LAPACK  3.11.0
LAPACK: Linear Algebra PACKage
slalsa.f
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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9 *> Download SLALSA + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slalsa.f">
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, dimension (N)
231 *> \endverbatim
232 *>
233 *> \param[out] IWORK
234 *> \verbatim
235 *> IWORK is INTEGER array, dimension (3*N)
236 *> \endverbatim
237 *>
238 *> \param[out] INFO
239 *> \verbatim
240 *> INFO is INTEGER
241 *> = 0: successful exit.
242 *> < 0: if INFO = -i, the i-th argument had an illegal value.
243 *> \endverbatim
244 *
245 * Authors:
246 * ========
247 *
248 *> \author Univ. of Tennessee
249 *> \author Univ. of California Berkeley
250 *> \author Univ. of Colorado Denver
251 *> \author NAG Ltd.
252 *
253 *> \ingroup lalsa
254 *
255 *> \par Contributors:
256 * ==================
257 *>
258 *> Ming Gu and Ren-Cang Li, Computer Science Division, University of
259 *> California at Berkeley, USA \n
260 *> Osni Marques, LBNL/NERSC, USA \n
261 *
262 * =====================================================================
263  SUBROUTINE slalsa( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
264  $ LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
265  $ GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
266  $ IWORK, INFO )
267 *
268 * -- LAPACK computational routine --
269 * -- LAPACK is a software package provided by Univ. of Tennessee, --
270 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
271 *
272 * .. Scalar Arguments ..
273  INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
274  $ smlsiz
275 * ..
276 * .. Array Arguments ..
277  INTEGER GIVCOL( ldgcol, * ), GIVPTR( * ), IWORK( * ),
278  $ k( * ), perm( ldgcol, * )
279  REAL B( ldb, * ), BX( ldbx, * ), C( * ),
280  $ difl( ldu, * ), difr( ldu, * ),
281  $ givnum( ldu, * ), poles( ldu, * ), s( * ),
282  $ u( ldu, * ), vt( ldu, * ), work( * ),
283  $ z( ldu, * )
284 * ..
285 *
286 * =====================================================================
287 *
288 * .. Parameters ..
289  REAL ZERO, ONE
290  parameter( zero = 0.0e0, one = 1.0e0 )
291 * ..
292 * .. Local Scalars ..
293  INTEGER I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2,
294  $ nd, ndb1, ndiml, ndimr, nl, nlf, nlp1, nlvl,
295  $ nr, nrf, nrp1, sqre
296 * ..
297 * .. External Subroutines ..
298  EXTERNAL scopy, sgemm, slals0, slasdt, xerbla
299 * ..
300 * .. Executable Statements ..
301 *
302 * Test the input parameters.
303 *
304  info = 0
305 *
306  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
307  info = -1
308  ELSE IF( smlsiz.LT.3 ) THEN
309  info = -2
310  ELSE IF( n.LT.smlsiz ) THEN
311  info = -3
312  ELSE IF( nrhs.LT.1 ) THEN
313  info = -4
314  ELSE IF( ldb.LT.n ) THEN
315  info = -6
316  ELSE IF( ldbx.LT.n ) THEN
317  info = -8
318  ELSE IF( ldu.LT.n ) THEN
319  info = -10
320  ELSE IF( ldgcol.LT.n ) THEN
321  info = -19
322  END IF
323  IF( info.NE.0 ) THEN
324  CALL xerbla( 'SLALSA', -info )
325  RETURN
326  END IF
327 *
328 * Book-keeping and setting up the computation tree.
329 *
330  inode = 1
331  ndiml = inode + n
332  ndimr = ndiml + n
333 *
334  CALL slasdt( n, nlvl, nd, iwork( inode ), iwork( ndiml ),
335  $ iwork( ndimr ), smlsiz )
336 *
337 * The following code applies back the left singular vector factors.
338 * For applying back the right singular vector factors, go to 50.
339 *
340  IF( icompq.EQ.1 ) THEN
341  GO TO 50
342  END IF
343 *
344 * The nodes on the bottom level of the tree were solved
345 * by SLASDQ. The corresponding left and right singular vector
346 * matrices are in explicit form. First apply back the left
347 * singular vector matrices.
348 *
349  ndb1 = ( nd+1 ) / 2
350  DO 10 i = ndb1, nd
351 *
352 * IC : center row of each node
353 * NL : number of rows of left subproblem
354 * NR : number of rows of right subproblem
355 * NLF: starting row of the left subproblem
356 * NRF: starting row of the right subproblem
357 *
358  i1 = i - 1
359  ic = iwork( inode+i1 )
360  nl = iwork( ndiml+i1 )
361  nr = iwork( ndimr+i1 )
362  nlf = ic - nl
363  nrf = ic + 1
364  CALL sgemm( 'T', 'N', nl, nrhs, nl, one, u( nlf, 1 ), ldu,
365  $ b( nlf, 1 ), ldb, zero, bx( nlf, 1 ), ldbx )
366  CALL sgemm( 'T', 'N', nr, nrhs, nr, one, u( nrf, 1 ), ldu,
367  $ b( nrf, 1 ), ldb, zero, bx( nrf, 1 ), ldbx )
368  10 CONTINUE
369 *
370 * Next copy the rows of B that correspond to unchanged rows
371 * in the bidiagonal matrix to BX.
372 *
373  DO 20 i = 1, nd
374  ic = iwork( inode+i-1 )
375  CALL scopy( nrhs, b( ic, 1 ), ldb, bx( ic, 1 ), ldbx )
376  20 CONTINUE
377 *
378 * Finally go through the left singular vector matrices of all
379 * the other subproblems bottom-up on the tree.
380 *
381  j = 2**nlvl
382  sqre = 0
383 *
384  DO 40 lvl = nlvl, 1, -1
385  lvl2 = 2*lvl - 1
386 *
387 * find the first node LF and last node LL on
388 * the current level LVL
389 *
390  IF( lvl.EQ.1 ) THEN
391  lf = 1
392  ll = 1
393  ELSE
394  lf = 2**( lvl-1 )
395  ll = 2*lf - 1
396  END IF
397  DO 30 i = lf, ll
398  im1 = i - 1
399  ic = iwork( inode+im1 )
400  nl = iwork( ndiml+im1 )
401  nr = iwork( ndimr+im1 )
402  nlf = ic - nl
403  nrf = ic + 1
404  j = j - 1
405  CALL slals0( icompq, nl, nr, sqre, nrhs, bx( nlf, 1 ), ldbx,
406  $ b( nlf, 1 ), ldb, perm( nlf, lvl ),
407  $ givptr( j ), givcol( nlf, lvl2 ), ldgcol,
408  $ givnum( nlf, lvl2 ), ldu, poles( nlf, lvl2 ),
409  $ difl( nlf, lvl ), difr( nlf, lvl2 ),
410  $ z( nlf, lvl ), k( j ), c( j ), s( j ), work,
411  $ info )
412  30 CONTINUE
413  40 CONTINUE
414  GO TO 90
415 *
416 * ICOMPQ = 1: applying back the right singular vector factors.
417 *
418  50 CONTINUE
419 *
420 * First now go through the right singular vector matrices of all
421 * the tree nodes top-down.
422 *
423  j = 0
424  DO 70 lvl = 1, nlvl
425  lvl2 = 2*lvl - 1
426 *
427 * Find the first node LF and last node LL on
428 * the current level LVL.
429 *
430  IF( lvl.EQ.1 ) THEN
431  lf = 1
432  ll = 1
433  ELSE
434  lf = 2**( lvl-1 )
435  ll = 2*lf - 1
436  END IF
437  DO 60 i = ll, lf, -1
438  im1 = i - 1
439  ic = iwork( inode+im1 )
440  nl = iwork( ndiml+im1 )
441  nr = iwork( ndimr+im1 )
442  nlf = ic - nl
443  nrf = ic + 1
444  IF( i.EQ.ll ) THEN
445  sqre = 0
446  ELSE
447  sqre = 1
448  END IF
449  j = j + 1
450  CALL slals0( icompq, nl, nr, sqre, nrhs, b( nlf, 1 ), ldb,
451  $ bx( nlf, 1 ), ldbx, perm( nlf, lvl ),
452  $ givptr( j ), givcol( nlf, lvl2 ), ldgcol,
453  $ givnum( nlf, lvl2 ), ldu, poles( nlf, lvl2 ),
454  $ difl( nlf, lvl ), difr( nlf, lvl2 ),
455  $ z( nlf, lvl ), k( j ), c( j ), s( j ), work,
456  $ info )
457  60 CONTINUE
458  70 CONTINUE
459 *
460 * The nodes on the bottom level of the tree were solved
461 * by SLASDQ. The corresponding right singular vector
462 * matrices are in explicit form. Apply them back.
463 *
464  ndb1 = ( nd+1 ) / 2
465  DO 80 i = ndb1, nd
466  i1 = i - 1
467  ic = iwork( inode+i1 )
468  nl = iwork( ndiml+i1 )
469  nr = iwork( ndimr+i1 )
470  nlp1 = nl + 1
471  IF( i.EQ.nd ) THEN
472  nrp1 = nr
473  ELSE
474  nrp1 = nr + 1
475  END IF
476  nlf = ic - nl
477  nrf = ic + 1
478  CALL sgemm( 'T', 'N', nlp1, nrhs, nlp1, one, vt( nlf, 1 ), ldu,
479  $ b( nlf, 1 ), ldb, zero, bx( nlf, 1 ), ldbx )
480  CALL sgemm( 'T', 'N', nrp1, nrhs, nrp1, one, vt( nrf, 1 ), ldu,
481  $ b( nrf, 1 ), ldb, zero, bx( nrf, 1 ), ldbx )
482  80 CONTINUE
483 *
484  90 CONTINUE
485 *
486  RETURN
487 *
488 * End of SLALSA
489 *
490  END
slalsa
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:267
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
sgemm
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
slals0
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:268
scopy
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
slasdt
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:105