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   *> \brief \b DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLASD6 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd6.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd6.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd6.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLASD6( 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 )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
   *      $                   NR, SQRE
   *       DOUBLE PRECISION   ALPHA, BETA, C, S
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
   *      $                   PERM( * )
   *       DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( * ),
   *      $                   GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
   *      $                   VF( * ), VL( * ), WORK( * ), Z( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLASD6 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, DLASD1, handles the case in which all singular
   *> values and singular vectors of the bidiagonal matrix are desired.
   *>
   *> DLASD6 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 DLASD6. 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 occurence the dimension of the
   *>       secular equation problem is reduced by one. This stage is
   *>       performed by the routine DLASD7.
   *>
   *>       The second stage consists of calculating the updated
   *>       singular values. This is done by finding the roots of the
   *>       secular equation via the routine DLASD4 (as called by DLASD8).
   *>       This routine also updates VF and VL and computes the distances
   *>       between the updated singular values and the old singular
   *>       values.
   *>
   *> DLASD6 is called from DLASDA.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] ICOMPQ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] NL
   *> \verbatim
   *>          NL is INTEGER
   *>         The row dimension of the upper block.  NL >= 1.
   *> \endverbatim
   *>
   *> \param[in] NR
   *> \verbatim
   *>          NR is INTEGER
   *>         The row dimension of the lower block.  NR >= 1.
   *> \endverbatim
   *>
   *> \param[in] SQRE
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in,out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in,out] VF
   *> \verbatim
   *>          VF is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in,out] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in,out] ALPHA
   *> \verbatim
   *>          ALPHA is DOUBLE PRECISION
   *>         Contains the diagonal element associated with the added row.
   *> \endverbatim
   *>
   *> \param[in,out] BETA
   *> \verbatim
   *>          BETA is DOUBLE PRECISION
   *>         Contains the off-diagonal element associated with the added
   *>         row.
   *> \endverbatim
   *>
   *> \param[out] IDXQ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] PERM
   *> \verbatim
   *>          PERM is INTEGER array, dimension ( N )
   *>         The permutations (from deflation and sorting) to be applied
   *>         to each block. Not referenced if ICOMPQ = 0.
   *> \endverbatim
   *>
   *> \param[out] GIVPTR
   *> \verbatim
   *>          GIVPTR is INTEGER
   *>         The number of Givens rotations which took place in this
   *>         subproblem. Not referenced if ICOMPQ = 0.
   *> \endverbatim
   *>
   *> \param[out] GIVCOL
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] LDGCOL
   *> \verbatim
   *>          LDGCOL is INTEGER
   *>         leading dimension of GIVCOL, must be at least N.
   *> \endverbatim
   *>
   *> \param[out] GIVNUM
   *> \verbatim
   *>          GIVNUM is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in] LDGNUM
   *> \verbatim
   *>          LDGNUM is INTEGER
   *>         The leading dimension of GIVNUM and POLES, must be at least N.
   *> \endverbatim
   *>
   *> \param[out] POLES
   *> \verbatim
   *>          POLES is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] DIFL
   *> \verbatim
   *>          DIFL is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] DIFR
   *> \verbatim
   *>          DIFR is DOUBLE PRECISION array,
   *>                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
   *>                  dimension ( N ) if ICOMPQ = 0.
   *>         On exit, DIFR(I, 1) is the distance between I-th updated
   *>         (undeflated) singular value and the I+1-th (undeflated) old
   *>         singular value.
   *>
   *>         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
   *>         normalizing factors for the right singular vector matrix.
   *>
   *>         See DLASD8 for details on DIFL and DIFR.
   *> \endverbatim
   *>
   *> \param[out] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension ( M )
   *>         The first elements of this array contain the components
   *>         of the deflation-adjusted updating row vector.
   *> \endverbatim
   *>
   *> \param[out] K
   *> \verbatim
   *>          K is INTEGER
   *>         Contains the dimension of the non-deflated matrix,
   *>         This is the order of the related secular equation. 1 <= K <=N.
   *> \endverbatim
   *>
   *> \param[out] C
   *> \verbatim
   *>          C is DOUBLE PRECISION
   *>         C contains garbage if SQRE =0 and the C-value of a Givens
   *>         rotation related to the right null space if SQRE = 1.
   *> \endverbatim
   *>
   *> \param[out] S
   *> \verbatim
   *>          S is DOUBLE PRECISION
   *>         S contains garbage if SQRE =0 and the S-value of a Givens
   *>         rotation related to the right null space if SQRE = 1.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension ( 4 * M )
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension ( 3 * N )
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          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
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date September 2012
   *
   *> \ingroup auxOTHERauxiliary
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ming Gu and Huan Ren, Computer Science Division, University of
   *>     California at Berkeley, USA
   *>
   *  =====================================================================
       SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,        SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
      $                   IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,       $                   IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
      $                   LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,       $                   LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
      $                   IWORK, INFO )       $                   IWORK, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine (version 3.4.2) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     September 2012
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,        INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
Line 21 Line 332
      $                   VF( * ), VL( * ), WORK( * ), Z( * )       $                   VF( * ), VL( * ), WORK( * ), Z( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLASD6 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, DLASD1, handles the case in which all singular  
 *  values and singular vectors of the bidiagonal matrix are desired.  
 *  
 *  DLASD6 computes the SVD as follows:  
 *  
 *                ( D1(in)  0    0     0 )  
 *    B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)  
 *                (   0     0   D2(in) 0 )  
 *  
 *      = U(out) * ( D(out) 0) * VT(out)  
 *  
 *  where Z' = (Z1' a Z2' b) = u' VT', 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 DLASD6. 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 occurence the dimension of the  
 *        secular equation problem is reduced by one. This stage is  
 *        performed by the routine DLASD7.  
 *  
 *        The second stage consists of calculating the updated  
 *        singular values. This is done by finding the roots of the  
 *        secular equation via the routine DLASD4 (as called by DLASD8).  
 *        This routine also updates VF and VL and computes the distances  
 *        between the updated singular values and the old singular  
 *        values.  
 *  
 *  DLASD6 is called from DLASDA.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  ICOMPQ (input) 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.  
 *  
 *  NL     (input) INTEGER  
 *         The row dimension of the upper block.  NL >= 1.  
 *  
 *  NR     (input) INTEGER  
 *         The row dimension of the lower block.  NR >= 1.  
 *  
 *  SQRE   (input) 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.  
 *  
 *  D      (input/output) DOUBLE PRECISION 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.  
 *  
 *  VF     (input/output) DOUBLE PRECISION 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.  
 *  
 *  VL     (input/output) DOUBLE PRECISION 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.  
 *  
 *  ALPHA  (input/output) DOUBLE PRECISION  
 *         Contains the diagonal element associated with the added row.  
 *  
 *  BETA   (input/output) DOUBLE PRECISION  
 *         Contains the off-diagonal element associated with the added  
 *         row.  
 *  
 *  IDXQ   (output) 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.  
 *  
 *  PERM   (output) INTEGER array, dimension ( N )  
 *         The permutations (from deflation and sorting) to be applied  
 *         to each block. Not referenced if ICOMPQ = 0.  
 *  
 *  GIVPTR (output) INTEGER  
 *         The number of Givens rotations which took place in this  
 *         subproblem. Not referenced if ICOMPQ = 0.  
 *  
 *  GIVCOL (output) 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.  
 *  
 *  LDGCOL (input) INTEGER  
 *         leading dimension of GIVCOL, must be at least N.  
 *  
 *  GIVNUM (output) DOUBLE PRECISION 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.  
 *  
 *  LDGNUM (input) INTEGER  
 *         The leading dimension of GIVNUM and POLES, must be at least N.  
 *  
 *  POLES  (output) DOUBLE PRECISION 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.  
 *  
 *  DIFL   (output) DOUBLE PRECISION 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.  
 *  
 *  DIFR   (output) DOUBLE PRECISION array,  
 *                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and  
 *                  dimension ( N ) if ICOMPQ = 0.  
 *         On exit, DIFR(I, 1) is the distance between I-th updated  
 *         (undeflated) singular value and the I+1-th (undeflated) old  
 *         singular value.  
 *  
 *         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the  
 *         normalizing factors for the right singular vector matrix.  
 *  
 *         See DLASD8 for details on DIFL and DIFR.  
 *  
 *  Z      (output) DOUBLE PRECISION array, dimension ( M )  
 *         The first elements of this array contain the components  
 *         of the deflation-adjusted updating row vector.  
 *  
 *  K      (output) INTEGER  
 *         Contains the dimension of the non-deflated matrix,  
 *         This is the order of the related secular equation. 1 <= K <=N.  
 *  
 *  C      (output) DOUBLE PRECISION  
 *         C contains garbage if SQRE =0 and the C-value of a Givens  
 *         rotation related to the right null space if SQRE = 1.  
 *  
 *  S      (output) DOUBLE PRECISION  
 *         S contains garbage if SQRE =0 and the S-value of a Givens  
 *         rotation related to the right null space if SQRE = 1.  
 *  
 *  WORK   (workspace) DOUBLE PRECISION array, dimension ( 4 * M )  
 *  
 *  IWORK  (workspace) INTEGER array, dimension ( 3 * N )  
 *  
 *  INFO   (output) INTEGER  
 *          = 0:  successful exit.  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *          > 0:  if INFO = 1, an singular value did not converge  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Ming Gu and Huan Ren, Computer Science Division, University of  
 *     California at Berkeley, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 282 Line 414
       CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,        CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
      $             WORK( ISIGMA ), WORK( IW ), INFO )       $             WORK( ISIGMA ), WORK( IW ), INFO )
 *  *
   *     Handle error returned
   *
         IF( INFO.NE.0 ) THEN
            CALL XERBLA( 'DLASD8', -INFO )
            RETURN
         END IF
   *
 *     Save the poles if ICOMPQ = 1.  *     Save the poles if ICOMPQ = 1.
 *  *
       IF( ICOMPQ.EQ.1 ) THEN        IF( ICOMPQ.EQ.1 ) THEN
Removed from v.1.3  
changed lines
  Added in v.1.16

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