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version 1.8, 2011/07/22 07:38:07 version 1.9, 2011/11/21 20:42:58
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   *> \brief \b DLASD2
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLASD2 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd2.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd2.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd2.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
   *                          LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
   *                          IDXC, IDXQ, COLTYP, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
   *       DOUBLE PRECISION   ALPHA, BETA
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
   *      $                   IDXQ( * )
   *       DOUBLE PRECISION   D( * ), DSIGMA( * ), U( LDU, * ),
   *      $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
   *      $                   Z( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLASD2 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.
   *>
   *> DLASD2 is called from DLASD1.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \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(N)
   *>         On exit Z contains the updating row vector in the secular
   *>         equation.
   *> \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[in,out] U
   *> \verbatim
   *>          U is DOUBLE PRECISION array, dimension(LDU,N)
   *>         On entry U contains the left singular vectors of two
   *>         submatrices in the two square blocks with corners at (1,1),
   *>         (NL, NL), and (NL+2, NL+2), (N,N).
   *>         On exit U contains the trailing (N-K) updated left singular
   *>         vectors (those which were deflated) in its last N-K columns.
   *> \endverbatim
   *>
   *> \param[in] LDU
   *> \verbatim
   *>          LDU is INTEGER
   *>         The leading dimension of the array U.  LDU >= N.
   *> \endverbatim
   *>
   *> \param[in,out] VT
   *> \verbatim
   *>          VT is DOUBLE PRECISION array, dimension(LDVT,M)
   *>         On entry VT**T contains the right singular vectors of two
   *>         submatrices in the two square blocks with corners at (1,1),
   *>         (NL+1, NL+1), and (NL+2, NL+2), (M,M).
   *>         On exit VT**T contains the trailing (N-K) updated right singular
   *>         vectors (those which were deflated) in its last N-K columns.
   *>         In case SQRE =1, the last row of VT spans the right null
   *>         space.
   *> \endverbatim
   *>
   *> \param[in] LDVT
   *> \verbatim
   *>          LDVT is INTEGER
   *>         The leading dimension of the array VT.  LDVT >= M.
   *> \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] U2
   *> \verbatim
   *>          U2 is DOUBLE PRECISION array, dimension(LDU2,N)
   *>         Contains a copy of the first K-1 left singular vectors which
   *>         will be used by DLASD3 in a matrix multiply (DGEMM) to solve
   *>         for the new left singular vectors. U2 is arranged into four
   *>         blocks. The first block contains a column with 1 at NL+1 and
   *>         zero everywhere else; the second block contains non-zero
   *>         entries only at and above NL; the third contains non-zero
   *>         entries only below NL+1; and the fourth is dense.
   *> \endverbatim
   *>
   *> \param[in] LDU2
   *> \verbatim
   *>          LDU2 is INTEGER
   *>         The leading dimension of the array U2.  LDU2 >= N.
   *> \endverbatim
   *>
   *> \param[out] VT2
   *> \verbatim
   *>          VT2 is DOUBLE PRECISION array, dimension(LDVT2,N)
   *>         VT2**T contains a copy of the first K right singular vectors
   *>         which will be used by DLASD3 in a matrix multiply (DGEMM) to
   *>         solve for the new right singular vectors. VT2 is arranged into
   *>         three blocks. The first block contains a row that corresponds
   *>         to the special 0 diagonal element in SIGMA; the second block
   *>         contains non-zeros only at and before NL +1; the third block
   *>         contains non-zeros only at and after  NL +2.
   *> \endverbatim
   *>
   *> \param[in] LDVT2
   *> \verbatim
   *>          LDVT2 is INTEGER
   *>         The leading dimension of the array VT2.  LDVT2 >= M.
   *> \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[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] IDXC
   *> \verbatim
   *>          IDXC is INTEGER array dimension(N)
   *>         This will contain the permutation used to arrange the columns
   *>         of the deflated U matrix into three groups:  the first group
   *>         contains non-zero entries only at and above NL, the second
   *>         contains non-zero entries only below NL+2, and the third is
   *>         dense.
   *> \endverbatim
   *>
   *> \param[in,out] 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 hlaf 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] COLTYP
   *> \verbatim
   *>          COLTYP is INTEGER array dimension(N)
   *>         As workspace, this will contain a label which will indicate
   *>         which of the following types a column in the U2 matrix or a
   *>         row in the VT2 matrix is:
   *>         1 : non-zero in the upper half only
   *>         2 : non-zero in the lower half only
   *>         3 : dense
   *>         4 : deflated
   *>
   *>         On exit, it is an array of dimension 4, with COLTYP(I) being
   *>         the dimension of the I-th type columns.
   *> \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 DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,        SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
      $                   LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,       $                   LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
      $                   IDXC, IDXQ, COLTYP, INFO )       $                   IDXC, IDXQ, COLTYP, 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            INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE        INTEGER            INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
Line 19 Line 286
      $                   Z( * )       $                   Z( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLASD2 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.  
 *  
 *  DLASD2 is called from DLASD1.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  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(N)  
 *         On exit Z contains the updating row vector in the secular  
 *         equation.  
 *  
 *  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.  
 *  
 *  U      (input/output) DOUBLE PRECISION array, dimension(LDU,N)  
 *         On entry U contains the left singular vectors of two  
 *         submatrices in the two square blocks with corners at (1,1),  
 *         (NL, NL), and (NL+2, NL+2), (N,N).  
 *         On exit U contains the trailing (N-K) updated left singular  
 *         vectors (those which were deflated) in its last N-K columns.  
 *  
 *  LDU    (input) INTEGER  
 *         The leading dimension of the array U.  LDU >= N.  
 *  
 *  VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)  
 *         On entry VT**T contains the right singular vectors of two  
 *         submatrices in the two square blocks with corners at (1,1),  
 *         (NL+1, NL+1), and (NL+2, NL+2), (M,M).  
 *         On exit VT**T contains the trailing (N-K) updated right singular  
 *         vectors (those which were deflated) in its last N-K columns.  
 *         In case SQRE =1, the last row of VT spans the right null  
 *         space.  
 *  
 *  LDVT   (input) INTEGER  
 *         The leading dimension of the array VT.  LDVT >= M.  
 *  
 *  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.  
 *  
 *  U2     (output) DOUBLE PRECISION array, dimension(LDU2,N)  
 *         Contains a copy of the first K-1 left singular vectors which  
 *         will be used by DLASD3 in a matrix multiply (DGEMM) to solve  
 *         for the new left singular vectors. U2 is arranged into four  
 *         blocks. The first block contains a column with 1 at NL+1 and  
 *         zero everywhere else; the second block contains non-zero  
 *         entries only at and above NL; the third contains non-zero  
 *         entries only below NL+1; and the fourth is dense.  
 *  
 *  LDU2   (input) INTEGER  
 *         The leading dimension of the array U2.  LDU2 >= N.  
 *  
 *  VT2    (output) DOUBLE PRECISION array, dimension(LDVT2,N)  
 *         VT2**T contains a copy of the first K right singular vectors  
 *         which will be used by DLASD3 in a matrix multiply (DGEMM) to  
 *         solve for the new right singular vectors. VT2 is arranged into  
 *         three blocks. The first block contains a row that corresponds  
 *         to the special 0 diagonal element in SIGMA; the second block  
 *         contains non-zeros only at and before NL +1; the third block  
 *         contains non-zeros only at and after  NL +2.  
 *  
 *  LDVT2  (input) INTEGER  
 *         The leading dimension of the array VT2.  LDVT2 >= M.  
 *  
 *  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.  
 *  
 *  IDX    (workspace) INTEGER array dimension(N)  
 *         This will contain the permutation used to sort the contents of  
 *         D into ascending order.  
 *  
 *  IDXC   (output) INTEGER array dimension(N)  
 *         This will contain the permutation used to arrange the columns  
 *         of the deflated U matrix into three groups:  the first group  
 *         contains non-zero entries only at and above NL, the second  
 *         contains non-zero entries only below NL+2, and the third is  
 *         dense.  
 *  
 *  IDXQ   (input/output) 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 hlaf 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.  
 *  
 *  COLTYP (workspace/output) INTEGER array dimension(N)  
 *         As workspace, this will contain a label which will indicate  
 *         which of the following types a column in the U2 matrix or a  
 *         row in the VT2 matrix is:  
 *         1 : non-zero in the upper half only  
 *         2 : non-zero in the lower half only  
 *         3 : dense  
 *         4 : deflated  
 *  
 *         On exit, it is an array of dimension 4, with COLTYP(I) being  
 *         the dimension of the I-th type columns.  
 *  
 *  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.1.8  
changed lines
  Added in v.1.9

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