 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zlalsa()

 subroutine zlalsa ( integer ICOMPQ, integer SMLSIZ, integer N, integer NRHS, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldbx, * ) BX, integer LDBX, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldu, * ) VT, integer, dimension( * ) K, double precision, dimension( ldu, * ) DIFL, double precision, dimension( ldu, * ) DIFR, double precision, dimension( ldu, * ) Z, double precision, dimension( ldu, * ) POLES, integer, dimension( * ) GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer LDGCOL, integer, dimension( ldgcol, * ) PERM, double precision, dimension( ldu, * ) GIVNUM, double precision, dimension( * ) C, double precision, dimension( * ) S, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO )

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

Download ZLALSA + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` ZLALSA 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, ZLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, ZLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in
compact form by ZLALSA.```
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 COMPLEX*16 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 COMPLEX*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ). On entry, VT**H 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 DOUBLE PRECISION array, dimension ( LDU, NLVL ). where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.``` [in] DIFR ``` DIFR is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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] RWORK ``` RWORK is DOUBLE PRECISION array, dimension at least MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ).``` [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 zlalsa.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  DOUBLE PRECISION C( * ), DIFL( LDU, * ), DIFR( LDU, * ),
280  \$ GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ),
281  \$ S( * ), U( LDU, * ), VT( LDU, * ), Z( LDU, * )
282  COMPLEX*16 B( LDB, * ), BX( LDBX, * )
283 * ..
284 *
285 * =====================================================================
286 *
287 * .. Parameters ..
288  DOUBLE PRECISION ZERO, ONE
289  parameter( zero = 0.0d0, one = 1.0d0 )
290 * ..
291 * .. Local Scalars ..
292  INTEGER I, I1, IC, IM1, INODE, J, JCOL, JIMAG, JREAL,
293  \$ JROW, LF, LL, LVL, LVL2, ND, NDB1, NDIML,
294  \$ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQRE
295 * ..
296 * .. External Subroutines ..
297  EXTERNAL dgemm, dlasdt, xerbla, zcopy, zlals0
298 * ..
299 * .. Intrinsic Functions ..
300  INTRINSIC dble, dcmplx, dimag
301 * ..
302 * .. Executable Statements ..
303 *
304 * Test the input parameters.
305 *
306  info = 0
307 *
308  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
309  info = -1
310  ELSE IF( smlsiz.LT.3 ) THEN
311  info = -2
312  ELSE IF( n.LT.smlsiz ) THEN
313  info = -3
314  ELSE IF( nrhs.LT.1 ) THEN
315  info = -4
316  ELSE IF( ldb.LT.n ) THEN
317  info = -6
318  ELSE IF( ldbx.LT.n ) THEN
319  info = -8
320  ELSE IF( ldu.LT.n ) THEN
321  info = -10
322  ELSE IF( ldgcol.LT.n ) THEN
323  info = -19
324  END IF
325  IF( info.NE.0 ) THEN
326  CALL xerbla( 'ZLALSA', -info )
327  RETURN
328  END IF
329 *
330 * Book-keeping and setting up the computation tree.
331 *
332  inode = 1
333  ndiml = inode + n
334  ndimr = ndiml + n
335 *
336  CALL dlasdt( n, nlvl, nd, iwork( inode ), iwork( ndiml ),
337  \$ iwork( ndimr ), smlsiz )
338 *
339 * The following code applies back the left singular vector factors.
340 * For applying back the right singular vector factors, go to 170.
341 *
342  IF( icompq.EQ.1 ) THEN
343  GO TO 170
344  END IF
345 *
346 * The nodes on the bottom level of the tree were solved
347 * by DLASDQ. The corresponding left and right singular vector
348 * matrices are in explicit form. First apply back the left
349 * singular vector matrices.
350 *
351  ndb1 = ( nd+1 ) / 2
352  DO 130 i = ndb1, nd
353 *
354 * IC : center row of each node
355 * NL : number of rows of left subproblem
356 * NR : number of rows of right subproblem
357 * NLF: starting row of the left subproblem
358 * NRF: starting row of the right subproblem
359 *
360  i1 = i - 1
361  ic = iwork( inode+i1 )
362  nl = iwork( ndiml+i1 )
363  nr = iwork( ndimr+i1 )
364  nlf = ic - nl
365  nrf = ic + 1
366 *
367 * Since B and BX are complex, the following call to DGEMM
368 * is performed in two steps (real and imaginary parts).
369 *
370 * CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
371 * \$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
372 *
373  j = nl*nrhs*2
374  DO 20 jcol = 1, nrhs
375  DO 10 jrow = nlf, nlf + nl - 1
376  j = j + 1
377  rwork( j ) = dble( b( jrow, jcol ) )
378  10 CONTINUE
379  20 CONTINUE
380  CALL dgemm( 'T', 'N', nl, nrhs, nl, one, u( nlf, 1 ), ldu,
381  \$ rwork( 1+nl*nrhs*2 ), nl, zero, rwork( 1 ), nl )
382  j = nl*nrhs*2
383  DO 40 jcol = 1, nrhs
384  DO 30 jrow = nlf, nlf + nl - 1
385  j = j + 1
386  rwork( j ) = dimag( b( jrow, jcol ) )
387  30 CONTINUE
388  40 CONTINUE
389  CALL dgemm( 'T', 'N', nl, nrhs, nl, one, u( nlf, 1 ), ldu,
390  \$ rwork( 1+nl*nrhs*2 ), nl, zero, rwork( 1+nl*nrhs ),
391  \$ nl )
392  jreal = 0
393  jimag = nl*nrhs
394  DO 60 jcol = 1, nrhs
395  DO 50 jrow = nlf, nlf + nl - 1
396  jreal = jreal + 1
397  jimag = jimag + 1
398  bx( jrow, jcol ) = dcmplx( rwork( jreal ),
399  \$ rwork( jimag ) )
400  50 CONTINUE
401  60 CONTINUE
402 *
403 * Since B and BX are complex, the following call to DGEMM
404 * is performed in two steps (real and imaginary parts).
405 *
406 * CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
407 * \$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
408 *
409  j = nr*nrhs*2
410  DO 80 jcol = 1, nrhs
411  DO 70 jrow = nrf, nrf + nr - 1
412  j = j + 1
413  rwork( j ) = dble( b( jrow, jcol ) )
414  70 CONTINUE
415  80 CONTINUE
416  CALL dgemm( 'T', 'N', nr, nrhs, nr, one, u( nrf, 1 ), ldu,
417  \$ rwork( 1+nr*nrhs*2 ), nr, zero, rwork( 1 ), nr )
418  j = nr*nrhs*2
419  DO 100 jcol = 1, nrhs
420  DO 90 jrow = nrf, nrf + nr - 1
421  j = j + 1
422  rwork( j ) = dimag( b( jrow, jcol ) )
423  90 CONTINUE
424  100 CONTINUE
425  CALL dgemm( 'T', 'N', nr, nrhs, nr, one, u( nrf, 1 ), ldu,
426  \$ rwork( 1+nr*nrhs*2 ), nr, zero, rwork( 1+nr*nrhs ),
427  \$ nr )
428  jreal = 0
429  jimag = nr*nrhs
430  DO 120 jcol = 1, nrhs
431  DO 110 jrow = nrf, nrf + nr - 1
432  jreal = jreal + 1
433  jimag = jimag + 1
434  bx( jrow, jcol ) = dcmplx( rwork( jreal ),
435  \$ rwork( jimag ) )
436  110 CONTINUE
437  120 CONTINUE
438 *
439  130 CONTINUE
440 *
441 * Next copy the rows of B that correspond to unchanged rows
442 * in the bidiagonal matrix to BX.
443 *
444  DO 140 i = 1, nd
445  ic = iwork( inode+i-1 )
446  CALL zcopy( nrhs, b( ic, 1 ), ldb, bx( ic, 1 ), ldbx )
447  140 CONTINUE
448 *
449 * Finally go through the left singular vector matrices of all
450 * the other subproblems bottom-up on the tree.
451 *
452  j = 2**nlvl
453  sqre = 0
454 *
455  DO 160 lvl = nlvl, 1, -1
456  lvl2 = 2*lvl - 1
457 *
458 * find the first node LF and last node LL on
459 * the current level LVL
460 *
461  IF( lvl.EQ.1 ) THEN
462  lf = 1
463  ll = 1
464  ELSE
465  lf = 2**( lvl-1 )
466  ll = 2*lf - 1
467  END IF
468  DO 150 i = lf, ll
469  im1 = i - 1
470  ic = iwork( inode+im1 )
471  nl = iwork( ndiml+im1 )
472  nr = iwork( ndimr+im1 )
473  nlf = ic - nl
474  nrf = ic + 1
475  j = j - 1
476  CALL zlals0( icompq, nl, nr, sqre, nrhs, bx( nlf, 1 ), ldbx,
477  \$ b( nlf, 1 ), ldb, perm( nlf, lvl ),
478  \$ givptr( j ), givcol( nlf, lvl2 ), ldgcol,
479  \$ givnum( nlf, lvl2 ), ldu, poles( nlf, lvl2 ),
480  \$ difl( nlf, lvl ), difr( nlf, lvl2 ),
481  \$ z( nlf, lvl ), k( j ), c( j ), s( j ), rwork,
482  \$ info )
483  150 CONTINUE
484  160 CONTINUE
485  GO TO 330
486 *
487 * ICOMPQ = 1: applying back the right singular vector factors.
488 *
489  170 CONTINUE
490 *
491 * First now go through the right singular vector matrices of all
492 * the tree nodes top-down.
493 *
494  j = 0
495  DO 190 lvl = 1, nlvl
496  lvl2 = 2*lvl - 1
497 *
498 * Find the first node LF and last node LL on
499 * the current level LVL.
500 *
501  IF( lvl.EQ.1 ) THEN
502  lf = 1
503  ll = 1
504  ELSE
505  lf = 2**( lvl-1 )
506  ll = 2*lf - 1
507  END IF
508  DO 180 i = ll, lf, -1
509  im1 = i - 1
510  ic = iwork( inode+im1 )
511  nl = iwork( ndiml+im1 )
512  nr = iwork( ndimr+im1 )
513  nlf = ic - nl
514  nrf = ic + 1
515  IF( i.EQ.ll ) THEN
516  sqre = 0
517  ELSE
518  sqre = 1
519  END IF
520  j = j + 1
521  CALL zlals0( icompq, nl, nr, sqre, nrhs, b( nlf, 1 ), ldb,
522  \$ bx( nlf, 1 ), ldbx, perm( nlf, lvl ),
523  \$ givptr( j ), givcol( nlf, lvl2 ), ldgcol,
524  \$ givnum( nlf, lvl2 ), ldu, poles( nlf, lvl2 ),
525  \$ difl( nlf, lvl ), difr( nlf, lvl2 ),
526  \$ z( nlf, lvl ), k( j ), c( j ), s( j ), rwork,
527  \$ info )
528  180 CONTINUE
529  190 CONTINUE
530 *
531 * The nodes on the bottom level of the tree were solved
532 * by DLASDQ. The corresponding right singular vector
533 * matrices are in explicit form. Apply them back.
534 *
535  ndb1 = ( nd+1 ) / 2
536  DO 320 i = ndb1, nd
537  i1 = i - 1
538  ic = iwork( inode+i1 )
539  nl = iwork( ndiml+i1 )
540  nr = iwork( ndimr+i1 )
541  nlp1 = nl + 1
542  IF( i.EQ.nd ) THEN
543  nrp1 = nr
544  ELSE
545  nrp1 = nr + 1
546  END IF
547  nlf = ic - nl
548  nrf = ic + 1
549 *
550 * Since B and BX are complex, the following call to DGEMM is
551 * performed in two steps (real and imaginary parts).
552 *
553 * CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU,
554 * \$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
555 *
556  j = nlp1*nrhs*2
557  DO 210 jcol = 1, nrhs
558  DO 200 jrow = nlf, nlf + nlp1 - 1
559  j = j + 1
560  rwork( j ) = dble( b( jrow, jcol ) )
561  200 CONTINUE
562  210 CONTINUE
563  CALL dgemm( 'T', 'N', nlp1, nrhs, nlp1, one, vt( nlf, 1 ), ldu,
564  \$ rwork( 1+nlp1*nrhs*2 ), nlp1, zero, rwork( 1 ),
565  \$ nlp1 )
566  j = nlp1*nrhs*2
567  DO 230 jcol = 1, nrhs
568  DO 220 jrow = nlf, nlf + nlp1 - 1
569  j = j + 1
570  rwork( j ) = dimag( b( jrow, jcol ) )
571  220 CONTINUE
572  230 CONTINUE
573  CALL dgemm( 'T', 'N', nlp1, nrhs, nlp1, one, vt( nlf, 1 ), ldu,
574  \$ rwork( 1+nlp1*nrhs*2 ), nlp1, zero,
575  \$ rwork( 1+nlp1*nrhs ), nlp1 )
576  jreal = 0
577  jimag = nlp1*nrhs
578  DO 250 jcol = 1, nrhs
579  DO 240 jrow = nlf, nlf + nlp1 - 1
580  jreal = jreal + 1
581  jimag = jimag + 1
582  bx( jrow, jcol ) = dcmplx( rwork( jreal ),
583  \$ rwork( jimag ) )
584  240 CONTINUE
585  250 CONTINUE
586 *
587 * Since B and BX are complex, the following call to DGEMM is
588 * performed in two steps (real and imaginary parts).
589 *
590 * CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU,
591 * \$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
592 *
593  j = nrp1*nrhs*2
594  DO 270 jcol = 1, nrhs
595  DO 260 jrow = nrf, nrf + nrp1 - 1
596  j = j + 1
597  rwork( j ) = dble( b( jrow, jcol ) )
598  260 CONTINUE
599  270 CONTINUE
600  CALL dgemm( 'T', 'N', nrp1, nrhs, nrp1, one, vt( nrf, 1 ), ldu,
601  \$ rwork( 1+nrp1*nrhs*2 ), nrp1, zero, rwork( 1 ),
602  \$ nrp1 )
603  j = nrp1*nrhs*2
604  DO 290 jcol = 1, nrhs
605  DO 280 jrow = nrf, nrf + nrp1 - 1
606  j = j + 1
607  rwork( j ) = dimag( b( jrow, jcol ) )
608  280 CONTINUE
609  290 CONTINUE
610  CALL dgemm( 'T', 'N', nrp1, nrhs, nrp1, one, vt( nrf, 1 ), ldu,
611  \$ rwork( 1+nrp1*nrhs*2 ), nrp1, zero,
612  \$ rwork( 1+nrp1*nrhs ), nrp1 )
613  jreal = 0
614  jimag = nrp1*nrhs
615  DO 310 jcol = 1, nrhs
616  DO 300 jrow = nrf, nrf + nrp1 - 1
617  jreal = jreal + 1
618  jimag = jimag + 1
619  bx( jrow, jcol ) = dcmplx( rwork( jreal ),
620  \$ rwork( jimag ) )
621  300 CONTINUE
622  310 CONTINUE
623 *
624  320 CONTINUE
625 *
626  330 CONTINUE
627 *
628  RETURN
629 *
630 * End of ZLALSA
631 *
subroutine dlasdt(N, LVL, ND, INODE, NDIML, NDIMR, MSUB)
DLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.
Definition: dlasdt.f:105
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zlals0(ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, RWORK, INFO)
ZLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer...
Definition: zlals0.f:270
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
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