LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
sggsvd.f
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1 *> \brief <b> SGGSVD 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
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sggsvd.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGGSVD( 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 * REAL 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 *> SGGSVD 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 REAL 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 REAL 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 REAL array, dimension (N)
209 *> \endverbatim
210 *>
211 *> \param[out] BETA
212 *> \verbatim
213 *> BETA is REAL 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 REAL 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 REAL 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 REAL 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 REAL array,
275 *> dimension (max(3*N,M,P)+N)
276 *> \endverbatim
277 *>
278 *> \param[out] IWORK
279 *> \verbatim
280 *> IWORK is INTEGER array, dimension (N)
281 *> On exit, IWORK stores the sorting information. More
282 *> precisely, the following loop will sort ALPHA
283 *> for I = K+1, min(M,K+L)
284 *> swap ALPHA(I) and ALPHA(IWORK(I))
285 *> endfor
286 *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
287 *> \endverbatim
288 *>
289 *> \param[out] INFO
290 *> \verbatim
291 *> INFO is INTEGER
292 *> = 0: successful exit
293 *> < 0: if INFO = -i, the i-th argument had an illegal value.
294 *> > 0: if INFO = 1, the Jacobi-type procedure failed to
295 *> converge. For further details, see subroutine STGSJA.
296 *> \endverbatim
297 *
298 *> \par Internal Parameters:
299 * =========================
300 *>
301 *> \verbatim
302 *> TOLA REAL
303 *> TOLB REAL
304 *> TOLA and TOLB are the thresholds to determine the effective
305 *> rank of (A**T,B**T)**T. Generally, they are set to
306 *> TOLA = MAX(M,N)*norm(A)*MACHEPS,
307 *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
308 *> The size of TOLA and TOLB may affect the size of backward
309 *> errors of the decomposition.
310 *> \endverbatim
311 *
312 * Authors:
313 * ========
314 *
315 *> \author Univ. of Tennessee
316 *> \author Univ. of California Berkeley
317 *> \author Univ. of Colorado Denver
318 *> \author NAG Ltd.
319 *
320 *> \date November 2011
321 *
322 *> \ingroup realOTHERsing
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 sggsvd( 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 (version 3.4.0) --
336 * -- LAPACK is a software package provided by Univ. of Tennessee, --
337 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
338 * November 2011
339 *
340 * .. Scalar Arguments ..
341  CHARACTER JOBQ, JOBU, JOBV
342  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
343 * ..
344 * .. Array Arguments ..
345  INTEGER IWORK( * )
346  REAL A( lda, * ), ALPHA( * ), B( ldb, * ),
347  $ beta( * ), q( ldq, * ), u( ldu, * ),
348  $ v( ldv, * ), work( * )
349 * ..
350 *
351 * =====================================================================
352 *
353 * .. Local Scalars ..
354  LOGICAL WANTQ, WANTU, WANTV
355  INTEGER I, IBND, ISUB, J, NCYCLE
356  REAL ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
357 * ..
358 * .. External Functions ..
359  LOGICAL LSAME
360  REAL SLAMCH, SLANGE
361  EXTERNAL lsame, slamch, slange
362 * ..
363 * .. External Subroutines ..
364  EXTERNAL scopy, sggsvp, stgsja, xerbla
365 * ..
366 * .. Intrinsic Functions ..
367  INTRINSIC max, min
368 * ..
369 * .. Executable Statements ..
370 *
371 * Test the input parameters
372 *
373  wantu = lsame( jobu, 'U' )
374  wantv = lsame( jobv, 'V' )
375  wantq = lsame( jobq, 'Q' )
376 *
377  info = 0
378  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
379  info = -1
380  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
381  info = -2
382  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
383  info = -3
384  ELSE IF( m.LT.0 ) THEN
385  info = -4
386  ELSE IF( n.LT.0 ) THEN
387  info = -5
388  ELSE IF( p.LT.0 ) THEN
389  info = -6
390  ELSE IF( lda.LT.max( 1, m ) ) THEN
391  info = -10
392  ELSE IF( ldb.LT.max( 1, p ) ) THEN
393  info = -12
394  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
395  info = -16
396  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
397  info = -18
398  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
399  info = -20
400  END IF
401  IF( info.NE.0 ) THEN
402  CALL xerbla( 'SGGSVD', -info )
403  RETURN
404  END IF
405 *
406 * Compute the Frobenius norm of matrices A and B
407 *
408  anorm = slange( '1', m, n, a, lda, work )
409  bnorm = slange( '1', p, n, b, ldb, work )
410 *
411 * Get machine precision and set up threshold for determining
412 * the effective numerical rank of the matrices A and B.
413 *
414  ulp = slamch( 'Precision' )
415  unfl = slamch( 'Safe Minimum' )
416  tola = max( m, n )*max( anorm, unfl )*ulp
417  tolb = max( p, n )*max( bnorm, unfl )*ulp
418 *
419 * Preprocessing
420 *
421  CALL sggsvp( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
422  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
423  $ work( n+1 ), info )
424 *
425 * Compute the GSVD of two upper "triangular" matrices
426 *
427  CALL stgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
428  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
429  $ work, ncycle, info )
430 *
431 * Sort the singular values and store the pivot indices in IWORK
432 * Copy ALPHA to WORK, then sort ALPHA in WORK
433 *
434  CALL scopy( n, alpha, 1, work, 1 )
435  ibnd = min( l, m-k )
436  DO 20 i = 1, ibnd
437 *
438 * Scan for largest ALPHA(K+I)
439 *
440  isub = i
441  smax = work( k+i )
442  DO 10 j = i + 1, ibnd
443  temp = work( k+j )
444  IF( temp.GT.smax ) THEN
445  isub = j
446  smax = temp
447  END IF
448  10 CONTINUE
449  IF( isub.NE.i ) THEN
450  work( k+isub ) = work( k+i )
451  work( k+i ) = smax
452  iwork( k+i ) = k + isub
453  ELSE
454  iwork( k+i ) = k + i
455  END IF
456  20 CONTINUE
457 *
458  RETURN
459 *
460 * End of SGGSVD
461 *
462  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine stgsja(JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
STGSJA
Definition: stgsja.f:380
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
subroutine sggsvp(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, INFO)
SGGSVP
Definition: sggsvp.f:256
subroutine sggsvd(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO)
SGGSVD computes the singular value decomposition (SVD) for OTHER matrices
Definition: sggsvd.f:334