LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ slasd2()

subroutine slasd2 ( integer  NL,
integer  NR,
integer  SQRE,
integer  K,
real, dimension( * )  D,
real, dimension( * )  Z,
real  ALPHA,
real  BETA,
real, dimension( ldu, * )  U,
integer  LDU,
real, dimension( ldvt, * )  VT,
integer  LDVT,
real, dimension( * )  DSIGMA,
real, dimension( ldu2, * )  U2,
integer  LDU2,
real, dimension( ldvt2, * )  VT2,
integer  LDVT2,
integer, dimension( * )  IDXP,
integer, dimension( * )  IDX,
integer, dimension( * )  IDXC,
integer, dimension( * )  IDXQ,
integer, dimension( * )  COLTYP,
integer  INFO 
)

SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.

Download SLASD2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SLASD2 merges the two sets of singular values together into a single
 sorted set.  Then it tries to deflate the size of the problem.
 There are two ways in which deflation can occur:  when two or more
 singular values are close together or if there is a tiny entry in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.

 SLASD2 is called from SLASD1.
Parameters
[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 N = NL + NR + 1 rows and
         M = N + SQRE >= N columns.
[out]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,out]D
          D is REAL array, dimension (N)
         On entry D contains the singular values of the two submatrices
         to be combined.  On exit D contains the trailing (N-K) updated
         singular values (those which were deflated) sorted into
         increasing order.
[out]Z
          Z is REAL array, dimension (N)
         On exit Z contains the updating row vector in the secular
         equation.
[in]ALPHA
          ALPHA is REAL
         Contains the diagonal element associated with the added row.
[in]BETA
          BETA is REAL
         Contains the off-diagonal element associated with the added
         row.
[in,out]U
          U is REAL array, dimension (LDU,N)
         On entry U contains the left singular vectors of two
         submatrices in the two square blocks with corners at (1,1),
         (NL, NL), and (NL+2, NL+2), (N,N).
         On exit U contains the trailing (N-K) updated left singular
         vectors (those which were deflated) in its last N-K columns.
[in]LDU
          LDU is INTEGER
         The leading dimension of the array U.  LDU >= N.
[in,out]VT
          VT is REAL array, dimension (LDVT,M)
         On entry VT**T contains the right singular vectors of two
         submatrices in the two square blocks with corners at (1,1),
         (NL+1, NL+1), and (NL+2, NL+2), (M,M).
         On exit VT**T contains the trailing (N-K) updated right singular
         vectors (those which were deflated) in its last N-K columns.
         In case SQRE =1, the last row of VT spans the right null
         space.
[in]LDVT
          LDVT is INTEGER
         The leading dimension of the array VT.  LDVT >= M.
[out]DSIGMA
          DSIGMA is REAL array, dimension (N)
         Contains a copy of the diagonal elements (K-1 singular values
         and one zero) in the secular equation.
[out]U2
          U2 is REAL array, dimension (LDU2,N)
         Contains a copy of the first K-1 left singular vectors which
         will be used by SLASD3 in a matrix multiply (SGEMM) to solve
         for the new left singular vectors. U2 is arranged into four
         blocks. The first block contains a column with 1 at NL+1 and
         zero everywhere else; the second block contains non-zero
         entries only at and above NL; the third contains non-zero
         entries only below NL+1; and the fourth is dense.
[in]LDU2
          LDU2 is INTEGER
         The leading dimension of the array U2.  LDU2 >= N.
[out]VT2
          VT2 is REAL array, dimension (LDVT2,N)
         VT2**T contains a copy of the first K right singular vectors
         which will be used by SLASD3 in a matrix multiply (SGEMM) to
         solve for the new right singular vectors. VT2 is arranged into
         three blocks. The first block contains a row that corresponds
         to the special 0 diagonal element in SIGMA; the second block
         contains non-zeros only at and before NL +1; the third block
         contains non-zeros only at and after  NL +2.
[in]LDVT2
          LDVT2 is INTEGER
         The leading dimension of the array VT2.  LDVT2 >= M.
[out]IDXP
          IDXP is INTEGER array, dimension (N)
         This will contain the permutation used to place deflated
         values of D at the end of the array. On output IDXP(2:K)
         points to the nondeflated D-values and IDXP(K+1:N)
         points to the deflated singular values.
[out]IDX
          IDX is INTEGER array, dimension (N)
         This will contain the permutation used to sort the contents of
         D into ascending order.
[out]IDXC
          IDXC is INTEGER array, dimension (N)
         This will contain the permutation used to arrange the columns
         of the deflated U matrix into three groups:  the first group
         contains non-zero entries only at and above NL, the second
         contains non-zero entries only below NL+2, and the third is
         dense.
[in,out]IDXQ
          IDXQ is INTEGER array, dimension (N)
         This contains the permutation which separately sorts the two
         sub-problems in D into ascending order.  Note that entries in
         the first hlaf of this permutation must first be moved one
         position backward; and entries in the second half
         must first have NL+1 added to their values.
[out]COLTYP
          COLTYP is INTEGER array, dimension (N)
         As workspace, this will contain a label which will indicate
         which of the following types a column in the U2 matrix or a
         row in the VT2 matrix is:
         1 : non-zero in the upper half only
         2 : non-zero in the lower half only
         3 : dense
         4 : deflated

         On exit, it is an array of dimension 4, with COLTYP(I) being
         the dimension of the I-th type columns.
[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 Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 266 of file slasd2.f.

269 *
270 * -- LAPACK auxiliary routine --
271 * -- LAPACK is a software package provided by Univ. of Tennessee, --
272 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273 *
274 * .. Scalar Arguments ..
275  INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
276  REAL ALPHA, BETA
277 * ..
278 * .. Array Arguments ..
279  INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
280  $ IDXQ( * )
281  REAL D( * ), DSIGMA( * ), U( LDU, * ),
282  $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
283  $ Z( * )
284 * ..
285 *
286 * =====================================================================
287 *
288 * .. Parameters ..
289  REAL ZERO, ONE, TWO, EIGHT
290  parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0,
291  $ eight = 8.0e+0 )
292 * ..
293 * .. Local Arrays ..
294  INTEGER CTOT( 4 ), PSM( 4 )
295 * ..
296 * .. Local Scalars ..
297  INTEGER CT, I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M,
298  $ N, NLP1, NLP2
299  REAL C, EPS, HLFTOL, S, TAU, TOL, Z1
300 * ..
301 * .. External Functions ..
302  REAL SLAMCH, SLAPY2
303  EXTERNAL slamch, slapy2
304 * ..
305 * .. External Subroutines ..
306  EXTERNAL scopy, slacpy, slamrg, slaset, srot, xerbla
307 * ..
308 * .. Intrinsic Functions ..
309  INTRINSIC abs, max
310 * ..
311 * .. Executable Statements ..
312 *
313 * Test the input parameters.
314 *
315  info = 0
316 *
317  IF( nl.LT.1 ) THEN
318  info = -1
319  ELSE IF( nr.LT.1 ) THEN
320  info = -2
321  ELSE IF( ( sqre.NE.1 ) .AND. ( sqre.NE.0 ) ) THEN
322  info = -3
323  END IF
324 *
325  n = nl + nr + 1
326  m = n + sqre
327 *
328  IF( ldu.LT.n ) THEN
329  info = -10
330  ELSE IF( ldvt.LT.m ) THEN
331  info = -12
332  ELSE IF( ldu2.LT.n ) THEN
333  info = -15
334  ELSE IF( ldvt2.LT.m ) THEN
335  info = -17
336  END IF
337  IF( info.NE.0 ) THEN
338  CALL xerbla( 'SLASD2', -info )
339  RETURN
340  END IF
341 *
342  nlp1 = nl + 1
343  nlp2 = nl + 2
344 *
345 * Generate the first part of the vector Z; and move the singular
346 * values in the first part of D one position backward.
347 *
348  z1 = alpha*vt( nlp1, nlp1 )
349  z( 1 ) = z1
350  DO 10 i = nl, 1, -1
351  z( i+1 ) = alpha*vt( i, nlp1 )
352  d( i+1 ) = d( i )
353  idxq( i+1 ) = idxq( i ) + 1
354  10 CONTINUE
355 *
356 * Generate the second part of the vector Z.
357 *
358  DO 20 i = nlp2, m
359  z( i ) = beta*vt( i, nlp2 )
360  20 CONTINUE
361 *
362 * Initialize some reference arrays.
363 *
364  DO 30 i = 2, nlp1
365  coltyp( i ) = 1
366  30 CONTINUE
367  DO 40 i = nlp2, n
368  coltyp( i ) = 2
369  40 CONTINUE
370 *
371 * Sort the singular values into increasing order
372 *
373  DO 50 i = nlp2, n
374  idxq( i ) = idxq( i ) + nlp1
375  50 CONTINUE
376 *
377 * DSIGMA, IDXC, IDXC, and the first column of U2
378 * are used as storage space.
379 *
380  DO 60 i = 2, n
381  dsigma( i ) = d( idxq( i ) )
382  u2( i, 1 ) = z( idxq( i ) )
383  idxc( i ) = coltyp( idxq( i ) )
384  60 CONTINUE
385 *
386  CALL slamrg( nl, nr, dsigma( 2 ), 1, 1, idx( 2 ) )
387 *
388  DO 70 i = 2, n
389  idxi = 1 + idx( i )
390  d( i ) = dsigma( idxi )
391  z( i ) = u2( idxi, 1 )
392  coltyp( i ) = idxc( idxi )
393  70 CONTINUE
394 *
395 * Calculate the allowable deflation tolerance
396 *
397  eps = slamch( 'Epsilon' )
398  tol = max( abs( alpha ), abs( beta ) )
399  tol = eight*eps*max( abs( d( n ) ), tol )
400 *
401 * There are 2 kinds of deflation -- first a value in the z-vector
402 * is small, second two (or more) singular values are very close
403 * together (their difference is small).
404 *
405 * If the value in the z-vector is small, we simply permute the
406 * array so that the corresponding singular value is moved to the
407 * end.
408 *
409 * If two values in the D-vector are close, we perform a two-sided
410 * rotation designed to make one of the corresponding z-vector
411 * entries zero, and then permute the array so that the deflated
412 * singular value is moved to the end.
413 *
414 * If there are multiple singular values then the problem deflates.
415 * Here the number of equal singular values are found. As each equal
416 * singular value is found, an elementary reflector is computed to
417 * rotate the corresponding singular subspace so that the
418 * corresponding components of Z are zero in this new basis.
419 *
420  k = 1
421  k2 = n + 1
422  DO 80 j = 2, n
423  IF( abs( z( j ) ).LE.tol ) THEN
424 *
425 * Deflate due to small z component.
426 *
427  k2 = k2 - 1
428  idxp( k2 ) = j
429  coltyp( j ) = 4
430  IF( j.EQ.n )
431  $ GO TO 120
432  ELSE
433  jprev = j
434  GO TO 90
435  END IF
436  80 CONTINUE
437  90 CONTINUE
438  j = jprev
439  100 CONTINUE
440  j = j + 1
441  IF( j.GT.n )
442  $ GO TO 110
443  IF( abs( z( j ) ).LE.tol ) THEN
444 *
445 * Deflate due to small z component.
446 *
447  k2 = k2 - 1
448  idxp( k2 ) = j
449  coltyp( j ) = 4
450  ELSE
451 *
452 * Check if singular values are close enough to allow deflation.
453 *
454  IF( abs( d( j )-d( jprev ) ).LE.tol ) THEN
455 *
456 * Deflation is possible.
457 *
458  s = z( jprev )
459  c = z( j )
460 *
461 * Find sqrt(a**2+b**2) without overflow or
462 * destructive underflow.
463 *
464  tau = slapy2( c, s )
465  c = c / tau
466  s = -s / tau
467  z( j ) = tau
468  z( jprev ) = zero
469 *
470 * Apply back the Givens rotation to the left and right
471 * singular vector matrices.
472 *
473  idxjp = idxq( idx( jprev )+1 )
474  idxj = idxq( idx( j )+1 )
475  IF( idxjp.LE.nlp1 ) THEN
476  idxjp = idxjp - 1
477  END IF
478  IF( idxj.LE.nlp1 ) THEN
479  idxj = idxj - 1
480  END IF
481  CALL srot( n, u( 1, idxjp ), 1, u( 1, idxj ), 1, c, s )
482  CALL srot( m, vt( idxjp, 1 ), ldvt, vt( idxj, 1 ), ldvt, c,
483  $ s )
484  IF( coltyp( j ).NE.coltyp( jprev ) ) THEN
485  coltyp( j ) = 3
486  END IF
487  coltyp( jprev ) = 4
488  k2 = k2 - 1
489  idxp( k2 ) = jprev
490  jprev = j
491  ELSE
492  k = k + 1
493  u2( k, 1 ) = z( jprev )
494  dsigma( k ) = d( jprev )
495  idxp( k ) = jprev
496  jprev = j
497  END IF
498  END IF
499  GO TO 100
500  110 CONTINUE
501 *
502 * Record the last singular value.
503 *
504  k = k + 1
505  u2( k, 1 ) = z( jprev )
506  dsigma( k ) = d( jprev )
507  idxp( k ) = jprev
508 *
509  120 CONTINUE
510 *
511 * Count up the total number of the various types of columns, then
512 * form a permutation which positions the four column types into
513 * four groups of uniform structure (although one or more of these
514 * groups may be empty).
515 *
516  DO 130 j = 1, 4
517  ctot( j ) = 0
518  130 CONTINUE
519  DO 140 j = 2, n
520  ct = coltyp( j )
521  ctot( ct ) = ctot( ct ) + 1
522  140 CONTINUE
523 *
524 * PSM(*) = Position in SubMatrix (of types 1 through 4)
525 *
526  psm( 1 ) = 2
527  psm( 2 ) = 2 + ctot( 1 )
528  psm( 3 ) = psm( 2 ) + ctot( 2 )
529  psm( 4 ) = psm( 3 ) + ctot( 3 )
530 *
531 * Fill out the IDXC array so that the permutation which it induces
532 * will place all type-1 columns first, all type-2 columns next,
533 * then all type-3's, and finally all type-4's, starting from the
534 * second column. This applies similarly to the rows of VT.
535 *
536  DO 150 j = 2, n
537  jp = idxp( j )
538  ct = coltyp( jp )
539  idxc( psm( ct ) ) = j
540  psm( ct ) = psm( ct ) + 1
541  150 CONTINUE
542 *
543 * Sort the singular values and corresponding singular vectors into
544 * DSIGMA, U2, and VT2 respectively. The singular values/vectors
545 * which were not deflated go into the first K slots of DSIGMA, U2,
546 * and VT2 respectively, while those which were deflated go into the
547 * last N - K slots, except that the first column/row will be treated
548 * separately.
549 *
550  DO 160 j = 2, n
551  jp = idxp( j )
552  dsigma( j ) = d( jp )
553  idxj = idxq( idx( idxp( idxc( j ) ) )+1 )
554  IF( idxj.LE.nlp1 ) THEN
555  idxj = idxj - 1
556  END IF
557  CALL scopy( n, u( 1, idxj ), 1, u2( 1, j ), 1 )
558  CALL scopy( m, vt( idxj, 1 ), ldvt, vt2( j, 1 ), ldvt2 )
559  160 CONTINUE
560 *
561 * Determine DSIGMA(1), DSIGMA(2) and Z(1)
562 *
563  dsigma( 1 ) = zero
564  hlftol = tol / two
565  IF( abs( dsigma( 2 ) ).LE.hlftol )
566  $ dsigma( 2 ) = hlftol
567  IF( m.GT.n ) THEN
568  z( 1 ) = slapy2( z1, z( m ) )
569  IF( z( 1 ).LE.tol ) THEN
570  c = one
571  s = zero
572  z( 1 ) = tol
573  ELSE
574  c = z1 / z( 1 )
575  s = z( m ) / z( 1 )
576  END IF
577  ELSE
578  IF( abs( z1 ).LE.tol ) THEN
579  z( 1 ) = tol
580  ELSE
581  z( 1 ) = z1
582  END IF
583  END IF
584 *
585 * Move the rest of the updating row to Z.
586 *
587  CALL scopy( k-1, u2( 2, 1 ), 1, z( 2 ), 1 )
588 *
589 * Determine the first column of U2, the first row of VT2 and the
590 * last row of VT.
591 *
592  CALL slaset( 'A', n, 1, zero, zero, u2, ldu2 )
593  u2( nlp1, 1 ) = one
594  IF( m.GT.n ) THEN
595  DO 170 i = 1, nlp1
596  vt( m, i ) = -s*vt( nlp1, i )
597  vt2( 1, i ) = c*vt( nlp1, i )
598  170 CONTINUE
599  DO 180 i = nlp2, m
600  vt2( 1, i ) = s*vt( m, i )
601  vt( m, i ) = c*vt( m, i )
602  180 CONTINUE
603  ELSE
604  CALL scopy( m, vt( nlp1, 1 ), ldvt, vt2( 1, 1 ), ldvt2 )
605  END IF
606  IF( m.GT.n ) THEN
607  CALL scopy( m, vt( m, 1 ), ldvt, vt2( m, 1 ), ldvt2 )
608  END IF
609 *
610 * The deflated singular values and their corresponding vectors go
611 * into the back of D, U, and V respectively.
612 *
613  IF( n.GT.k ) THEN
614  CALL scopy( n-k, dsigma( k+1 ), 1, d( k+1 ), 1 )
615  CALL slacpy( 'A', n, n-k, u2( 1, k+1 ), ldu2, u( 1, k+1 ),
616  $ ldu )
617  CALL slacpy( 'A', n-k, m, vt2( k+1, 1 ), ldvt2, vt( k+1, 1 ),
618  $ ldvt )
619  END IF
620 *
621 * Copy CTOT into COLTYP for referencing in SLASD3.
622 *
623  DO 190 j = 1, 4
624  coltyp( j ) = ctot( j )
625  190 CONTINUE
626 *
627  RETURN
628 *
629 * End of SLASD2
630 *
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
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
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:63
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slamrg(N1, N2, A, STRD1, STRD2, INDEX)
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition: slamrg.f:99
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
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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