 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ slalsa()

 subroutine slalsa ( integer ICOMPQ, integer SMLSIZ, integer N, integer NRHS, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldbx, * ) BX, integer LDBX, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldu, * ) VT, integer, dimension( * ) K, real, dimension( ldu, * ) DIFL, real, dimension( ldu, * ) DIFR, real, dimension( ldu, * ) Z, real, dimension( ldu, * ) POLES, integer, dimension( * ) GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer LDGCOL, integer, dimension( ldgcol, * ) PERM, real, dimension( ldu, * ) GIVNUM, real, dimension( * ) C, real, dimension( * ) S, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Purpose:
``` SLALSA is an itermediate step in solving the least squares problem
by computing the SVD of the coefficient matrix in compact form (The
singular vectors are computed as products of simple orthorgonal
matrices.).

If ICOMPQ = 0, SLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in
compact form by SLALSA.```
Parameters
 [in] ICOMPQ ``` ICOMPQ is INTEGER Specifies whether the left or the right singular vector matrix is involved. = 0: Left singular vector matrix = 1: Right singular vector matrix``` [in] SMLSIZ ``` SMLSIZ is INTEGER The maximum size of the subproblems at the bottom of the computation tree.``` [in] N ``` N is INTEGER The row and column dimensions of the upper bidiagonal matrix.``` [in] NRHS ``` NRHS is INTEGER The number of columns of B and BX. NRHS must be at least 1.``` [in,out] B ``` B is REAL array, dimension ( LDB, NRHS ) On input, B contains the right hand sides of the least squares problem in rows 1 through M. On output, B contains the solution X in rows 1 through N.``` [in] LDB ``` LDB is INTEGER The leading dimension of B in the calling subprogram. LDB must be at least max(1,MAX( M, N ) ).``` [out] BX ``` BX is REAL array, dimension ( LDBX, NRHS ) On exit, the result of applying the left or right singular vector matrix to B.``` [in] LDBX ``` LDBX is INTEGER The leading dimension of BX.``` [in] U ``` U is REAL array, dimension ( LDU, SMLSIZ ). On entry, U contains the left singular vector matrices of all subproblems at the bottom level.``` [in] LDU ``` LDU is INTEGER, LDU = > N. The leading dimension of arrays U, VT, DIFL, DIFR, POLES, GIVNUM, and Z.``` [in] VT ``` VT is REAL array, dimension ( LDU, SMLSIZ+1 ). On entry, VT**T contains the right singular vector matrices of all subproblems at the bottom level.``` [in] K ` K is INTEGER array, dimension ( N ).` [in] DIFL ``` DIFL is REAL array, dimension ( LDU, NLVL ). where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.``` [in] DIFR ``` DIFR is REAL array, dimension ( LDU, 2 * NLVL ). On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record distances between singular values on the I-th level and singular values on the (I -1)-th level, and DIFR(*, 2 * I) record the normalizing factors of the right singular vectors matrices of subproblems on I-th level.``` [in] Z ``` Z is REAL array, dimension ( LDU, NLVL ). On entry, Z(1, I) contains the components of the deflation- adjusted updating row vector for subproblems on the I-th level.``` [in] POLES ``` POLES is REAL array, dimension ( LDU, 2 * NLVL ). On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old singular values involved in the secular equations on the I-th level.``` [in] GIVPTR ``` GIVPTR is INTEGER array, dimension ( N ). On entry, GIVPTR( I ) records the number of Givens rotations performed on the I-th problem on the computation tree.``` [in] GIVCOL ``` GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ). On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the locations of Givens rotations performed on the I-th level on the computation tree.``` [in] LDGCOL ``` LDGCOL is INTEGER, LDGCOL = > N. The leading dimension of arrays GIVCOL and PERM.``` [in] PERM ``` PERM is INTEGER array, dimension ( LDGCOL, NLVL ). On entry, PERM(*, I) records permutations done on the I-th level of the computation tree.``` [in] GIVNUM ``` GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ). On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S- values of Givens rotations performed on the I-th level on the computation tree.``` [in] C ``` C is REAL array, dimension ( N ). On entry, if the I-th subproblem is not square, C( I ) contains the C-value of a Givens rotation related to the right null space of the I-th subproblem.``` [in] S ``` S is REAL array, dimension ( N ). On entry, if the I-th subproblem is not square, S( I ) contains the S-value of a Givens rotation related to the right null space of the I-th subproblem.``` [out] WORK ` WORK is REAL array, dimension (N)` [out] IWORK ` IWORK is INTEGER array, dimension (3*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```
Contributors:
Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Definition at line 263 of file slalsa.f.

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 *
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
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
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