LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ slals0()

subroutine slals0 ( integer  ICOMPQ,
integer  NL,
integer  NR,
integer  SQRE,
integer  NRHS,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( ldbx, * )  BX,
integer  LDBX,
integer, dimension( * )  PERM,
integer  GIVPTR,
integer, dimension( ldgcol, * )  GIVCOL,
integer  LDGCOL,
real, dimension( ldgnum, * )  GIVNUM,
integer  LDGNUM,
real, dimension( ldgnum, * )  POLES,
real, dimension( * )  DIFL,
real, dimension( ldgnum, * )  DIFR,
real, dimension( * )  Z,
integer  K,
real  C,
real  S,
real, dimension( * )  WORK,
integer  INFO 
)

SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Download SLALS0 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SLALS0 applies back the multiplying factors of either the left or the
 right singular vector matrix of a diagonal matrix appended by a row
 to the right hand side matrix B in solving the least squares problem
 using the divide-and-conquer SVD approach.

 For the left singular vector matrix, three types of orthogonal
 matrices are involved:

 (1L) Givens rotations: the number of such rotations is GIVPTR; the
      pairs of columns/rows they were applied to are stored in GIVCOL;
      and the C- and S-values of these rotations are stored in GIVNUM.

 (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
      J-th row.

 (3L) The left singular vector matrix of the remaining matrix.

 For the right singular vector matrix, four types of orthogonal
 matrices are involved:

 (1R) The right singular vector matrix of the remaining matrix.

 (2R) If SQRE = 1, one extra Givens rotation to generate the right
      null space.

 (3R) The inverse transformation of (2L).

 (4R) The inverse transformation of (1L).
Parameters
[in]ICOMPQ
          ICOMPQ is INTEGER
         Specifies whether singular vectors are to be computed in
         factored form:
         = 0: Left singular vector matrix.
         = 1: Right singular vector matrix.
[in]NL
          NL is INTEGER
         The row dimension of the upper block. NL >= 1.
[in]NR
          NR is INTEGER
         The row dimension of the lower block. NR >= 1.
[in]SQRE
          SQRE is INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.
[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. LDB must be at least
         max(1,MAX( M, N ) ).
[out]BX
          BX is REAL array, dimension ( LDBX, NRHS )
[in]LDBX
          LDBX is INTEGER
         The leading dimension of BX.
[in]PERM
          PERM is INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) applied
         to the two blocks.
[in]GIVPTR
          GIVPTR is INTEGER
         The number of Givens rotations which took place in this
         subproblem.
[in]GIVCOL
          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of rows/columns
         involved in a Givens rotation.
[in]LDGCOL
          LDGCOL is INTEGER
         The leading dimension of GIVCOL, must be at least N.
[in]GIVNUM
          GIVNUM is REAL array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value used in the
         corresponding Givens rotation.
[in]LDGNUM
          LDGNUM is INTEGER
         The leading dimension of arrays DIFR, POLES and
         GIVNUM, must be at least K.
[in]POLES
          POLES is REAL array, dimension ( LDGNUM, 2 )
         On entry, POLES(1:K, 1) contains the new singular
         values obtained from solving the secular equation, and
         POLES(1:K, 2) is an array containing the poles in the secular
         equation.
[in]DIFL
          DIFL is REAL array, dimension ( K ).
         On entry, DIFL(I) is the distance between I-th updated
         (undeflated) singular value and the I-th (undeflated) old
         singular value.
[in]DIFR
          DIFR is REAL array, dimension ( LDGNUM, 2 ).
         On entry, DIFR(I, 1) contains the distances between I-th
         updated (undeflated) singular value and the I+1-th
         (undeflated) old singular value. And DIFR(I, 2) is the
         normalizing factor for the I-th right singular vector.
[in]Z
          Z is REAL array, dimension ( K )
         Contain the components of the deflation-adjusted updating row
         vector.
[in]K
          K is INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.
[in]C
          C is REAL
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.
[in]S
          S is REAL
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.
[out]WORK
          WORK is REAL array, dimension ( K )
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Definition at line 265 of file slals0.f.

268 *
269 * -- LAPACK computational routine --
270 * -- LAPACK is a software package provided by Univ. of Tennessee, --
271 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
272 *
273 * .. Scalar Arguments ..
274  INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
275  $ LDGNUM, NL, NR, NRHS, SQRE
276  REAL C, S
277 * ..
278 * .. Array Arguments ..
279  INTEGER GIVCOL( LDGCOL, * ), PERM( * )
280  REAL B( LDB, * ), BX( LDBX, * ), DIFL( * ),
281  $ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
282  $ POLES( LDGNUM, * ), WORK( * ), Z( * )
283 * ..
284 *
285 * =====================================================================
286 *
287 * .. Parameters ..
288  REAL ONE, ZERO, NEGONE
289  parameter( one = 1.0e0, zero = 0.0e0, negone = -1.0e0 )
290 * ..
291 * .. Local Scalars ..
292  INTEGER I, J, M, N, NLP1
293  REAL DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
294 * ..
295 * .. External Subroutines ..
296  EXTERNAL scopy, sgemv, slacpy, slascl, srot, sscal,
297  $ xerbla
298 * ..
299 * .. External Functions ..
300  REAL SLAMC3, SNRM2
301  EXTERNAL slamc3, snrm2
302 * ..
303 * .. Intrinsic Functions ..
304  INTRINSIC max
305 * ..
306 * .. Executable Statements ..
307 *
308 * Test the input parameters.
309 *
310  info = 0
311  n = nl + nr + 1
312 *
313  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
314  info = -1
315  ELSE IF( nl.LT.1 ) THEN
316  info = -2
317  ELSE IF( nr.LT.1 ) THEN
318  info = -3
319  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
320  info = -4
321  ELSE IF( nrhs.LT.1 ) THEN
322  info = -5
323  ELSE IF( ldb.LT.n ) THEN
324  info = -7
325  ELSE IF( ldbx.LT.n ) THEN
326  info = -9
327  ELSE IF( givptr.LT.0 ) THEN
328  info = -11
329  ELSE IF( ldgcol.LT.n ) THEN
330  info = -13
331  ELSE IF( ldgnum.LT.n ) THEN
332  info = -15
333  ELSE IF( k.LT.1 ) THEN
334  info = -20
335  END IF
336  IF( info.NE.0 ) THEN
337  CALL xerbla( 'SLALS0', -info )
338  RETURN
339  END IF
340 *
341  m = n + sqre
342  nlp1 = nl + 1
343 *
344  IF( icompq.EQ.0 ) THEN
345 *
346 * Apply back orthogonal transformations from the left.
347 *
348 * Step (1L): apply back the Givens rotations performed.
349 *
350  DO 10 i = 1, givptr
351  CALL srot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
352  $ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
353  $ givnum( i, 1 ) )
354  10 CONTINUE
355 *
356 * Step (2L): permute rows of B.
357 *
358  CALL scopy( nrhs, b( nlp1, 1 ), ldb, bx( 1, 1 ), ldbx )
359  DO 20 i = 2, n
360  CALL scopy( nrhs, b( perm( i ), 1 ), ldb, bx( i, 1 ), ldbx )
361  20 CONTINUE
362 *
363 * Step (3L): apply the inverse of the left singular vector
364 * matrix to BX.
365 *
366  IF( k.EQ.1 ) THEN
367  CALL scopy( nrhs, bx, ldbx, b, ldb )
368  IF( z( 1 ).LT.zero ) THEN
369  CALL sscal( nrhs, negone, b, ldb )
370  END IF
371  ELSE
372  DO 50 j = 1, k
373  diflj = difl( j )
374  dj = poles( j, 1 )
375  dsigj = -poles( j, 2 )
376  IF( j.LT.k ) THEN
377  difrj = -difr( j, 1 )
378  dsigjp = -poles( j+1, 2 )
379  END IF
380  IF( ( z( j ).EQ.zero ) .OR. ( poles( j, 2 ).EQ.zero ) )
381  $ THEN
382  work( j ) = zero
383  ELSE
384  work( j ) = -poles( j, 2 )*z( j ) / diflj /
385  $ ( poles( j, 2 )+dj )
386  END IF
387  DO 30 i = 1, j - 1
388  IF( ( z( i ).EQ.zero ) .OR.
389  $ ( poles( i, 2 ).EQ.zero ) ) THEN
390  work( i ) = zero
391  ELSE
392  work( i ) = poles( i, 2 )*z( i ) /
393  $ ( slamc3( poles( i, 2 ), dsigj )-
394  $ diflj ) / ( poles( i, 2 )+dj )
395  END IF
396  30 CONTINUE
397  DO 40 i = j + 1, k
398  IF( ( z( i ).EQ.zero ) .OR.
399  $ ( poles( i, 2 ).EQ.zero ) ) THEN
400  work( i ) = zero
401  ELSE
402  work( i ) = poles( i, 2 )*z( i ) /
403  $ ( slamc3( poles( i, 2 ), dsigjp )+
404  $ difrj ) / ( poles( i, 2 )+dj )
405  END IF
406  40 CONTINUE
407  work( 1 ) = negone
408  temp = snrm2( k, work, 1 )
409  CALL sgemv( 'T', k, nrhs, one, bx, ldbx, work, 1, zero,
410  $ b( j, 1 ), ldb )
411  CALL slascl( 'G', 0, 0, temp, one, 1, nrhs, b( j, 1 ),
412  $ ldb, info )
413  50 CONTINUE
414  END IF
415 *
416 * Move the deflated rows of BX to B also.
417 *
418  IF( k.LT.max( m, n ) )
419  $ CALL slacpy( 'A', n-k, nrhs, bx( k+1, 1 ), ldbx,
420  $ b( k+1, 1 ), ldb )
421  ELSE
422 *
423 * Apply back the right orthogonal transformations.
424 *
425 * Step (1R): apply back the new right singular vector matrix
426 * to B.
427 *
428  IF( k.EQ.1 ) THEN
429  CALL scopy( nrhs, b, ldb, bx, ldbx )
430  ELSE
431  DO 80 j = 1, k
432  dsigj = poles( j, 2 )
433  IF( z( j ).EQ.zero ) THEN
434  work( j ) = zero
435  ELSE
436  work( j ) = -z( j ) / difl( j ) /
437  $ ( dsigj+poles( j, 1 ) ) / difr( j, 2 )
438  END IF
439  DO 60 i = 1, j - 1
440  IF( z( j ).EQ.zero ) THEN
441  work( i ) = zero
442  ELSE
443  work( i ) = z( j ) / ( slamc3( dsigj, -poles( i+1,
444  $ 2 ) )-difr( i, 1 ) ) /
445  $ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
446  END IF
447  60 CONTINUE
448  DO 70 i = j + 1, k
449  IF( z( j ).EQ.zero ) THEN
450  work( i ) = zero
451  ELSE
452  work( i ) = z( j ) / ( slamc3( dsigj, -poles( i,
453  $ 2 ) )-difl( i ) ) /
454  $ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
455  END IF
456  70 CONTINUE
457  CALL sgemv( 'T', k, nrhs, one, b, ldb, work, 1, zero,
458  $ bx( j, 1 ), ldbx )
459  80 CONTINUE
460  END IF
461 *
462 * Step (2R): if SQRE = 1, apply back the rotation that is
463 * related to the right null space of the subproblem.
464 *
465  IF( sqre.EQ.1 ) THEN
466  CALL scopy( nrhs, b( m, 1 ), ldb, bx( m, 1 ), ldbx )
467  CALL srot( nrhs, bx( 1, 1 ), ldbx, bx( m, 1 ), ldbx, c, s )
468  END IF
469  IF( k.LT.max( m, n ) )
470  $ CALL slacpy( 'A', n-k, nrhs, b( k+1, 1 ), ldb, bx( k+1, 1 ),
471  $ ldbx )
472 *
473 * Step (3R): permute rows of B.
474 *
475  CALL scopy( nrhs, bx( 1, 1 ), ldbx, b( nlp1, 1 ), ldb )
476  IF( sqre.EQ.1 ) THEN
477  CALL scopy( nrhs, bx( m, 1 ), ldbx, b( m, 1 ), ldb )
478  END IF
479  DO 90 i = 2, n
480  CALL scopy( nrhs, bx( i, 1 ), ldbx, b( perm( i ), 1 ), ldb )
481  90 CONTINUE
482 *
483 * Step (4R): apply back the Givens rotations performed.
484 *
485  DO 100 i = givptr, 1, -1
486  CALL srot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
487  $ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
488  $ -givnum( i, 1 ) )
489  100 CONTINUE
490  END IF
491 *
492  RETURN
493 *
494 * End of SLALS0
495 *
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:143
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:92
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
real function slamc3(A, B)
SLAMC3
Definition: slamch.f:169
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