LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
slalsa.f
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1 *> \brief \b SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
22 * LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
23 * GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
24 * IWORK, INFO )
25 *
26 * .. Scalar Arguments ..
27 * INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
28 * $ SMLSIZ
29 * ..
30 * .. Array Arguments ..
31 * INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
32 * $ K( * ), PERM( LDGCOL, * )
33 * REAL B( LDB, * ), BX( LDBX, * ), C( * ),
34 * $ DIFL( LDU, * ), DIFR( LDU, * ),
35 * $ GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
36 * $ U( LDU, * ), VT( LDU, * ), WORK( * ),
37 * $ Z( LDU, * )
38 * ..
39 *
40 *
41 *> \par Purpose:
42 * =============
43 *>
44 *> \verbatim
45 *>
46 *> SLALSA is an itermediate step in solving the least squares problem
47 *> by computing the SVD of the coefficient matrix in compact form (The
48 *> singular vectors are computed as products of simple orthorgonal
49 *> matrices.).
50 *>
51 *> If ICOMPQ = 0, SLALSA applies the inverse of the left singular vector
52 *> matrix of an upper bidiagonal matrix to the right hand side; and if
53 *> ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
54 *> right hand side. The singular vector matrices were generated in
55 *> compact form by SLALSA.
56 *> \endverbatim
57 *
58 * Arguments:
59 * ==========
60 *
61 *> \param[in] ICOMPQ
62 *> \verbatim
63 *> ICOMPQ is INTEGER
64 *> Specifies whether the left or the right singular vector
65 *> matrix is involved.
66 *> = 0: Left singular vector matrix
67 *> = 1: Right singular vector matrix
68 *> \endverbatim
69 *>
70 *> \param[in] SMLSIZ
71 *> \verbatim
72 *> SMLSIZ is INTEGER
73 *> The maximum size of the subproblems at the bottom of the
74 *> computation tree.
75 *> \endverbatim
76 *>
77 *> \param[in] N
78 *> \verbatim
79 *> N is INTEGER
80 *> The row and column dimensions of the upper bidiagonal matrix.
81 *> \endverbatim
82 *>
83 *> \param[in] NRHS
84 *> \verbatim
85 *> NRHS is INTEGER
86 *> The number of columns of B and BX. NRHS must be at least 1.
87 *> \endverbatim
88 *>
89 *> \param[in,out] B
90 *> \verbatim
91 *> B is REAL array, dimension ( LDB, NRHS )
92 *> On input, B contains the right hand sides of the least
93 *> squares problem in rows 1 through M.
94 *> On output, B contains the solution X in rows 1 through N.
95 *> \endverbatim
96 *>
97 *> \param[in] LDB
98 *> \verbatim
99 *> LDB is INTEGER
100 *> The leading dimension of B in the calling subprogram.
101 *> LDB must be at least max(1,MAX( M, N ) ).
102 *> \endverbatim
103 *>
104 *> \param[out] BX
105 *> \verbatim
106 *> BX is REAL array, dimension ( LDBX, NRHS )
107 *> On exit, the result of applying the left or right singular
108 *> vector matrix to B.
109 *> \endverbatim
110 *>
111 *> \param[in] LDBX
112 *> \verbatim
113 *> LDBX is INTEGER
114 *> The leading dimension of BX.
115 *> \endverbatim
116 *>
117 *> \param[in] U
118 *> \verbatim
119 *> U is REAL array, dimension ( LDU, SMLSIZ ).
120 *> On entry, U contains the left singular vector matrices of all
121 *> subproblems at the bottom level.
122 *> \endverbatim
123 *>
124 *> \param[in] LDU
125 *> \verbatim
126 *> LDU is INTEGER, LDU = > N.
127 *> The leading dimension of arrays U, VT, DIFL, DIFR,
128 *> POLES, GIVNUM, and Z.
129 *> \endverbatim
130 *>
131 *> \param[in] VT
132 *> \verbatim
133 *> VT is REAL array, dimension ( LDU, SMLSIZ+1 ).
134 *> On entry, VT**T contains the right singular vector matrices of
135 *> all subproblems at the bottom level.
136 *> \endverbatim
137 *>
138 *> \param[in] K
139 *> \verbatim
140 *> K is INTEGER array, dimension ( N ).
141 *> \endverbatim
142 *>
143 *> \param[in] DIFL
144 *> \verbatim
145 *> DIFL is REAL array, dimension ( LDU, NLVL ).
146 *> where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
147 *> \endverbatim
148 *>
149 *> \param[in] DIFR
150 *> \verbatim
151 *> DIFR is REAL array, dimension ( LDU, 2 * NLVL ).
152 *> On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
153 *> distances between singular values on the I-th level and
154 *> singular values on the (I -1)-th level, and DIFR(*, 2 * I)
155 *> record the normalizing factors of the right singular vectors
156 *> matrices of subproblems on I-th level.
157 *> \endverbatim
158 *>
159 *> \param[in] Z
160 *> \verbatim
161 *> Z is REAL array, dimension ( LDU, NLVL ).
162 *> On entry, Z(1, I) contains the components of the deflation-
163 *> adjusted updating row vector for subproblems on the I-th
164 *> level.
165 *> \endverbatim
166 *>
167 *> \param[in] POLES
168 *> \verbatim
169 *> POLES is REAL array, dimension ( LDU, 2 * NLVL ).
170 *> On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
171 *> singular values involved in the secular equations on the I-th
172 *> level.
173 *> \endverbatim
174 *>
175 *> \param[in] GIVPTR
176 *> \verbatim
177 *> GIVPTR is INTEGER array, dimension ( N ).
178 *> On entry, GIVPTR( I ) records the number of Givens
179 *> rotations performed on the I-th problem on the computation
180 *> tree.
181 *> \endverbatim
182 *>
183 *> \param[in] GIVCOL
184 *> \verbatim
185 *> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
186 *> On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
187 *> locations of Givens rotations performed on the I-th level on
188 *> the computation tree.
189 *> \endverbatim
190 *>
191 *> \param[in] LDGCOL
192 *> \verbatim
193 *> LDGCOL is INTEGER, LDGCOL = > N.
194 *> The leading dimension of arrays GIVCOL and PERM.
195 *> \endverbatim
196 *>
197 *> \param[in] PERM
198 *> \verbatim
199 *> PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
200 *> On entry, PERM(*, I) records permutations done on the I-th
201 *> level of the computation tree.
202 *> \endverbatim
203 *>
204 *> \param[in] GIVNUM
205 *> \verbatim
206 *> GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ).
207 *> On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
208 *> values of Givens rotations performed on the I-th level on the
209 *> computation tree.
210 *> \endverbatim
211 *>
212 *> \param[in] C
213 *> \verbatim
214 *> C is REAL array, dimension ( N ).
215 *> On entry, if the I-th subproblem is not square,
216 *> C( I ) contains the C-value of a Givens rotation related to
217 *> the right null space of the I-th subproblem.
218 *> \endverbatim
219 *>
220 *> \param[in] S
221 *> \verbatim
222 *> S is REAL array, dimension ( N ).
223 *> On entry, if the I-th subproblem is not square,
224 *> S( I ) contains the S-value of a Givens rotation related to
225 *> the right null space of the I-th subproblem.
226 *> \endverbatim
227 *>
228 *> \param[out] WORK
229 *> \verbatim
230 *> WORK is REAL array.
231 *> The dimension must be at least N.
232 *> \endverbatim
233 *>
234 *> \param[out] IWORK
235 *> \verbatim
236 *> IWORK is INTEGER array.
237 *> The dimension must be at least 3 * N
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 slalsa( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
268  $ ldu, vt, k, difl, difr, z, poles, givptr,
269  $ givcol, ldgcol, perm, givnum, c, s, work,
270  $ iwork, info )
271 *
272 * -- LAPACK computational routine (version 3.4.2) --
273 * -- LAPACK is a software package provided by Univ. of Tennessee, --
274 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
275 * September 2012
276 *
277 * .. Scalar Arguments ..
278  INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
279  $ smlsiz
280 * ..
281 * .. Array Arguments ..
282  INTEGER GIVCOL( ldgcol, * ), GIVPTR( * ), IWORK( * ),
283  $ k( * ), perm( ldgcol, * )
284  REAL B( ldb, * ), BX( ldbx, * ), C( * ),
285  $ difl( ldu, * ), difr( ldu, * ),
286  $ givnum( ldu, * ), poles( ldu, * ), s( * ),
287  $ u( ldu, * ), vt( ldu, * ), work( * ),
288  $ z( ldu, * )
289 * ..
290 *
291 * =====================================================================
292 *
293 * .. Parameters ..
294  REAL ZERO, ONE
295  parameter ( zero = 0.0e0, one = 1.0e0 )
296 * ..
297 * .. Local Scalars ..
298  INTEGER I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2,
299  $ nd, ndb1, ndiml, ndimr, nl, nlf, nlp1, nlvl,
300  $ nr, nrf, nrp1, sqre
301 * ..
302 * .. External Subroutines ..
303  EXTERNAL scopy, sgemm, slals0, slasdt, xerbla
304 * ..
305 * .. Executable Statements ..
306 *
307 * Test the input parameters.
308 *
309  info = 0
310 *
311  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
312  info = -1
313  ELSE IF( smlsiz.LT.3 ) THEN
314  info = -2
315  ELSE IF( n.LT.smlsiz ) THEN
316  info = -3
317  ELSE IF( nrhs.LT.1 ) THEN
318  info = -4
319  ELSE IF( ldb.LT.n ) THEN
320  info = -6
321  ELSE IF( ldbx.LT.n ) THEN
322  info = -8
323  ELSE IF( ldu.LT.n ) THEN
324  info = -10
325  ELSE IF( ldgcol.LT.n ) THEN
326  info = -19
327  END IF
328  IF( info.NE.0 ) THEN
329  CALL xerbla( 'SLALSA', -info )
330  RETURN
331  END IF
332 *
333 * Book-keeping and setting up the computation tree.
334 *
335  inode = 1
336  ndiml = inode + n
337  ndimr = ndiml + n
338 *
339  CALL slasdt( n, nlvl, nd, iwork( inode ), iwork( ndiml ),
340  $ iwork( ndimr ), smlsiz )
341 *
342 * The following code applies back the left singular vector factors.
343 * For applying back the right singular vector factors, go to 50.
344 *
345  IF( icompq.EQ.1 ) THEN
346  GO TO 50
347  END IF
348 *
349 * The nodes on the bottom level of the tree were solved
350 * by SLASDQ. The corresponding left and right singular vector
351 * matrices are in explicit form. First apply back the left
352 * singular vector matrices.
353 *
354  ndb1 = ( nd+1 ) / 2
355  DO 10 i = ndb1, nd
356 *
357 * IC : center row of each node
358 * NL : number of rows of left subproblem
359 * NR : number of rows of right subproblem
360 * NLF: starting row of the left subproblem
361 * NRF: starting row of the right subproblem
362 *
363  i1 = i - 1
364  ic = iwork( inode+i1 )
365  nl = iwork( ndiml+i1 )
366  nr = iwork( ndimr+i1 )
367  nlf = ic - nl
368  nrf = ic + 1
369  CALL sgemm( 'T', 'N', nl, nrhs, nl, one, u( nlf, 1 ), ldu,
370  $ b( nlf, 1 ), ldb, zero, bx( nlf, 1 ), ldbx )
371  CALL sgemm( 'T', 'N', nr, nrhs, nr, one, u( nrf, 1 ), ldu,
372  $ b( nrf, 1 ), ldb, zero, bx( nrf, 1 ), ldbx )
373  10 CONTINUE
374 *
375 * Next copy the rows of B that correspond to unchanged rows
376 * in the bidiagonal matrix to BX.
377 *
378  DO 20 i = 1, nd
379  ic = iwork( inode+i-1 )
380  CALL scopy( nrhs, b( ic, 1 ), ldb, bx( ic, 1 ), ldbx )
381  20 CONTINUE
382 *
383 * Finally go through the left singular vector matrices of all
384 * the other subproblems bottom-up on the tree.
385 *
386  j = 2**nlvl
387  sqre = 0
388 *
389  DO 40 lvl = nlvl, 1, -1
390  lvl2 = 2*lvl - 1
391 *
392 * find the first node LF and last node LL on
393 * the current level LVL
394 *
395  IF( lvl.EQ.1 ) THEN
396  lf = 1
397  ll = 1
398  ELSE
399  lf = 2**( lvl-1 )
400  ll = 2*lf - 1
401  END IF
402  DO 30 i = lf, ll
403  im1 = i - 1
404  ic = iwork( inode+im1 )
405  nl = iwork( ndiml+im1 )
406  nr = iwork( ndimr+im1 )
407  nlf = ic - nl
408  nrf = ic + 1
409  j = j - 1
410  CALL slals0( icompq, nl, nr, sqre, nrhs, bx( nlf, 1 ), ldbx,
411  $ b( nlf, 1 ), ldb, perm( nlf, lvl ),
412  $ givptr( j ), givcol( nlf, lvl2 ), ldgcol,
413  $ givnum( nlf, lvl2 ), ldu, poles( nlf, lvl2 ),
414  $ difl( nlf, lvl ), difr( nlf, lvl2 ),
415  $ z( nlf, lvl ), k( j ), c( j ), s( j ), work,
416  $ info )
417  30 CONTINUE
418  40 CONTINUE
419  GO TO 90
420 *
421 * ICOMPQ = 1: applying back the right singular vector factors.
422 *
423  50 CONTINUE
424 *
425 * First now go through the right singular vector matrices of all
426 * the tree nodes top-down.
427 *
428  j = 0
429  DO 70 lvl = 1, nlvl
430  lvl2 = 2*lvl - 1
431 *
432 * Find the first node LF and last node LL on
433 * the current level LVL.
434 *
435  IF( lvl.EQ.1 ) THEN
436  lf = 1
437  ll = 1
438  ELSE
439  lf = 2**( lvl-1 )
440  ll = 2*lf - 1
441  END IF
442  DO 60 i = ll, lf, -1
443  im1 = i - 1
444  ic = iwork( inode+im1 )
445  nl = iwork( ndiml+im1 )
446  nr = iwork( ndimr+im1 )
447  nlf = ic - nl
448  nrf = ic + 1
449  IF( i.EQ.ll ) THEN
450  sqre = 0
451  ELSE
452  sqre = 1
453  END IF
454  j = j + 1
455  CALL slals0( icompq, nl, nr, sqre, nrhs, b( nlf, 1 ), ldb,
456  $ bx( nlf, 1 ), ldbx, perm( nlf, lvl ),
457  $ givptr( j ), givcol( nlf, lvl2 ), ldgcol,
458  $ givnum( nlf, lvl2 ), ldu, poles( nlf, lvl2 ),
459  $ difl( nlf, lvl ), difr( nlf, lvl2 ),
460  $ z( nlf, lvl ), k( j ), c( j ), s( j ), work,
461  $ info )
462  60 CONTINUE
463  70 CONTINUE
464 *
465 * The nodes on the bottom level of the tree were solved
466 * by SLASDQ. The corresponding right singular vector
467 * matrices are in explicit form. Apply them back.
468 *
469  ndb1 = ( nd+1 ) / 2
470  DO 80 i = ndb1, nd
471  i1 = i - 1
472  ic = iwork( inode+i1 )
473  nl = iwork( ndiml+i1 )
474  nr = iwork( ndimr+i1 )
475  nlp1 = nl + 1
476  IF( i.EQ.nd ) THEN
477  nrp1 = nr
478  ELSE
479  nrp1 = nr + 1
480  END IF
481  nlf = ic - nl
482  nrf = ic + 1
483  CALL sgemm( 'T', 'N', nlp1, nrhs, nlp1, one, vt( nlf, 1 ), ldu,
484  $ b( nlf, 1 ), ldb, zero, bx( nlf, 1 ), ldbx )
485  CALL sgemm( 'T', 'N', nrp1, nrhs, nrp1, one, vt( nrf, 1 ), ldu,
486  $ b( nrf, 1 ), ldb, zero, bx( nrf, 1 ), ldbx )
487  80 CONTINUE
488 *
489  90 CONTINUE
490 *
491  RETURN
492 *
493 * End of SLALSA
494 *
495  END
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:270
subroutine slasdt(N, LVL, ND, INODE, NDIML, NDIMR, MSUB)
SLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.
Definition: slasdt.f:107
subroutine slalsa(ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)
SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
Definition: slalsa.f:271
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53