and has been replaced by routine DGGSVD3.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