LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ 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*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
scopy
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
slacpy
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
slamch
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
slamrg
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
slapy2
real function slapy2(x, y)
SLAPY2 returns sqrt(x2+y2).
Definition slapy2.f:63
slaset
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
srot
subroutine srot(n, sx, incx, sy, incy, c, s)
SROT
Definition srot.f:92
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