 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.

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```
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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