LAPACK  3.4.2
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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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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