LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine slasd6 ( integer ICOMPQ, integer NL, integer NR, integer SQRE, real, dimension( * ) D, real, dimension( * ) VF, real, dimension( * ) VL, real ALPHA, real BETA, integer, dimension( * ) IDXQ, 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( * ) DIFR, real, dimension( * ) Z, integer K, real C, real S, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.

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Purpose:
``` SLASD6 computes the SVD of an updated upper bidiagonal matrix B
obtained by merging two smaller ones by appending a row. This
routine is used only for the problem which requires all singular
values and optionally singular vector matrices in factored form.
B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
A related subroutine, SLASD1, handles the case in which all singular
values and singular vectors of the bidiagonal matrix are desired.

SLASD6 computes the SVD as follows:

( D1(in)    0    0       0 )
B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
(   0       0   D2(in)   0 )

= U(out) * ( D(out) 0) * VT(out)

where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
elsewhere; and the entry b is empty if SQRE = 0.

The singular values of B can be computed using D1, D2, the first
components of all the right singular vectors of the lower block, and
the last components of all the right singular vectors of the upper
block. These components are stored and updated in VF and VL,
respectively, in SLASD6. Hence U and VT are not explicitly
referenced.

The singular values are stored in D. The algorithm consists of two
stages:

The first stage consists of deflating the size of the problem
when there are multiple singular values or if there is a zero
in the Z vector. For each such occurrence the dimension of the
secular equation problem is reduced by one. This stage is
performed by the routine SLASD7.

The second stage consists of calculating the updated
singular values. This is done by finding the roots of the
secular equation via the routine SLASD4 (as called by SLASD8).
This routine also updates VF and VL and computes the distances
between the updated singular values and the old singular
values.

SLASD6 is called from SLASDA.```
Parameters
 [in] ICOMPQ ``` ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well.``` [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,out] D ``` D is REAL array, dimension (NL+NR+1). On entry D(1:NL,1:NL) contains the singular values of the upper block, and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix.``` [in,out] VF ``` VF is REAL array, dimension (M) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix.``` [in,out] VL ``` VL is REAL array, dimension (M) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix.``` [in,out] ALPHA ``` ALPHA is REAL Contains the diagonal element associated with the added row.``` [in,out] BETA ``` BETA is REAL Contains the off-diagonal element associated with the added row.``` [in,out] IDXQ ``` IDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order.``` [out] PERM ``` PERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each block. Not referenced if ICOMPQ = 0.``` [out] GIVPTR ``` GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0.``` [out] GIVCOL ``` GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0.``` [in] LDGCOL ``` LDGCOL is INTEGER leading dimension of GIVCOL, must be at least N.``` [out] GIVNUM ``` GIVNUM is REAL array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0.``` [in] LDGNUM ``` LDGNUM is INTEGER The leading dimension of GIVNUM and POLES, must be at least N.``` [out] POLES ``` POLES is REAL array, dimension ( LDGNUM, 2 ) On exit, POLES(1,*) is an array containing the new singular values obtained from solving the secular equation, and POLES(2,*) is an array containing the poles in the secular equation. Not referenced if ICOMPQ = 0.``` [out] DIFL ``` DIFL is REAL array, dimension ( N ) On exit, DIFL(I) is the distance between I-th updated (undeflated) singular value and the I-th (undeflated) old singular value.``` [out] DIFR ``` DIFR is REAL array, dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and dimension ( K ) if ICOMPQ = 0. On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix. See SLASD8 for details on DIFL and DIFR.``` [out] Z ``` Z is REAL array, dimension ( M ) The first elements of this array contain the components of the deflation-adjusted updating row vector.``` [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.``` [out] 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.``` [out] 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 ( 4 * M )` [out] IWORK ` IWORK is INTEGER array, dimension ( 3 * 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```
Date
June 2016
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 315 of file slasd6.f.

315 *
316 * -- LAPACK auxiliary routine (version 3.6.1) --
317 * -- LAPACK is a software package provided by Univ. of Tennessee, --
318 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
319 * June 2016
320 *
321 * .. Scalar Arguments ..
322  INTEGER givptr, icompq, info, k, ldgcol, ldgnum, nl,
323  \$ nr, sqre
324  REAL alpha, beta, c, s
325 * ..
326 * .. Array Arguments ..
327  INTEGER givcol( ldgcol, * ), idxq( * ), iwork( * ),
328  \$ perm( * )
329  REAL d( * ), difl( * ), difr( * ),
330  \$ givnum( ldgnum, * ), poles( ldgnum, * ),
331  \$ vf( * ), vl( * ), work( * ), z( * )
332 * ..
333 *
334 * =====================================================================
335 *
336 * .. Parameters ..
337  REAL one, zero
338  parameter ( one = 1.0e+0, zero = 0.0e+0 )
339 * ..
340 * .. Local Scalars ..
341  INTEGER i, idx, idxc, idxp, isigma, ivfw, ivlw, iw, m,
342  \$ n, n1, n2
343  REAL orgnrm
344 * ..
345 * .. External Subroutines ..
346  EXTERNAL scopy, slamrg, slascl, slasd7, slasd8, xerbla
347 * ..
348 * .. Intrinsic Functions ..
349  INTRINSIC abs, max
350 * ..
351 * .. Executable Statements ..
352 *
353 * Test the input parameters.
354 *
355  info = 0
356  n = nl + nr + 1
357  m = n + sqre
358 *
359  IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
360  info = -1
361  ELSE IF( nl.LT.1 ) THEN
362  info = -2
363  ELSE IF( nr.LT.1 ) THEN
364  info = -3
365  ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
366  info = -4
367  ELSE IF( ldgcol.LT.n ) THEN
368  info = -14
369  ELSE IF( ldgnum.LT.n ) THEN
370  info = -16
371  END IF
372  IF( info.NE.0 ) THEN
373  CALL xerbla( 'SLASD6', -info )
374  RETURN
375  END IF
376 *
377 * The following values are for bookkeeping purposes only. They are
378 * integer pointers which indicate the portion of the workspace
379 * used by a particular array in SLASD7 and SLASD8.
380 *
381  isigma = 1
382  iw = isigma + n
383  ivfw = iw + m
384  ivlw = ivfw + m
385 *
386  idx = 1
387  idxc = idx + n
388  idxp = idxc + n
389 *
390 * Scale.
391 *
392  orgnrm = max( abs( alpha ), abs( beta ) )
393  d( nl+1 ) = zero
394  DO 10 i = 1, n
395  IF( abs( d( i ) ).GT.orgnrm ) THEN
396  orgnrm = abs( d( i ) )
397  END IF
398  10 CONTINUE
399  CALL slascl( 'G', 0, 0, orgnrm, one, n, 1, d, n, info )
400  alpha = alpha / orgnrm
401  beta = beta / orgnrm
402 *
403 * Sort and Deflate singular values.
404 *
405  CALL slasd7( icompq, nl, nr, sqre, k, d, z, work( iw ), vf,
406  \$ work( ivfw ), vl, work( ivlw ), alpha, beta,
407  \$ work( isigma ), iwork( idx ), iwork( idxp ), idxq,
408  \$ perm, givptr, givcol, ldgcol, givnum, ldgnum, c, s,
409  \$ info )
410 *
411 * Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
412 *
413  CALL slasd8( icompq, k, d, z, vf, vl, difl, difr, ldgnum,
414  \$ work( isigma ), work( iw ), info )
415 *
416 * Report the possible convergence failure.
417 *
418  IF( info.NE.0 ) THEN
419  RETURN
420  END IF
421 *
422 * Save the poles if ICOMPQ = 1.
423 *
424  IF( icompq.EQ.1 ) THEN
425  CALL scopy( k, d, 1, poles( 1, 1 ), 1 )
426  CALL scopy( k, work( isigma ), 1, poles( 1, 2 ), 1 )
427  END IF
428 *
429 * Unscale.
430 *
431  CALL slascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, info )
432 *
433 * Prepare the IDXQ sorting permutation.
434 *
435  n1 = k
436  n2 = n - k
437  CALL slamrg( n1, n2, d, 1, -1, idxq )
438 *
439  RETURN
440 *
441 * End of SLASD6
442 *
subroutine slasd8(ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, DSIGMA, WORK, INFO)
SLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D...
Definition: slasd8.f:168
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:145
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:101
subroutine slasd7(ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, INFO)
SLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to def...
Definition: slasd7.f:282
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53

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