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   *> \brief \b DLASDA
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLASDA + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasda.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasda.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasda.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
   *                          DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
   *                          PERM, GIVNUM, C, S, WORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
   *      $                   K( * ), PERM( LDGCOL, * )
   *       DOUBLE PRECISION   C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
   *      $                   E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
   *      $                   S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
   *      $                   Z( LDU, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> Using a divide and conquer approach, DLASDA 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, DLASD0, computes the singular values and
   *> the singular vectors in explicit form.
   *> \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] SMLSIZ
   *> \verbatim
   *>          SMLSIZ is INTEGER
   *>         The maximum size of the subproblems at the bottom of the
   *>         computation tree.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>         The row dimension of the upper bidiagonal matrix. This is
   *>         also the dimension of the main diagonal array D.
   *> \endverbatim
   *>
   *> \param[in] SQRE
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in,out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension ( N )
   *>         On entry D contains the main diagonal of the bidiagonal
   *>         matrix. On exit D, if INFO = 0, contains its singular values.
   *> \endverbatim
   *>
   *> \param[in] E
   *> \verbatim
   *>          E is DOUBLE PRECISION array, dimension ( M-1 )
   *>         Contains the subdiagonal entries of the bidiagonal matrix.
   *>         On exit, E has been destroyed.
   *> \endverbatim
   *>
   *> \param[out] U
   *> \verbatim
   *>          U is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in] LDU
   *> \verbatim
   *>          LDU is INTEGER, LDU = > N.
   *>         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
   *>         GIVNUM, and Z.
   *> \endverbatim
   *>
   *> \param[out] VT
   *> \verbatim
   *>          VT is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] K
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] DIFL
   *> \verbatim
   *>          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ),
   *>         where NLVL = floor(log_2 (N/SMLSIZ))).
   *> \endverbatim
   *>
   *> \param[out] DIFR
   *> \verbatim
   *>          DIFR is DOUBLE PRECISION 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 DLASD8 for details.
   *> \endverbatim
   *>
   *> \param[out] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] POLES
   *> \verbatim
   *>          POLES is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] GIVPTR
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] GIVCOL
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] LDGCOL
   *> \verbatim
   *>          LDGCOL is INTEGER, LDGCOL = > N.
   *>         The leading dimension of arrays GIVCOL and PERM.
   *> \endverbatim
   *>
   *> \param[out] PERM
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] GIVNUM
   *> \verbatim
   *>          GIVNUM is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] C
   *> \verbatim
   *>          C is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] S
   *> \verbatim
   *>          S is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension
   *>         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array.
   *>         Dimension must be at least (7 * 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 November 2011
   *
   *> \ingroup auxOTHERauxiliary
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ming Gu and Huan Ren, Computer Science Division, University of
   *>     California at Berkeley, USA
   *>
   *  =====================================================================
       SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,        SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
      $                   DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,       $                   DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
      $                   PERM, GIVNUM, C, S, WORK, IWORK, INFO )       $                   PERM, GIVNUM, C, S, WORK, IWORK, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2.2) --  *  -- 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..--
 *     June 2010  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE        INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
Line 19 Line 291
      $                   Z( LDU, * )       $                   Z( LDU, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  Using a divide and conquer approach, DLASDA 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, DLASD0, computes the singular values and  
 *  the singular vectors in explicit form.  
 *  
 *  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.  
 *  
 *  SMLSIZ (input) INTEGER  
 *         The maximum size of the subproblems at the bottom of the  
 *         computation tree.  
 *  
 *  N      (input) INTEGER  
 *         The row dimension of the upper bidiagonal matrix. This is  
 *         also the dimension of the main diagonal array D.  
 *  
 *  SQRE   (input) 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.  
 *  
 *  D      (input/output) DOUBLE PRECISION array, dimension ( N )  
 *         On entry D contains the main diagonal of the bidiagonal  
 *         matrix. On exit D, if INFO = 0, contains its singular values.  
 *  
 *  E      (input) DOUBLE PRECISION array, dimension ( M-1 )  
 *         Contains the subdiagonal entries of the bidiagonal matrix.  
 *         On exit, E has been destroyed.  
 *  
 *  U      (output) DOUBLE PRECISION 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.  
 *  
 *  LDU    (input) INTEGER, LDU = > N.  
 *         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,  
 *         GIVNUM, and Z.  
 *  
 *  VT     (output) DOUBLE PRECISION 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.  
 *  
 *  K      (output) 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.  
 *  
 *  DIFL   (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),  
 *         where NLVL = floor(log_2 (N/SMLSIZ))).  
 *  
 *  DIFR   (output) DOUBLE PRECISION 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 DLASD8 for details.  
 *  
 *  Z      (output) DOUBLE PRECISION 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.  
 *  
 *  POLES  (output) DOUBLE PRECISION 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.  
 *  
 *  GIVPTR (output) 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.  
 *  
 *  GIVCOL (output) 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.  
 *  
 *  LDGCOL (input) INTEGER, LDGCOL = > N.  
 *         The leading dimension of arrays GIVCOL and PERM.  
 *  
 *  PERM   (output) 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.  
 *  
 *  GIVNUM (output) DOUBLE PRECISION 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.  
 *  
 *  C      (output) DOUBLE PRECISION 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.  
 *  
 *  S      (output) DOUBLE PRECISION 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.  
 *  
 *  WORK   (workspace) DOUBLE PRECISION array, dimension  
 *         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).  
 *  
 *  IWORK  (workspace) INTEGER array.  
 *         Dimension must be at least (7 * N).  
 *  
 *  INFO   (output) 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  
 *  
 *  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.1.9  
changed lines
  Added in v.1.10

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