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version 1.9, 2011/07/22 07:38:07 version 1.10, 2011/11/21 20:42:58
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   *> \brief \b DLASD3
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLASD3 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd3.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd3.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd3.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
   *                          LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
   *                          INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
   *      $                   SQRE
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            CTOT( * ), IDXC( * )
   *       DOUBLE PRECISION   D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
   *      $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
   *      $                   Z( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLASD3 finds all the square roots of the roots of the secular
   *> equation, as defined by the values in D and Z.  It makes the
   *> appropriate calls to DLASD4 and then updates the singular
   *> vectors by matrix multiplication.
   *>
   *> This code makes very mild assumptions about floating point
   *> arithmetic. It will work on machines with a guard digit in
   *> add/subtract, or on those binary machines without guard digits
   *> which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
   *> It could conceivably fail on hexadecimal or decimal machines
   *> without guard digits, but we know of none.
   *>
   *> DLASD3 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[in] K
   *> \verbatim
   *>          K is INTEGER
   *>         The size of the secular equation, 1 =< K = < N.
   *> \endverbatim
   *>
   *> \param[out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension(K)
   *>         On exit the square roots of the roots of the secular equation,
   *>         in ascending order.
   *> \endverbatim
   *>
   *> \param[out] Q
   *> \verbatim
   *>          Q is DOUBLE PRECISION array,
   *>                     dimension at least (LDQ,K).
   *> \endverbatim
   *>
   *> \param[in] LDQ
   *> \verbatim
   *>          LDQ is INTEGER
   *>         The leading dimension of the array Q.  LDQ >= K.
   *> \endverbatim
   *>
   *> \param[in] DSIGMA
   *> \verbatim
   *>          DSIGMA is DOUBLE PRECISION array, dimension(K)
   *>         The first K elements of this array contain the old roots
   *>         of the deflated updating problem.  These are the poles
   *>         of the secular equation.
   *> \endverbatim
   *>
   *> \param[out] U
   *> \verbatim
   *>          U is DOUBLE PRECISION array, dimension (LDU, N)
   *>         The last N - K columns of this matrix contain the deflated
   *>         left singular vectors.
   *> \endverbatim
   *>
   *> \param[in] LDU
   *> \verbatim
   *>          LDU is INTEGER
   *>         The leading dimension of the array U.  LDU >= N.
   *> \endverbatim
   *>
   *> \param[in,out] U2
   *> \verbatim
   *>          U2 is DOUBLE PRECISION array, dimension (LDU2, N)
   *>         The first K columns of this matrix contain the non-deflated
   *>         left singular vectors for the split problem.
   *> \endverbatim
   *>
   *> \param[in] LDU2
   *> \verbatim
   *>          LDU2 is INTEGER
   *>         The leading dimension of the array U2.  LDU2 >= N.
   *> \endverbatim
   *>
   *> \param[out] VT
   *> \verbatim
   *>          VT is DOUBLE PRECISION array, dimension (LDVT, M)
   *>         The last M - K columns of VT**T contain the deflated
   *>         right singular vectors.
   *> \endverbatim
   *>
   *> \param[in] LDVT
   *> \verbatim
   *>          LDVT is INTEGER
   *>         The leading dimension of the array VT.  LDVT >= N.
   *> \endverbatim
   *>
   *> \param[in,out] VT2
   *> \verbatim
   *>          VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
   *>         The first K columns of VT2**T contain the non-deflated
   *>         right singular vectors for the split problem.
   *> \endverbatim
   *>
   *> \param[in] LDVT2
   *> \verbatim
   *>          LDVT2 is INTEGER
   *>         The leading dimension of the array VT2.  LDVT2 >= N.
   *> \endverbatim
   *>
   *> \param[in] IDXC
   *> \verbatim
   *>          IDXC is INTEGER array, dimension ( N )
   *>         The permutation used to arrange the columns of U (and rows of
   *>         VT) into three groups:  the first group contains non-zero
   *>         entries only at and above (or before) NL +1; the second
   *>         contains non-zero entries only at and below (or after) NL+2;
   *>         and the third is dense. The first column of U and the row of
   *>         VT are treated separately, however.
   *>
   *>         The rows of the singular vectors found by DLASD4
   *>         must be likewise permuted before the matrix multiplies can
   *>         take place.
   *> \endverbatim
   *>
   *> \param[in] CTOT
   *> \verbatim
   *>          CTOT is INTEGER array, dimension ( 4 )
   *>         A count of the total number of the various types of columns
   *>         in U (or rows in VT), as described in IDXC. The fourth column
   *>         type is any column which has been deflated.
   *> \endverbatim
   *>
   *> \param[in] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension (K)
   *>         The first K elements of this array contain the components
   *>         of the deflation-adjusted updating row vector.
   *> \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 DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,        SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
      $                   LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,       $                   LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
      $                   INFO )       $                   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            INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,        INTEGER            INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
Line 18 Line 241
      $                   Z( * )       $                   Z( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLASD3 finds all the square roots of the roots of the secular  
 *  equation, as defined by the values in D and Z.  It makes the  
 *  appropriate calls to DLASD4 and then updates the singular  
 *  vectors by matrix multiplication.  
 *  
 *  This code makes very mild assumptions about floating point  
 *  arithmetic. It will work on machines with a guard digit in  
 *  add/subtract, or on those binary machines without guard digits  
 *  which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.  
 *  It could conceivably fail on hexadecimal or decimal machines  
 *  without guard digits, but we know of none.  
 *  
 *  DLASD3 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      (input) INTEGER  
 *         The size of the secular equation, 1 =< K = < N.  
 *  
 *  D      (output) DOUBLE PRECISION array, dimension(K)  
 *         On exit the square roots of the roots of the secular equation,  
 *         in ascending order.  
 *  
 *  Q      (workspace) DOUBLE PRECISION array,  
 *                     dimension at least (LDQ,K).  
 *  
 *  LDQ    (input) INTEGER  
 *         The leading dimension of the array Q.  LDQ >= K.  
 *  
 *  DSIGMA (input) DOUBLE PRECISION array, dimension(K)  
 *         The first K elements of this array contain the old roots  
 *         of the deflated updating problem.  These are the poles  
 *         of the secular equation.  
 *  
 *  U      (output) DOUBLE PRECISION array, dimension (LDU, N)  
 *         The last N - K columns of this matrix contain the deflated  
 *         left singular vectors.  
 *  
 *  LDU    (input) INTEGER  
 *         The leading dimension of the array U.  LDU >= N.  
 *  
 *  U2     (input/output) DOUBLE PRECISION array, dimension (LDU2, N)  
 *         The first K columns of this matrix contain the non-deflated  
 *         left singular vectors for the split problem.  
 *  
 *  LDU2   (input) INTEGER  
 *         The leading dimension of the array U2.  LDU2 >= N.  
 *  
 *  VT     (output) DOUBLE PRECISION array, dimension (LDVT, M)  
 *         The last M - K columns of VT**T contain the deflated  
 *         right singular vectors.  
 *  
 *  LDVT   (input) INTEGER  
 *         The leading dimension of the array VT.  LDVT >= N.  
 *  
 *  VT2    (input/output) DOUBLE PRECISION array, dimension (LDVT2, N)  
 *         The first K columns of VT2**T contain the non-deflated  
 *         right singular vectors for the split problem.  
 *  
 *  LDVT2  (input) INTEGER  
 *         The leading dimension of the array VT2.  LDVT2 >= N.  
 *  
 *  IDXC   (input) INTEGER array, dimension ( N )  
 *         The permutation used to arrange the columns of U (and rows of  
 *         VT) into three groups:  the first group contains non-zero  
 *         entries only at and above (or before) NL +1; the second  
 *         contains non-zero entries only at and below (or after) NL+2;  
 *         and the third is dense. The first column of U and the row of  
 *         VT are treated separately, however.  
 *  
 *         The rows of the singular vectors found by DLASD4  
 *         must be likewise permuted before the matrix multiplies can  
 *         take place.  
 *  
 *  CTOT   (input) INTEGER array, dimension ( 4 )  
 *         A count of the total number of the various types of columns  
 *         in U (or rows in VT), as described in IDXC. The fourth column  
 *         type is any column which has been deflated.  
 *  
 *  Z      (input) DOUBLE PRECISION array, dimension (K)  
 *         The first K elements of this array contain the components  
 *         of the deflation-adjusted updating row vector.  
 *  
 *  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  
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  Added in v.1.10

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