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version 1.7, 2010/12/21 13:53:33 version 1.8, 2011/11/21 20:42:59
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   *> \brief \b DLASD7
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLASD7 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd7.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd7.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd7.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLASD7( 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 )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
   *      $                   NR, SQRE
   *       DOUBLE PRECISION   ALPHA, BETA, C, S
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ),
   *      $                   IDXQ( * ), PERM( * )
   *       DOUBLE PRECISION   D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ),
   *      $                   VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ),
   *      $                   ZW( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLASD7 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.
   *>
   *> DLASD7 is called from DLASD6.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] ICOMPQ
   *> \verbatim
   *>          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.
   *> \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
   *>         N = NL + NR + 1 rows and
   *>         M = N + SQRE >= N columns.
   *> \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[in,out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension ( M )
   *>         On exit Z contains the updating row vector in the secular
   *>         equation.
   *> \endverbatim
   *>
   *> \param[out] ZW
   *> \verbatim
   *>          ZW is DOUBLE PRECISION array, dimension ( M )
   *>         Workspace for Z.
   *> \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[out] VFW
   *> \verbatim
   *>          VFW is DOUBLE PRECISION array, dimension ( M )
   *>         Workspace for VF.
   *> \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[out] VLW
   *> \verbatim
   *>          VLW is DOUBLE PRECISION array, dimension ( M )
   *>         Workspace for VL.
   *> \endverbatim
   *>
   *> \param[in] ALPHA
   *> \verbatim
   *>          ALPHA is DOUBLE PRECISION
   *>         Contains the diagonal element associated with the added row.
   *> \endverbatim
   *>
   *> \param[in] BETA
   *> \verbatim
   *>          BETA is DOUBLE PRECISION
   *>         Contains the off-diagonal element associated with the added
   *>         row.
   *> \endverbatim
   *>
   *> \param[out] DSIGMA
   *> \verbatim
   *>          DSIGMA is DOUBLE PRECISION array, dimension ( N )
   *>         Contains a copy of the diagonal elements (K-1 singular values
   *>         and one zero) in the secular equation.
   *> \endverbatim
   *>
   *> \param[out] IDX
   *> \verbatim
   *>          IDX is INTEGER array, dimension ( N )
   *>         This will contain the permutation used to sort the contents of
   *>         D into ascending order.
   *> \endverbatim
   *>
   *> \param[out] IDXP
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] IDXQ
   *> \verbatim
   *>          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 half 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.
   *> \endverbatim
   *>
   *> \param[out] PERM
   *> \verbatim
   *>          PERM is INTEGER array, dimension ( N )
   *>         The permutations (from deflation and sorting) to be applied
   *>         to each singular 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
   *>         The 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, must be at least 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] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>         = 0:  successful exit.
   *>         < 0:  if INFO = -i, the i-th argument had an illegal value.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup auxOTHERauxiliary
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ming Gu and Huan Ren, Computer Science Division, University of
   *>     California at Berkeley, USA
   *>
   *  =====================================================================
       SUBROUTINE DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL,        SUBROUTINE DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL,
      $                   VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ,       $                   VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ,
      $                   PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,       $                   PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
      $                   C, S, INFO )       $                   C, S, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine (version 3.4.0) --
 *  -- 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  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,        INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
Line 21 Line 298
      $                   ZW( * )       $                   ZW( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLASD7 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.  
 *  
 *  DLASD7 is called from DLASD6.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  ICOMPQ  (input) 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.  
 *  
 *  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  
 *         N = NL + NR + 1 rows and  
 *         M = N + SQRE >= N columns.  
 *  
 *  K      (output) INTEGER  
 *         Contains the dimension of the non-deflated matrix, this is  
 *         the order of the related secular equation. 1 <= K <=N.  
 *  
 *  D      (input/output) DOUBLE PRECISION 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.  
 *  
 *  Z      (output) DOUBLE PRECISION array, dimension ( M )  
 *         On exit Z contains the updating row vector in the secular  
 *         equation.  
 *  
 *  ZW     (workspace) DOUBLE PRECISION array, dimension ( M )  
 *         Workspace for Z.  
 *  
 *  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.  
 *  
 *  VFW    (workspace) DOUBLE PRECISION array, dimension ( M )  
 *         Workspace for VF.  
 *  
 *  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.  
 *  
 *  VLW    (workspace) DOUBLE PRECISION array, dimension ( M )  
 *         Workspace for VL.  
 *  
 *  ALPHA  (input) DOUBLE PRECISION  
 *         Contains the diagonal element associated with the added row.  
 *  
 *  BETA   (input) DOUBLE PRECISION  
 *         Contains the off-diagonal element associated with the added  
 *         row.  
 *  
 *  DSIGMA (output) DOUBLE PRECISION array, dimension ( N )  
 *         Contains a copy of the diagonal elements (K-1 singular values  
 *         and one zero) in the secular equation.  
 *  
 *  IDX    (workspace) INTEGER array, dimension ( N )  
 *         This will contain the permutation used to sort the contents of  
 *         D into ascending order.  
 *  
 *  IDXP   (workspace) 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.  
 *  
 *  IDXQ   (input) 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 half 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.  
 *  
 *  PERM   (output) INTEGER array, dimension ( N )  
 *         The permutations (from deflation and sorting) to be applied  
 *         to each singular 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  
 *         The 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, must be at least 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.  
 *  
 *  INFO   (output) INTEGER  
 *         = 0:  successful exit.  
 *         < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Ming Gu and Huan Ren, Computer Science Division, University of  
 *     California at Berkeley, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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