LAPACK  3.9.0
LAPACK: Linear Algebra PACKage
dlals0.f
Go to the documentation of this file.
1 *> \brief \b DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DLALS0 + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlals0.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlals0.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlals0.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
22 * PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
23 * POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
27 * $ LDGNUM, NL, NR, NRHS, SQRE
28 * DOUBLE PRECISION C, S
29 * ..
30 * .. Array Arguments ..
31 * INTEGER GIVCOL( LDGCOL, * ), PERM( * )
32 * DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
33 * $ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
34 * $ POLES( LDGNUM, * ), WORK( * ), Z( * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> DLALS0 applies back the multiplying factors of either the left or the
44 *> right singular vector matrix of a diagonal matrix appended by a row
45 *> to the right hand side matrix B in solving the least squares problem
46 *> using the divide-and-conquer SVD approach.
47 *>
48 *> For the left singular vector matrix, three types of orthogonal
49 *> matrices are involved:
50 *>
51 *> (1L) Givens rotations: the number of such rotations is GIVPTR; the
52 *> pairs of columns/rows they were applied to are stored in GIVCOL;
53 *> and the C- and S-values of these rotations are stored in GIVNUM.
54 *>
55 *> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
56 *> row, and for J=2:N, PERM(J)-th row of B is to be moved to the
57 *> J-th row.
58 *>
59 *> (3L) The left singular vector matrix of the remaining matrix.
60 *>
61 *> For the right singular vector matrix, four types of orthogonal
62 *> matrices are involved:
63 *>
64 *> (1R) The right singular vector matrix of the remaining matrix.
65 *>
66 *> (2R) If SQRE = 1, one extra Givens rotation to generate the right
67 *> null space.
68 *>
69 *> (3R) The inverse transformation of (2L).
70 *>
71 *> (4R) The inverse transformation of (1L).
72 *> \endverbatim
73 *
74 * Arguments:
75 * ==========
76 *
77 *> \param[in] ICOMPQ
78 *> \verbatim
79 *> ICOMPQ is INTEGER
80 *> Specifies whether singular vectors are to be computed in
81 *> factored form:
82 *> = 0: Left singular vector matrix.
83 *> = 1: Right singular vector matrix.
84 *> \endverbatim
85 *>
86 *> \param[in] NL
87 *> \verbatim
88 *> NL is INTEGER
89 *> The row dimension of the upper block. NL >= 1.
90 *> \endverbatim
91 *>
92 *> \param[in] NR
93 *> \verbatim
94 *> NR is INTEGER
95 *> The row dimension of the lower block. NR >= 1.
96 *> \endverbatim
97 *>
98 *> \param[in] SQRE
99 *> \verbatim
100 *> SQRE is INTEGER
101 *> = 0: the lower block is an NR-by-NR square matrix.
102 *> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
103 *>
104 *> The bidiagonal matrix has row dimension N = NL + NR + 1,
105 *> and column dimension M = N + SQRE.
106 *> \endverbatim
107 *>
108 *> \param[in] NRHS
109 *> \verbatim
110 *> NRHS is INTEGER
111 *> The number of columns of B and BX. NRHS must be at least 1.
112 *> \endverbatim
113 *>
114 *> \param[in,out] B
115 *> \verbatim
116 *> B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
117 *> On input, B contains the right hand sides of the least
118 *> squares problem in rows 1 through M. On output, B contains
119 *> the solution X in rows 1 through N.
120 *> \endverbatim
121 *>
122 *> \param[in] LDB
123 *> \verbatim
124 *> LDB is INTEGER
125 *> The leading dimension of B. LDB must be at least
126 *> max(1,MAX( M, N ) ).
127 *> \endverbatim
128 *>
129 *> \param[out] BX
130 *> \verbatim
131 *> BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
132 *> \endverbatim
133 *>
134 *> \param[in] LDBX
135 *> \verbatim
136 *> LDBX is INTEGER
137 *> The leading dimension of BX.
138 *> \endverbatim
139 *>
140 *> \param[in] PERM
141 *> \verbatim
142 *> PERM is INTEGER array, dimension ( N )
143 *> The permutations (from deflation and sorting) applied
144 *> to the two blocks.
145 *> \endverbatim
146 *>
147 *> \param[in] GIVPTR
148 *> \verbatim
149 *> GIVPTR is INTEGER
150 *> The number of Givens rotations which took place in this
151 *> subproblem.
152 *> \endverbatim
153 *>
154 *> \param[in] GIVCOL
155 *> \verbatim
156 *> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
157 *> Each pair of numbers indicates a pair of rows/columns
158 *> involved in a Givens rotation.
159 *> \endverbatim
160 *>
161 *> \param[in] LDGCOL
162 *> \verbatim
163 *> LDGCOL is INTEGER
164 *> The leading dimension of GIVCOL, must be at least N.
165 *> \endverbatim
166 *>
167 *> \param[in] GIVNUM
168 *> \verbatim
169 *> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
170 *> Each number indicates the C or S value used in the
171 *> corresponding Givens rotation.
172 *> \endverbatim
173 *>
174 *> \param[in] LDGNUM
175 *> \verbatim
176 *> LDGNUM is INTEGER
177 *> The leading dimension of arrays DIFR, POLES and
178 *> GIVNUM, must be at least K.
179 *> \endverbatim
180 *>
181 *> \param[in] POLES
182 *> \verbatim
183 *> POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
184 *> On entry, POLES(1:K, 1) contains the new singular
185 *> values obtained from solving the secular equation, and
186 *> POLES(1:K, 2) is an array containing the poles in the secular
187 *> equation.
188 *> \endverbatim
189 *>
190 *> \param[in] DIFL
191 *> \verbatim
192 *> DIFL is DOUBLE PRECISION array, dimension ( K ).
193 *> On entry, DIFL(I) is the distance between I-th updated
194 *> (undeflated) singular value and the I-th (undeflated) old
195 *> singular value.
196 *> \endverbatim
197 *>
198 *> \param[in] DIFR
199 *> \verbatim
200 *> DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
201 *> On entry, DIFR(I, 1) contains the distances between I-th
202 *> updated (undeflated) singular value and the I+1-th
203 *> (undeflated) old singular value. And DIFR(I, 2) is the
204 *> normalizing factor for the I-th right singular vector.
205 *> \endverbatim
206 *>
207 *> \param[in] Z
208 *> \verbatim
209 *> Z is DOUBLE PRECISION array, dimension ( K )
210 *> Contain the components of the deflation-adjusted updating row
211 *> vector.
212 *> \endverbatim
213 *>
214 *> \param[in] K
215 *> \verbatim
216 *> K is INTEGER
217 *> Contains the dimension of the non-deflated matrix,
218 *> This is the order of the related secular equation. 1 <= K <=N.
219 *> \endverbatim
220 *>
221 *> \param[in] C
222 *> \verbatim
223 *> C is DOUBLE PRECISION
224 *> C contains garbage if SQRE =0 and the C-value of a Givens
225 *> rotation related to the right null space if SQRE = 1.
226 *> \endverbatim
227 *>
228 *> \param[in] S
229 *> \verbatim
230 *> S is DOUBLE PRECISION
231 *> S contains garbage if SQRE =0 and the S-value of a Givens
232 *> rotation related to the right null space if SQRE = 1.
233 *> \endverbatim
234 *>
235 *> \param[out] WORK
236 *> \verbatim
237 *> WORK is DOUBLE PRECISION array, dimension ( K )
238 *> \endverbatim
239 *>
240 *> \param[out] INFO
241 *> \verbatim
242 *> INFO is INTEGER
243 *> = 0: successful exit.
244 *> < 0: if INFO = -i, the i-th argument had an illegal value.
245 *> \endverbatim
246 *
247 * Authors:
248 * ========
249 *
250 *> \author Univ. of Tennessee
251 *> \author Univ. of California Berkeley
252 *> \author Univ. of Colorado Denver
253 *> \author NAG Ltd.
254 *
255 *> \date December 2016
256 *
257 *> \ingroup doubleOTHERcomputational
258 *
259 *> \par Contributors:
260 * ==================
261 *>
262 *> Ming Gu and Ren-Cang Li, Computer Science Division, University of
263 *> California at Berkeley, USA \n
264 *> Osni Marques, LBNL/NERSC, USA \n
265 *
266 * =====================================================================
267  SUBROUTINE dlals0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
268  $ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
269  $ POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
270 *
271 * -- LAPACK computational routine (version 3.7.0) --
272 * -- LAPACK is a software package provided by Univ. of Tennessee, --
273 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
274 * December 2016
275 *
276 * .. Scalar Arguments ..
277  INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
278  $ LDGNUM, NL, NR, NRHS, SQRE
279  DOUBLE PRECISION C, S
280 * ..
281 * .. Array Arguments ..
282  INTEGER GIVCOL( LDGCOL, * ), PERM( * )
283  DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
284  $ difr( ldgnum, * ), givnum( ldgnum, * ),
285  $ poles( ldgnum, * ), work( * ), z( * )
286 * ..
287 *
288 * =====================================================================
289 *
290 * .. Parameters ..
291  DOUBLE PRECISION ONE, ZERO, NEGONE
292  PARAMETER ( ONE = 1.0d0, zero = 0.0d0, negone = -1.0d0 )
293 * ..
294 * .. Local Scalars ..
295  INTEGER I, J, M, N, NLP1
296  DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
297 * ..
298 * .. External Subroutines ..
299  EXTERNAL dcopy, dgemv, dlacpy, dlascl, drot, dscal,
300  $ xerbla
301 * ..
302 * .. External Functions ..
303  DOUBLE PRECISION DLAMC3, DNRM2
304  EXTERNAL DLAMC3, DNRM2
305 * ..
306 * .. Intrinsic Functions ..
307  INTRINSIC max
308 * ..
309 * .. Executable Statements ..
310 *
311 * Test the input parameters.
312 *
313  info = 0
314  n = nl + nr + 1
315 *
316  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
317  info = -1
318  ELSE IF( nl.LT.1 ) THEN
319  info = -2
320  ELSE IF( nr.LT.1 ) THEN
321  info = -3
322  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
323  info = -4
324  ELSE IF( nrhs.LT.1 ) THEN
325  info = -5
326  ELSE IF( ldb.LT.n ) THEN
327  info = -7
328  ELSE IF( ldbx.LT.n ) THEN
329  info = -9
330  ELSE IF( givptr.LT.0 ) THEN
331  info = -11
332  ELSE IF( ldgcol.LT.n ) THEN
333  info = -13
334  ELSE IF( ldgnum.LT.n ) THEN
335  info = -15
336  ELSE IF( k.LT.1 ) THEN
337  info = -20
338  END IF
339  IF( info.NE.0 ) THEN
340  CALL xerbla( 'DLALS0', -info )
341  RETURN
342  END IF
343 *
344  m = n + sqre
345  nlp1 = nl + 1
346 *
347  IF( icompq.EQ.0 ) THEN
348 *
349 * Apply back orthogonal transformations from the left.
350 *
351 * Step (1L): apply back the Givens rotations performed.
352 *
353  DO 10 i = 1, givptr
354  CALL drot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
355  $ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
356  $ givnum( i, 1 ) )
357  10 CONTINUE
358 *
359 * Step (2L): permute rows of B.
360 *
361  CALL dcopy( nrhs, b( nlp1, 1 ), ldb, bx( 1, 1 ), ldbx )
362  DO 20 i = 2, n
363  CALL dcopy( nrhs, b( perm( i ), 1 ), ldb, bx( i, 1 ), ldbx )
364  20 CONTINUE
365 *
366 * Step (3L): apply the inverse of the left singular vector
367 * matrix to BX.
368 *
369  IF( k.EQ.1 ) THEN
370  CALL dcopy( nrhs, bx, ldbx, b, ldb )
371  IF( z( 1 ).LT.zero ) THEN
372  CALL dscal( nrhs, negone, b, ldb )
373  END IF
374  ELSE
375  DO 50 j = 1, k
376  diflj = difl( j )
377  dj = poles( j, 1 )
378  dsigj = -poles( j, 2 )
379  IF( j.LT.k ) THEN
380  difrj = -difr( j, 1 )
381  dsigjp = -poles( j+1, 2 )
382  END IF
383  IF( ( z( j ).EQ.zero ) .OR. ( poles( j, 2 ).EQ.zero ) )
384  $ THEN
385  work( j ) = zero
386  ELSE
387  work( j ) = -poles( j, 2 )*z( j ) / diflj /
388  $ ( poles( j, 2 )+dj )
389  END IF
390  DO 30 i = 1, j - 1
391  IF( ( z( i ).EQ.zero ) .OR.
392  $ ( poles( i, 2 ).EQ.zero ) ) THEN
393  work( i ) = zero
394  ELSE
395  work( i ) = poles( i, 2 )*z( i ) /
396  $ ( dlamc3( poles( i, 2 ), dsigj )-
397  $ diflj ) / ( poles( i, 2 )+dj )
398  END IF
399  30 CONTINUE
400  DO 40 i = j + 1, k
401  IF( ( z( i ).EQ.zero ) .OR.
402  $ ( poles( i, 2 ).EQ.zero ) ) THEN
403  work( i ) = zero
404  ELSE
405  work( i ) = poles( i, 2 )*z( i ) /
406  $ ( dlamc3( poles( i, 2 ), dsigjp )+
407  $ difrj ) / ( poles( i, 2 )+dj )
408  END IF
409  40 CONTINUE
410  work( 1 ) = negone
411  temp = dnrm2( k, work, 1 )
412  CALL dgemv( 'T', k, nrhs, one, bx, ldbx, work, 1, zero,
413  $ b( j, 1 ), ldb )
414  CALL dlascl( 'G', 0, 0, temp, one, 1, nrhs, b( j, 1 ),
415  $ ldb, info )
416  50 CONTINUE
417  END IF
418 *
419 * Move the deflated rows of BX to B also.
420 *
421  IF( k.LT.max( m, n ) )
422  $ CALL dlacpy( 'A', n-k, nrhs, bx( k+1, 1 ), ldbx,
423  $ b( k+1, 1 ), ldb )
424  ELSE
425 *
426 * Apply back the right orthogonal transformations.
427 *
428 * Step (1R): apply back the new right singular vector matrix
429 * to B.
430 *
431  IF( k.EQ.1 ) THEN
432  CALL dcopy( nrhs, b, ldb, bx, ldbx )
433  ELSE
434  DO 80 j = 1, k
435  dsigj = poles( j, 2 )
436  IF( z( j ).EQ.zero ) THEN
437  work( j ) = zero
438  ELSE
439  work( j ) = -z( j ) / difl( j ) /
440  $ ( dsigj+poles( j, 1 ) ) / difr( j, 2 )
441  END IF
442  DO 60 i = 1, j - 1
443  IF( z( j ).EQ.zero ) THEN
444  work( i ) = zero
445  ELSE
446  work( i ) = z( j ) / ( dlamc3( dsigj, -poles( i+1,
447  $ 2 ) )-difr( i, 1 ) ) /
448  $ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
449  END IF
450  60 CONTINUE
451  DO 70 i = j + 1, k
452  IF( z( j ).EQ.zero ) THEN
453  work( i ) = zero
454  ELSE
455  work( i ) = z( j ) / ( dlamc3( dsigj, -poles( i,
456  $ 2 ) )-difl( i ) ) /
457  $ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
458  END IF
459  70 CONTINUE
460  CALL dgemv( 'T', k, nrhs, one, b, ldb, work, 1, zero,
461  $ bx( j, 1 ), ldbx )
462  80 CONTINUE
463  END IF
464 *
465 * Step (2R): if SQRE = 1, apply back the rotation that is
466 * related to the right null space of the subproblem.
467 *
468  IF( sqre.EQ.1 ) THEN
469  CALL dcopy( nrhs, b( m, 1 ), ldb, bx( m, 1 ), ldbx )
470  CALL drot( nrhs, bx( 1, 1 ), ldbx, bx( m, 1 ), ldbx, c, s )
471  END IF
472  IF( k.LT.max( m, n ) )
473  $ CALL dlacpy( 'A', n-k, nrhs, b( k+1, 1 ), ldb, bx( k+1, 1 ),
474  $ ldbx )
475 *
476 * Step (3R): permute rows of B.
477 *
478  CALL dcopy( nrhs, bx( 1, 1 ), ldbx, b( nlp1, 1 ), ldb )
479  IF( sqre.EQ.1 ) THEN
480  CALL dcopy( nrhs, bx( m, 1 ), ldbx, b( m, 1 ), ldb )
481  END IF
482  DO 90 i = 2, n
483  CALL dcopy( nrhs, bx( i, 1 ), ldbx, b( perm( i ), 1 ), ldb )
484  90 CONTINUE
485 *
486 * Step (4R): apply back the Givens rotations performed.
487 *
488  DO 100 i = givptr, 1, -1
489  CALL drot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
490  $ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
491  $ -givnum( i, 1 ) )
492  100 CONTINUE
493  END IF
494 *
495  RETURN
496 *
497 * End of DLALS0
498 *
499  END
dlascl
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:145
dlals0
subroutine dlals0(ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO)
DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer...
Definition: dlals0.f:270
dcopy
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
drot
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:94
dgemv
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:158
dscal
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81
dlacpy
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105