LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ slasda()

subroutine slasda ( integer  ICOMPQ,
integer  SMLSIZ,
integer  N,
integer  SQRE,
real, dimension( * )  D,
real, dimension( * )  E,
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 
)

SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.

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

Purpose:
 Using a divide and conquer approach, SLASDA computes the singular
 value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
 B with diagonal D and offdiagonal E, where M = N + SQRE. The
 algorithm computes the singular values in the SVD B = U * S * VT.
 The orthogonal matrices U and VT are optionally computed in
 compact form.

 A related subroutine, SLASD0, computes the singular values and
 the singular vectors in explicit form.
Parameters
[in]ICOMPQ
          ICOMPQ is INTEGER
         Specifies whether singular vectors are to be computed
         in compact form, as follows
         = 0: Compute singular values only.
         = 1: Compute singular vectors of upper bidiagonal
              matrix in compact form.
[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 dimension of the upper bidiagonal matrix. This is
         also the dimension of the main diagonal array D.
[in]SQRE
          SQRE is INTEGER
         Specifies the column dimension of the bidiagonal matrix.
         = 0: The bidiagonal matrix has column dimension M = N;
         = 1: The bidiagonal matrix has column dimension M = N + 1.
[in,out]D
          D is REAL array, dimension ( N )
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit D, if INFO = 0, contains its singular values.
[in]E
          E is REAL array, dimension ( M-1 )
         Contains the subdiagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.
[out]U
          U is REAL array,
         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, 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.
[out]VT
          VT is REAL array,
         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
         singular vector matrices of all subproblems at the bottom
         level.
[out]K
          K is INTEGER array, dimension ( N )
         if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
         secular equation on the computation tree.
[out]DIFL
          DIFL is REAL array, dimension ( LDU, NLVL ),
         where NLVL = floor(log_2 (N/SMLSIZ))).
[out]DIFR
          DIFR is REAL array,
                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
         record distances between singular values on the I-th
         level and singular values on the (I -1)-th level, and
         DIFR(1:N, 2 * I ) contains the normalizing factors for
         the right singular vector matrix. See SLASD8 for details.
[out]Z
          Z is REAL array,
                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
                  dimension ( N ) if ICOMPQ = 0.
         The first K elements of Z(1, I) contain the components of
         the deflation-adjusted updating row vector for subproblems
         on the I-th level.
[out]POLES
          POLES is REAL array,
         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
         POLES(1, 2*I) contain  the new and old singular values
         involved in the secular equations on the I-th level.
[out]GIVPTR
          GIVPTR is INTEGER array,
         dimension ( N ) if ICOMPQ = 1, and not referenced if
         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
         the number of Givens rotations performed on the I-th
         problem on the computation tree.
[out]GIVCOL
          GIVCOL is INTEGER array,
         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record 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.
[out]PERM
          PERM is INTEGER array, dimension ( LDGCOL, NLVL )
         if ICOMPQ = 1, and not referenced
         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
         permutations done on the I-th level of the computation tree.
[out]GIVNUM
          GIVNUM is REAL array,
         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
         values of Givens rotations performed on the I-th level on
         the computation tree.
[out]C
          C is REAL array,
         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
         C( I ) contains the C-value of a Givens rotation related to
         the right null space of the I-th subproblem.
[out]S
          S is REAL array, dimension ( N ) if
         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
         and the I-th subproblem is not square, on exit, 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
         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
[out]IWORK
          IWORK is INTEGER array, dimension (7*N).
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, a singular value did not converge
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 270 of file slasda.f.

273 *
274 * -- LAPACK auxiliary routine --
275 * -- LAPACK is a software package provided by Univ. of Tennessee, --
276 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
277 *
278 * .. Scalar Arguments ..
279  INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
280 * ..
281 * .. Array Arguments ..
282  INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
283  $ K( * ), PERM( LDGCOL, * )
284  REAL C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
285  $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
286  $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
287  $ Z( LDU, * )
288 * ..
289 *
290 * =====================================================================
291 *
292 * .. Parameters ..
293  REAL ZERO, ONE
294  parameter( zero = 0.0e+0, one = 1.0e+0 )
295 * ..
296 * .. Local Scalars ..
297  INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,
298  $ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,
299  $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,
300  $ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI
301  REAL ALPHA, BETA
302 * ..
303 * .. External Subroutines ..
304  EXTERNAL scopy, slasd6, slasdq, slasdt, slaset, xerbla
305 * ..
306 * .. Executable Statements ..
307 *
308 * Test the input parameters.
309 *
310  info = 0
311 *
312  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
313  info = -1
314  ELSE IF( smlsiz.LT.3 ) THEN
315  info = -2
316  ELSE IF( n.LT.0 ) THEN
317  info = -3
318  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
319  info = -4
320  ELSE IF( ldu.LT.( n+sqre ) ) THEN
321  info = -8
322  ELSE IF( ldgcol.LT.n ) THEN
323  info = -17
324  END IF
325  IF( info.NE.0 ) THEN
326  CALL xerbla( 'SLASDA', -info )
327  RETURN
328  END IF
329 *
330  m = n + sqre
331 *
332 * If the input matrix is too small, call SLASDQ to find the SVD.
333 *
334  IF( n.LE.smlsiz ) THEN
335  IF( icompq.EQ.0 ) THEN
336  CALL slasdq( 'U', sqre, n, 0, 0, 0, d, e, vt, ldu, u, ldu,
337  $ u, ldu, work, info )
338  ELSE
339  CALL slasdq( 'U', sqre, n, m, n, 0, d, e, vt, ldu, u, ldu,
340  $ u, ldu, work, info )
341  END IF
342  RETURN
343  END IF
344 *
345 * Book-keeping and set up the computation tree.
346 *
347  inode = 1
348  ndiml = inode + n
349  ndimr = ndiml + n
350  idxq = ndimr + n
351  iwk = idxq + n
352 *
353  ncc = 0
354  nru = 0
355 *
356  smlszp = smlsiz + 1
357  vf = 1
358  vl = vf + m
359  nwork1 = vl + m
360  nwork2 = nwork1 + smlszp*smlszp
361 *
362  CALL slasdt( n, nlvl, nd, iwork( inode ), iwork( ndiml ),
363  $ iwork( ndimr ), smlsiz )
364 *
365 * for the nodes on bottom level of the tree, solve
366 * their subproblems by SLASDQ.
367 *
368  ndb1 = ( nd+1 ) / 2
369  DO 30 i = ndb1, nd
370 *
371 * IC : center row of each node
372 * NL : number of rows of left subproblem
373 * NR : number of rows of right subproblem
374 * NLF: starting row of the left subproblem
375 * NRF: starting row of the right subproblem
376 *
377  i1 = i - 1
378  ic = iwork( inode+i1 )
379  nl = iwork( ndiml+i1 )
380  nlp1 = nl + 1
381  nr = iwork( ndimr+i1 )
382  nlf = ic - nl
383  nrf = ic + 1
384  idxqi = idxq + nlf - 2
385  vfi = vf + nlf - 1
386  vli = vl + nlf - 1
387  sqrei = 1
388  IF( icompq.EQ.0 ) THEN
389  CALL slaset( 'A', nlp1, nlp1, zero, one, work( nwork1 ),
390  $ smlszp )
391  CALL slasdq( 'U', sqrei, nl, nlp1, nru, ncc, d( nlf ),
392  $ e( nlf ), work( nwork1 ), smlszp,
393  $ work( nwork2 ), nl, work( nwork2 ), nl,
394  $ work( nwork2 ), info )
395  itemp = nwork1 + nl*smlszp
396  CALL scopy( nlp1, work( nwork1 ), 1, work( vfi ), 1 )
397  CALL scopy( nlp1, work( itemp ), 1, work( vli ), 1 )
398  ELSE
399  CALL slaset( 'A', nl, nl, zero, one, u( nlf, 1 ), ldu )
400  CALL slaset( 'A', nlp1, nlp1, zero, one, vt( nlf, 1 ), ldu )
401  CALL slasdq( 'U', sqrei, nl, nlp1, nl, ncc, d( nlf ),
402  $ e( nlf ), vt( nlf, 1 ), ldu, u( nlf, 1 ), ldu,
403  $ u( nlf, 1 ), ldu, work( nwork1 ), info )
404  CALL scopy( nlp1, vt( nlf, 1 ), 1, work( vfi ), 1 )
405  CALL scopy( nlp1, vt( nlf, nlp1 ), 1, work( vli ), 1 )
406  END IF
407  IF( info.NE.0 ) THEN
408  RETURN
409  END IF
410  DO 10 j = 1, nl
411  iwork( idxqi+j ) = j
412  10 CONTINUE
413  IF( ( i.EQ.nd ) .AND. ( sqre.EQ.0 ) ) THEN
414  sqrei = 0
415  ELSE
416  sqrei = 1
417  END IF
418  idxqi = idxqi + nlp1
419  vfi = vfi + nlp1
420  vli = vli + nlp1
421  nrp1 = nr + sqrei
422  IF( icompq.EQ.0 ) THEN
423  CALL slaset( 'A', nrp1, nrp1, zero, one, work( nwork1 ),
424  $ smlszp )
425  CALL slasdq( 'U', sqrei, nr, nrp1, nru, ncc, d( nrf ),
426  $ e( nrf ), work( nwork1 ), smlszp,
427  $ work( nwork2 ), nr, work( nwork2 ), nr,
428  $ work( nwork2 ), info )
429  itemp = nwork1 + ( nrp1-1 )*smlszp
430  CALL scopy( nrp1, work( nwork1 ), 1, work( vfi ), 1 )
431  CALL scopy( nrp1, work( itemp ), 1, work( vli ), 1 )
432  ELSE
433  CALL slaset( 'A', nr, nr, zero, one, u( nrf, 1 ), ldu )
434  CALL slaset( 'A', nrp1, nrp1, zero, one, vt( nrf, 1 ), ldu )
435  CALL slasdq( 'U', sqrei, nr, nrp1, nr, ncc, d( nrf ),
436  $ e( nrf ), vt( nrf, 1 ), ldu, u( nrf, 1 ), ldu,
437  $ u( nrf, 1 ), ldu, work( nwork1 ), info )
438  CALL scopy( nrp1, vt( nrf, 1 ), 1, work( vfi ), 1 )
439  CALL scopy( nrp1, vt( nrf, nrp1 ), 1, work( vli ), 1 )
440  END IF
441  IF( info.NE.0 ) THEN
442  RETURN
443  END IF
444  DO 20 j = 1, nr
445  iwork( idxqi+j ) = j
446  20 CONTINUE
447  30 CONTINUE
448 *
449 * Now conquer each subproblem bottom-up.
450 *
451  j = 2**nlvl
452  DO 50 lvl = nlvl, 1, -1
453  lvl2 = lvl*2 - 1
454 *
455 * Find the first node LF and last node LL on
456 * the current level LVL.
457 *
458  IF( lvl.EQ.1 ) THEN
459  lf = 1
460  ll = 1
461  ELSE
462  lf = 2**( lvl-1 )
463  ll = 2*lf - 1
464  END IF
465  DO 40 i = lf, ll
466  im1 = i - 1
467  ic = iwork( inode+im1 )
468  nl = iwork( ndiml+im1 )
469  nr = iwork( ndimr+im1 )
470  nlf = ic - nl
471  nrf = ic + 1
472  IF( i.EQ.ll ) THEN
473  sqrei = sqre
474  ELSE
475  sqrei = 1
476  END IF
477  vfi = vf + nlf - 1
478  vli = vl + nlf - 1
479  idxqi = idxq + nlf - 1
480  alpha = d( ic )
481  beta = e( ic )
482  IF( icompq.EQ.0 ) THEN
483  CALL slasd6( icompq, nl, nr, sqrei, d( nlf ),
484  $ work( vfi ), work( vli ), alpha, beta,
485  $ iwork( idxqi ), perm, givptr( 1 ), givcol,
486  $ ldgcol, givnum, ldu, poles, difl, difr, z,
487  $ k( 1 ), c( 1 ), s( 1 ), work( nwork1 ),
488  $ iwork( iwk ), info )
489  ELSE
490  j = j - 1
491  CALL slasd6( icompq, nl, nr, sqrei, d( nlf ),
492  $ work( vfi ), work( vli ), alpha, beta,
493  $ iwork( idxqi ), perm( nlf, lvl ),
494  $ givptr( j ), givcol( nlf, lvl2 ), ldgcol,
495  $ givnum( nlf, lvl2 ), ldu,
496  $ poles( nlf, lvl2 ), difl( nlf, lvl ),
497  $ difr( nlf, lvl2 ), z( nlf, lvl ), k( j ),
498  $ c( j ), s( j ), work( nwork1 ),
499  $ iwork( iwk ), info )
500  END IF
501  IF( info.NE.0 ) THEN
502  RETURN
503  END IF
504  40 CONTINUE
505  50 CONTINUE
506 *
507  RETURN
508 *
509 * End of SLASDA
510 *
subroutine slasd6(ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO)
SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by...
Definition: slasd6.f:313
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 slasdq(UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)
SLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e....
Definition: slasdq.f:211
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 scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
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