LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
sbdsdc.f
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1 *> \brief \b SBDSDC
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SBDSDC + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sbdsdc.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sbdsdc.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sbdsdc.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
22 * WORK, IWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER COMPQ, UPLO
26 * INTEGER INFO, LDU, LDVT, N
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IQ( * ), IWORK( * )
30 * REAL D( * ), E( * ), Q( * ), U( LDU, * ),
31 * $ VT( LDVT, * ), WORK( * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> SBDSDC computes the singular value decomposition (SVD) of a real
41 *> N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT,
42 *> using a divide and conquer method, where S is a diagonal matrix
43 *> with non-negative diagonal elements (the singular values of B), and
44 *> U and VT are orthogonal matrices of left and right singular vectors,
45 *> respectively. SBDSDC can be used to compute all singular values,
46 *> and optionally, singular vectors or singular vectors in compact form.
47 *>
48 *> This code makes very mild assumptions about floating point
49 *> arithmetic. It will work on machines with a guard digit in
50 *> add/subtract, or on those binary machines without guard digits
51 *> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
52 *> It could conceivably fail on hexadecimal or decimal machines
53 *> without guard digits, but we know of none. See SLASD3 for details.
54 *>
55 *> The code currently calls SLASDQ if singular values only are desired.
56 *> However, it can be slightly modified to compute singular values
57 *> using the divide and conquer method.
58 *> \endverbatim
59 *
60 * Arguments:
61 * ==========
62 *
63 *> \param[in] UPLO
64 *> \verbatim
65 *> UPLO is CHARACTER*1
66 *> = 'U': B is upper bidiagonal.
67 *> = 'L': B is lower bidiagonal.
68 *> \endverbatim
69 *>
70 *> \param[in] COMPQ
71 *> \verbatim
72 *> COMPQ is CHARACTER*1
73 *> Specifies whether singular vectors are to be computed
74 *> as follows:
75 *> = 'N': Compute singular values only;
76 *> = 'P': Compute singular values and compute singular
77 *> vectors in compact form;
78 *> = 'I': Compute singular values and singular vectors.
79 *> \endverbatim
80 *>
81 *> \param[in] N
82 *> \verbatim
83 *> N is INTEGER
84 *> The order of the matrix B. N >= 0.
85 *> \endverbatim
86 *>
87 *> \param[in,out] D
88 *> \verbatim
89 *> D is REAL array, dimension (N)
90 *> On entry, the n diagonal elements of the bidiagonal matrix B.
91 *> On exit, if INFO=0, the singular values of B.
92 *> \endverbatim
93 *>
94 *> \param[in,out] E
95 *> \verbatim
96 *> E is REAL array, dimension (N-1)
97 *> On entry, the elements of E contain the offdiagonal
98 *> elements of the bidiagonal matrix whose SVD is desired.
99 *> On exit, E has been destroyed.
100 *> \endverbatim
101 *>
102 *> \param[out] U
103 *> \verbatim
104 *> U is REAL array, dimension (LDU,N)
105 *> If COMPQ = 'I', then:
106 *> On exit, if INFO = 0, U contains the left singular vectors
107 *> of the bidiagonal matrix.
108 *> For other values of COMPQ, U is not referenced.
109 *> \endverbatim
110 *>
111 *> \param[in] LDU
112 *> \verbatim
113 *> LDU is INTEGER
114 *> The leading dimension of the array U. LDU >= 1.
115 *> If singular vectors are desired, then LDU >= max( 1, N ).
116 *> \endverbatim
117 *>
118 *> \param[out] VT
119 *> \verbatim
120 *> VT is REAL array, dimension (LDVT,N)
121 *> If COMPQ = 'I', then:
122 *> On exit, if INFO = 0, VT**T contains the right singular
123 *> vectors of the bidiagonal matrix.
124 *> For other values of COMPQ, VT is not referenced.
125 *> \endverbatim
126 *>
127 *> \param[in] LDVT
128 *> \verbatim
129 *> LDVT is INTEGER
130 *> The leading dimension of the array VT. LDVT >= 1.
131 *> If singular vectors are desired, then LDVT >= max( 1, N ).
132 *> \endverbatim
133 *>
134 *> \param[out] Q
135 *> \verbatim
136 *> Q is REAL array, dimension (LDQ)
137 *> If COMPQ = 'P', then:
138 *> On exit, if INFO = 0, Q and IQ contain the left
139 *> and right singular vectors in a compact form,
140 *> requiring O(N log N) space instead of 2*N**2.
141 *> In particular, Q contains all the REAL data in
142 *> LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
143 *> words of memory, where SMLSIZ is returned by ILAENV and
144 *> is equal to the maximum size of the subproblems at the
145 *> bottom of the computation tree (usually about 25).
146 *> For other values of COMPQ, Q is not referenced.
147 *> \endverbatim
148 *>
149 *> \param[out] IQ
150 *> \verbatim
151 *> IQ is INTEGER array, dimension (LDIQ)
152 *> If COMPQ = 'P', then:
153 *> On exit, if INFO = 0, Q and IQ contain the left
154 *> and right singular vectors in a compact form,
155 *> requiring O(N log N) space instead of 2*N**2.
156 *> In particular, IQ contains all INTEGER data in
157 *> LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
158 *> words of memory, where SMLSIZ is returned by ILAENV and
159 *> is equal to the maximum size of the subproblems at the
160 *> bottom of the computation tree (usually about 25).
161 *> For other values of COMPQ, IQ is not referenced.
162 *> \endverbatim
163 *>
164 *> \param[out] WORK
165 *> \verbatim
166 *> WORK is REAL array, dimension (MAX(1,LWORK))
167 *> If COMPQ = 'N' then LWORK >= (4 * N).
168 *> If COMPQ = 'P' then LWORK >= (6 * N).
169 *> If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
170 *> \endverbatim
171 *>
172 *> \param[out] IWORK
173 *> \verbatim
174 *> IWORK is INTEGER array, dimension (8*N)
175 *> \endverbatim
176 *>
177 *> \param[out] INFO
178 *> \verbatim
179 *> INFO is INTEGER
180 *> = 0: successful exit.
181 *> < 0: if INFO = -i, the i-th argument had an illegal value.
182 *> > 0: The algorithm failed to compute a singular value.
183 *> The update process of divide and conquer failed.
184 *> \endverbatim
185 *
186 * Authors:
187 * ========
188 *
189 *> \author Univ. of Tennessee
190 *> \author Univ. of California Berkeley
191 *> \author Univ. of Colorado Denver
192 *> \author NAG Ltd.
193 *
194 *> \date November 2011
195 *
196 *> \ingroup auxOTHERcomputational
197 *
198 *> \par Contributors:
199 * ==================
200 *>
201 *> Ming Gu and Huan Ren, Computer Science Division, University of
202 *> California at Berkeley, USA
203 *>
204 * =====================================================================
205  SUBROUTINE sbdsdc( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
206  $ work, iwork, info )
207 *
208 * -- LAPACK computational routine (version 3.4.0) --
209 * -- LAPACK is a software package provided by Univ. of Tennessee, --
210 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
211 * November 2011
212 *
213 * .. Scalar Arguments ..
214  CHARACTER COMPQ, UPLO
215  INTEGER INFO, LDU, LDVT, N
216 * ..
217 * .. Array Arguments ..
218  INTEGER IQ( * ), IWORK( * )
219  REAL D( * ), E( * ), Q( * ), U( ldu, * ),
220  $ vt( ldvt, * ), work( * )
221 * ..
222 *
223 * =====================================================================
224 * Changed dimension statement in comment describing E from (N) to
225 * (N-1). Sven, 17 Feb 05.
226 * =====================================================================
227 *
228 * .. Parameters ..
229  REAL ZERO, ONE, TWO
230  parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0 )
231 * ..
232 * .. Local Scalars ..
233  INTEGER DIFL, DIFR, GIVCOL, GIVNUM, GIVPTR, I, IC,
234  $ icompq, ierr, ii, is, iu, iuplo, ivt, j, k, kk,
235  $ mlvl, nm1, nsize, perm, poles, qstart, smlsiz,
236  $ smlszp, sqre, start, wstart, z
237  REAL CS, EPS, ORGNRM, P, R, SN
238 * ..
239 * .. External Functions ..
240  LOGICAL LSAME
241  INTEGER ILAENV
242  REAL SLAMCH, SLANST
243  EXTERNAL slamch, slanst, ilaenv, lsame
244 * ..
245 * .. External Subroutines ..
246  EXTERNAL scopy, slartg, slascl, slasd0, slasda, slasdq,
247  $ slaset, slasr, sswap, xerbla
248 * ..
249 * .. Intrinsic Functions ..
250  INTRINSIC REAL, ABS, INT, LOG, SIGN
251 * ..
252 * .. Executable Statements ..
253 *
254 * Test the input parameters.
255 *
256  info = 0
257 *
258  iuplo = 0
259  IF( lsame( uplo, 'U' ) )
260  $ iuplo = 1
261  IF( lsame( uplo, 'L' ) )
262  $ iuplo = 2
263  IF( lsame( compq, 'N' ) ) THEN
264  icompq = 0
265  ELSE IF( lsame( compq, 'P' ) ) THEN
266  icompq = 1
267  ELSE IF( lsame( compq, 'I' ) ) THEN
268  icompq = 2
269  ELSE
270  icompq = -1
271  END IF
272  IF( iuplo.EQ.0 ) THEN
273  info = -1
274  ELSE IF( icompq.LT.0 ) THEN
275  info = -2
276  ELSE IF( n.LT.0 ) THEN
277  info = -3
278  ELSE IF( ( ldu.LT.1 ) .OR. ( ( icompq.EQ.2 ) .AND. ( ldu.LT.
279  $ n ) ) ) THEN
280  info = -7
281  ELSE IF( ( ldvt.LT.1 ) .OR. ( ( icompq.EQ.2 ) .AND. ( ldvt.LT.
282  $ n ) ) ) THEN
283  info = -9
284  END IF
285  IF( info.NE.0 ) THEN
286  CALL xerbla( 'SBDSDC', -info )
287  RETURN
288  END IF
289 *
290 * Quick return if possible
291 *
292  IF( n.EQ.0 )
293  $ RETURN
294  smlsiz = ilaenv( 9, 'SBDSDC', ' ', 0, 0, 0, 0 )
295  IF( n.EQ.1 ) THEN
296  IF( icompq.EQ.1 ) THEN
297  q( 1 ) = sign( one, d( 1 ) )
298  q( 1+smlsiz*n ) = one
299  ELSE IF( icompq.EQ.2 ) THEN
300  u( 1, 1 ) = sign( one, d( 1 ) )
301  vt( 1, 1 ) = one
302  END IF
303  d( 1 ) = abs( d( 1 ) )
304  RETURN
305  END IF
306  nm1 = n - 1
307 *
308 * If matrix lower bidiagonal, rotate to be upper bidiagonal
309 * by applying Givens rotations on the left
310 *
311  wstart = 1
312  qstart = 3
313  IF( icompq.EQ.1 ) THEN
314  CALL scopy( n, d, 1, q( 1 ), 1 )
315  CALL scopy( n-1, e, 1, q( n+1 ), 1 )
316  END IF
317  IF( iuplo.EQ.2 ) THEN
318  qstart = 5
319  wstart = 2*n - 1
320  DO 10 i = 1, n - 1
321  CALL slartg( d( i ), e( i ), cs, sn, r )
322  d( i ) = r
323  e( i ) = sn*d( i+1 )
324  d( i+1 ) = cs*d( i+1 )
325  IF( icompq.EQ.1 ) THEN
326  q( i+2*n ) = cs
327  q( i+3*n ) = sn
328  ELSE IF( icompq.EQ.2 ) THEN
329  work( i ) = cs
330  work( nm1+i ) = -sn
331  END IF
332  10 CONTINUE
333  END IF
334 *
335 * If ICOMPQ = 0, use SLASDQ to compute the singular values.
336 *
337  IF( icompq.EQ.0 ) THEN
338  CALL slasdq( 'U', 0, n, 0, 0, 0, d, e, vt, ldvt, u, ldu, u,
339  $ ldu, work( wstart ), info )
340  GO TO 40
341  END IF
342 *
343 * If N is smaller than the minimum divide size SMLSIZ, then solve
344 * the problem with another solver.
345 *
346  IF( n.LE.smlsiz ) THEN
347  IF( icompq.EQ.2 ) THEN
348  CALL slaset( 'A', n, n, zero, one, u, ldu )
349  CALL slaset( 'A', n, n, zero, one, vt, ldvt )
350  CALL slasdq( 'U', 0, n, n, n, 0, d, e, vt, ldvt, u, ldu, u,
351  $ ldu, work( wstart ), info )
352  ELSE IF( icompq.EQ.1 ) THEN
353  iu = 1
354  ivt = iu + n
355  CALL slaset( 'A', n, n, zero, one, q( iu+( qstart-1 )*n ),
356  $ n )
357  CALL slaset( 'A', n, n, zero, one, q( ivt+( qstart-1 )*n ),
358  $ n )
359  CALL slasdq( 'U', 0, n, n, n, 0, d, e,
360  $ q( ivt+( qstart-1 )*n ), n,
361  $ q( iu+( qstart-1 )*n ), n,
362  $ q( iu+( qstart-1 )*n ), n, work( wstart ),
363  $ info )
364  END IF
365  GO TO 40
366  END IF
367 *
368  IF( icompq.EQ.2 ) THEN
369  CALL slaset( 'A', n, n, zero, one, u, ldu )
370  CALL slaset( 'A', n, n, zero, one, vt, ldvt )
371  END IF
372 *
373 * Scale.
374 *
375  orgnrm = slanst( 'M', n, d, e )
376  IF( orgnrm.EQ.zero )
377  $ RETURN
378  CALL slascl( 'G', 0, 0, orgnrm, one, n, 1, d, n, ierr )
379  CALL slascl( 'G', 0, 0, orgnrm, one, nm1, 1, e, nm1, ierr )
380 *
381  eps = slamch( 'Epsilon' )
382 *
383  mlvl = int( log( REAL( N ) / REAL( SMLSIZ+1 ) ) / log( TWO ) ) + 1
384  smlszp = smlsiz + 1
385 *
386  IF( icompq.EQ.1 ) THEN
387  iu = 1
388  ivt = 1 + smlsiz
389  difl = ivt + smlszp
390  difr = difl + mlvl
391  z = difr + mlvl*2
392  ic = z + mlvl
393  is = ic + 1
394  poles = is + 1
395  givnum = poles + 2*mlvl
396 *
397  k = 1
398  givptr = 2
399  perm = 3
400  givcol = perm + mlvl
401  END IF
402 *
403  DO 20 i = 1, n
404  IF( abs( d( i ) ).LT.eps ) THEN
405  d( i ) = sign( eps, d( i ) )
406  END IF
407  20 CONTINUE
408 *
409  start = 1
410  sqre = 0
411 *
412  DO 30 i = 1, nm1
413  IF( ( abs( e( i ) ).LT.eps ) .OR. ( i.EQ.nm1 ) ) THEN
414 *
415 * Subproblem found. First determine its size and then
416 * apply divide and conquer on it.
417 *
418  IF( i.LT.nm1 ) THEN
419 *
420 * A subproblem with E(I) small for I < NM1.
421 *
422  nsize = i - start + 1
423  ELSE IF( abs( e( i ) ).GE.eps ) THEN
424 *
425 * A subproblem with E(NM1) not too small but I = NM1.
426 *
427  nsize = n - start + 1
428  ELSE
429 *
430 * A subproblem with E(NM1) small. This implies an
431 * 1-by-1 subproblem at D(N). Solve this 1-by-1 problem
432 * first.
433 *
434  nsize = i - start + 1
435  IF( icompq.EQ.2 ) THEN
436  u( n, n ) = sign( one, d( n ) )
437  vt( n, n ) = one
438  ELSE IF( icompq.EQ.1 ) THEN
439  q( n+( qstart-1 )*n ) = sign( one, d( n ) )
440  q( n+( smlsiz+qstart-1 )*n ) = one
441  END IF
442  d( n ) = abs( d( n ) )
443  END IF
444  IF( icompq.EQ.2 ) THEN
445  CALL slasd0( nsize, sqre, d( start ), e( start ),
446  $ u( start, start ), ldu, vt( start, start ),
447  $ ldvt, smlsiz, iwork, work( wstart ), info )
448  ELSE
449  CALL slasda( icompq, smlsiz, nsize, sqre, d( start ),
450  $ e( start ), q( start+( iu+qstart-2 )*n ), n,
451  $ q( start+( ivt+qstart-2 )*n ),
452  $ iq( start+k*n ), q( start+( difl+qstart-2 )*
453  $ n ), q( start+( difr+qstart-2 )*n ),
454  $ q( start+( z+qstart-2 )*n ),
455  $ q( start+( poles+qstart-2 )*n ),
456  $ iq( start+givptr*n ), iq( start+givcol*n ),
457  $ n, iq( start+perm*n ),
458  $ q( start+( givnum+qstart-2 )*n ),
459  $ q( start+( ic+qstart-2 )*n ),
460  $ q( start+( is+qstart-2 )*n ),
461  $ work( wstart ), iwork, info )
462  END IF
463  IF( info.NE.0 ) THEN
464  RETURN
465  END IF
466  start = i + 1
467  END IF
468  30 CONTINUE
469 *
470 * Unscale
471 *
472  CALL slascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, ierr )
473  40 CONTINUE
474 *
475 * Use Selection Sort to minimize swaps of singular vectors
476 *
477  DO 60 ii = 2, n
478  i = ii - 1
479  kk = i
480  p = d( i )
481  DO 50 j = ii, n
482  IF( d( j ).GT.p ) THEN
483  kk = j
484  p = d( j )
485  END IF
486  50 CONTINUE
487  IF( kk.NE.i ) THEN
488  d( kk ) = d( i )
489  d( i ) = p
490  IF( icompq.EQ.1 ) THEN
491  iq( i ) = kk
492  ELSE IF( icompq.EQ.2 ) THEN
493  CALL sswap( n, u( 1, i ), 1, u( 1, kk ), 1 )
494  CALL sswap( n, vt( i, 1 ), ldvt, vt( kk, 1 ), ldvt )
495  END IF
496  ELSE IF( icompq.EQ.1 ) THEN
497  iq( i ) = i
498  END IF
499  60 CONTINUE
500 *
501 * If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO
502 *
503  IF( icompq.EQ.1 ) THEN
504  IF( iuplo.EQ.1 ) THEN
505  iq( n ) = 1
506  ELSE
507  iq( n ) = 0
508  END IF
509  END IF
510 *
511 * If B is lower bidiagonal, update U by those Givens rotations
512 * which rotated B to be upper bidiagonal
513 *
514  IF( ( iuplo.EQ.2 ) .AND. ( icompq.EQ.2 ) )
515  $ CALL slasr( 'L', 'V', 'B', n, n, work( 1 ), work( n ), u, ldu )
516 *
517  RETURN
518 *
519 * End of SBDSDC
520 *
521  END
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: slaset.f:112
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:53
subroutine sbdsdc(UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ, WORK, IWORK, INFO)
SBDSDC
Definition: sbdsdc.f:207
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slasdq(UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)
SLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e...
Definition: slasdq.f:213
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
subroutine slasd0(N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK, WORK, INFO)
SLASD0 computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and of...
Definition: slasd0.f:152
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:141
subroutine slasda(ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)
SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagona...
Definition: slasda.f:275
subroutine slasr(SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)
SLASR applies a sequence of plane rotations to a general rectangular matrix.
Definition: slasr.f:201
subroutine slartg(F, G, CS, SN, R)
SLARTG generates a plane rotation with real cosine and real sine.
Definition: slartg.f:99