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slals0.f
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1 *> \brief \b SLALS0 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 SLALS0 + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slals0.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slals0.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slals0.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLALS0( 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 * REAL C, S
29 * ..
30 * .. Array Arguments ..
31 * INTEGER GIVCOL( LDGCOL, * ), PERM( * )
32 * REAL 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 *> SLALS0 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
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 REAL
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 REAL 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 September 2012
256 *
257 *> \ingroup realOTHERcomputational
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 slals0( 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 3.4.2) --
272 * -- LAPACK is a software package provided by Univ. of Tennessee, --
273 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
274 * September 2012
275 *
276 * .. Scalar Arguments ..
277  INTEGER givptr, icompq, info, k, ldb, ldbx, ldgcol,
278  $ ldgnum, nl, nr, nrhs, sqre
279  REAL c, s
280 * ..
281 * .. Array Arguments ..
282  INTEGER givcol( ldgcol, * ), perm( * )
283  REAL b( ldb, * ), bx( ldbx, * ), difl( * ),
284  $ difr( ldgnum, * ), givnum( ldgnum, * ),
285  $ poles( ldgnum, * ), work( * ), z( * )
286 * ..
287 *
288 * =====================================================================
289 *
290 * .. Parameters ..
291  REAL one, zero, negone
292  parameter( one = 1.0e0, zero = 0.0e0, negone = -1.0e0 )
293 * ..
294 * .. Local Scalars ..
295  INTEGER i, j, m, n, nlp1
296  REAL diflj, difrj, dj, dsigj, dsigjp, temp
297 * ..
298 * .. External Subroutines ..
299  EXTERNAL scopy, sgemv, slacpy, slascl, srot, sscal,
300  $ xerbla
301 * ..
302 * .. External Functions ..
303  REAL slamc3, snrm2
304  EXTERNAL slamc3, snrm2
305 * ..
306 * .. Intrinsic Functions ..
307  INTRINSIC max
308 * ..
309 * .. Executable Statements ..
310 *
311 * Test the input parameters.
312 *
313  info = 0
314 *
315  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
316  info = -1
317  ELSE IF( nl.LT.1 ) THEN
318  info = -2
319  ELSE IF( nr.LT.1 ) THEN
320  info = -3
321  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
322  info = -4
323  END IF
324 *
325  n = nl + nr + 1
326 *
327  IF( nrhs.LT.1 ) THEN
328  info = -5
329  ELSE IF( ldb.LT.n ) THEN
330  info = -7
331  ELSE IF( ldbx.LT.n ) THEN
332  info = -9
333  ELSE IF( givptr.LT.0 ) THEN
334  info = -11
335  ELSE IF( ldgcol.LT.n ) THEN
336  info = -13
337  ELSE IF( ldgnum.LT.n ) THEN
338  info = -15
339  ELSE IF( k.LT.1 ) THEN
340  info = -20
341  END IF
342  IF( info.NE.0 ) THEN
343  CALL xerbla( 'SLALS0', -info )
344  RETURN
345  END IF
346 *
347  m = n + sqre
348  nlp1 = nl + 1
349 *
350  IF( icompq.EQ.0 ) THEN
351 *
352 * Apply back orthogonal transformations from the left.
353 *
354 * Step (1L): apply back the Givens rotations performed.
355 *
356  DO 10 i = 1, givptr
357  CALL srot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
358  $ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
359  $ givnum( i, 1 ) )
360  10 CONTINUE
361 *
362 * Step (2L): permute rows of B.
363 *
364  CALL scopy( nrhs, b( nlp1, 1 ), ldb, bx( 1, 1 ), ldbx )
365  DO 20 i = 2, n
366  CALL scopy( nrhs, b( perm( i ), 1 ), ldb, bx( i, 1 ), ldbx )
367  20 CONTINUE
368 *
369 * Step (3L): apply the inverse of the left singular vector
370 * matrix to BX.
371 *
372  IF( k.EQ.1 ) THEN
373  CALL scopy( nrhs, bx, ldbx, b, ldb )
374  IF( z( 1 ).LT.zero ) THEN
375  CALL sscal( nrhs, negone, b, ldb )
376  END IF
377  ELSE
378  DO 50 j = 1, k
379  diflj = difl( j )
380  dj = poles( j, 1 )
381  dsigj = -poles( j, 2 )
382  IF( j.LT.k ) THEN
383  difrj = -difr( j, 1 )
384  dsigjp = -poles( j+1, 2 )
385  END IF
386  IF( ( z( j ).EQ.zero ) .OR. ( poles( j, 2 ).EQ.zero ) )
387  $ THEN
388  work( j ) = zero
389  ELSE
390  work( j ) = -poles( j, 2 )*z( j ) / diflj /
391  $ ( poles( j, 2 )+dj )
392  END IF
393  DO 30 i = 1, j - 1
394  IF( ( z( i ).EQ.zero ) .OR.
395  $ ( poles( i, 2 ).EQ.zero ) ) THEN
396  work( i ) = zero
397  ELSE
398  work( i ) = poles( i, 2 )*z( i ) /
399  $ ( slamc3( poles( i, 2 ), dsigj )-
400  $ diflj ) / ( poles( i, 2 )+dj )
401  END IF
402  30 CONTINUE
403  DO 40 i = j + 1, k
404  IF( ( z( i ).EQ.zero ) .OR.
405  $ ( poles( i, 2 ).EQ.zero ) ) THEN
406  work( i ) = zero
407  ELSE
408  work( i ) = poles( i, 2 )*z( i ) /
409  $ ( slamc3( poles( i, 2 ), dsigjp )+
410  $ difrj ) / ( poles( i, 2 )+dj )
411  END IF
412  40 CONTINUE
413  work( 1 ) = negone
414  temp = snrm2( k, work, 1 )
415  CALL sgemv( 'T', k, nrhs, one, bx, ldbx, work, 1, zero,
416  $ b( j, 1 ), ldb )
417  CALL slascl( 'G', 0, 0, temp, one, 1, nrhs, b( j, 1 ),
418  $ ldb, info )
419  50 CONTINUE
420  END IF
421 *
422 * Move the deflated rows of BX to B also.
423 *
424  IF( k.LT.max( m, n ) )
425  $ CALL slacpy( 'A', n-k, nrhs, bx( k+1, 1 ), ldbx,
426  $ b( k+1, 1 ), ldb )
427  ELSE
428 *
429 * Apply back the right orthogonal transformations.
430 *
431 * Step (1R): apply back the new right singular vector matrix
432 * to B.
433 *
434  IF( k.EQ.1 ) THEN
435  CALL scopy( nrhs, b, ldb, bx, ldbx )
436  ELSE
437  DO 80 j = 1, k
438  dsigj = poles( j, 2 )
439  IF( z( j ).EQ.zero ) THEN
440  work( j ) = zero
441  ELSE
442  work( j ) = -z( j ) / difl( j ) /
443  $ ( dsigj+poles( j, 1 ) ) / difr( j, 2 )
444  END IF
445  DO 60 i = 1, j - 1
446  IF( z( j ).EQ.zero ) THEN
447  work( i ) = zero
448  ELSE
449  work( i ) = z( j ) / ( slamc3( dsigj, -poles( i+1,
450  $ 2 ) )-difr( i, 1 ) ) /
451  $ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
452  END IF
453  60 CONTINUE
454  DO 70 i = j + 1, k
455  IF( z( j ).EQ.zero ) THEN
456  work( i ) = zero
457  ELSE
458  work( i ) = z( j ) / ( slamc3( dsigj, -poles( i,
459  $ 2 ) )-difl( i ) ) /
460  $ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
461  END IF
462  70 CONTINUE
463  CALL sgemv( 'T', k, nrhs, one, b, ldb, work, 1, zero,
464  $ bx( j, 1 ), ldbx )
465  80 CONTINUE
466  END IF
467 *
468 * Step (2R): if SQRE = 1, apply back the rotation that is
469 * related to the right null space of the subproblem.
470 *
471  IF( sqre.EQ.1 ) THEN
472  CALL scopy( nrhs, b( m, 1 ), ldb, bx( m, 1 ), ldbx )
473  CALL srot( nrhs, bx( 1, 1 ), ldbx, bx( m, 1 ), ldbx, c, s )
474  END IF
475  IF( k.LT.max( m, n ) )
476  $ CALL slacpy( 'A', n-k, nrhs, b( k+1, 1 ), ldb, bx( k+1, 1 ),
477  $ ldbx )
478 *
479 * Step (3R): permute rows of B.
480 *
481  CALL scopy( nrhs, bx( 1, 1 ), ldbx, b( nlp1, 1 ), ldb )
482  IF( sqre.EQ.1 ) THEN
483  CALL scopy( nrhs, bx( m, 1 ), ldbx, b( m, 1 ), ldb )
484  END IF
485  DO 90 i = 2, n
486  CALL scopy( nrhs, bx( i, 1 ), ldbx, b( perm( i ), 1 ), ldb )
487  90 CONTINUE
488 *
489 * Step (4R): apply back the Givens rotations performed.
490 *
491  DO 100 i = givptr, 1, -1
492  CALL srot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
493  $ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
494  $ -givnum( i, 1 ) )
495  100 CONTINUE
496  END IF
497 *
498  RETURN
499 *
500 * End of SLALS0
501 *
502  END
real function slamc3(A, B)
SLAMC3
Definition: slamch.f:171
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:61
set ue cd $ADTTMP cat<< EOF > tmp f Program LinearEquations Implicit none Real b(3) integer i
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:52
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:157
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:140
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:104
subroutine slals0(ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO)
SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer...
Definition: slals0.f:267
real function snrm2(N, X, INCX)
SNRM2
Definition: snrm2.f:55
set ue cd $ADTTMP cat<< EOF > tmp f Program LinearEquations Implicit none Real j
Definition: xerbla-fortran:9
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:54
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:52