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version 1.8, 2010/12/21 13:53:50 version 1.14, 2012/12/14 14:22:50
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   *> \brief \b ZLALSD uses the singular value decomposition of A to solve the least squares problem.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZLALSD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlalsd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlalsd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlalsd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
   *                          RANK, WORK, RWORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            INFO, LDB, N, NRHS, RANK, SMLSIZ
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   D( * ), E( * ), RWORK( * )
   *       COMPLEX*16         B( LDB, * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZLALSD uses the singular value decomposition of A to solve the least
   *> squares problem of finding X to minimize the Euclidean norm of each
   *> column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
   *> are N-by-NRHS. The solution X overwrites B.
   *>
   *> The singular values of A smaller than RCOND times the largest
   *> singular value are treated as zero in solving the least squares
   *> problem; in this case a minimum norm solution is returned.
   *> The actual singular values are returned in D in ascending order.
   *>
   *> 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>         = 'U': D and E define an upper bidiagonal matrix.
   *>         = 'L': D and E define a  lower bidiagonal matrix.
   *> \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 dimension of the  bidiagonal matrix.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] NRHS
   *> \verbatim
   *>          NRHS is INTEGER
   *>         The number of columns of B. NRHS must be at least 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, if INFO = 0, D contains its singular values.
   *> \endverbatim
   *>
   *> \param[in,out] E
   *> \verbatim
   *>          E is DOUBLE PRECISION array, dimension (N-1)
   *>         Contains the super-diagonal entries of the bidiagonal matrix.
   *>         On exit, E has been destroyed.
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
   *>         On input, B contains the right hand sides of the least
   *>         squares problem. On output, B contains the solution X.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>         The leading dimension of B in the calling subprogram.
   *>         LDB must be at least max(1,N).
   *> \endverbatim
   *>
   *> \param[in] RCOND
   *> \verbatim
   *>          RCOND is DOUBLE PRECISION
   *>         The singular values of A less than or equal to RCOND times
   *>         the largest singular value are treated as zero in solving
   *>         the least squares problem. If RCOND is negative,
   *>         machine precision is used instead.
   *>         For example, if diag(S)*X=B were the least squares problem,
   *>         where diag(S) is a diagonal matrix of singular values, the
   *>         solution would be X(i) = B(i) / S(i) if S(i) is greater than
   *>         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
   *>         RCOND*max(S).
   *> \endverbatim
   *>
   *> \param[out] RANK
   *> \verbatim
   *>          RANK is INTEGER
   *>         The number of singular values of A greater than RCOND times
   *>         the largest singular value.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension at least
   *>         (N * NRHS).
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension at least
   *>         (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
   *>         MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),
   *>         where
   *>         NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension at least
   *>         (3*N*NLVL + 11*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:  The algorithm failed to compute a singular value while
   *>               working on the submatrix lying in rows and columns
   *>               INFO/(N+1) through MOD(INFO,N+1).
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date September 2012
   *
   *> \ingroup complex16OTHERcomputational
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ming Gu and Ren-Cang Li, Computer Science Division, University of
   *>       California at Berkeley, USA \n
   *>     Osni Marques, LBNL/NERSC, USA \n
   *
   *  =====================================================================
       SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,        SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
      $                   RANK, WORK, RWORK, IWORK, INFO )       $                   RANK, WORK, RWORK, IWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2.2) --  *  -- LAPACK computational 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..--
 *     June 2010  *     September 2012
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
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       COMPLEX*16         B( LDB, * ), WORK( * )        COMPLEX*16         B( LDB, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZLALSD uses the singular value decomposition of A to solve the least  
 *  squares problem of finding X to minimize the Euclidean norm of each  
 *  column of A*X-B, where A is N-by-N upper bidiagonal, and X and B  
 *  are N-by-NRHS. The solution X overwrites B.  
 *  
 *  The singular values of A smaller than RCOND times the largest  
 *  singular value are treated as zero in solving the least squares  
 *  problem; in this case a minimum norm solution is returned.  
 *  The actual singular values are returned in D in ascending order.  
 *  
 *  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.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  UPLO   (input) CHARACTER*1  
 *         = 'U': D and E define an upper bidiagonal matrix.  
 *         = 'L': D and E define a  lower bidiagonal matrix.  
 *  
 *  SMLSIZ (input) INTEGER  
 *         The maximum size of the subproblems at the bottom of the  
 *         computation tree.  
 *  
 *  N      (input) INTEGER  
 *         The dimension of the  bidiagonal matrix.  N >= 0.  
 *  
 *  NRHS   (input) INTEGER  
 *         The number of columns of B. NRHS must be at least 1.  
 *  
 *  D      (input/output) DOUBLE PRECISION array, dimension (N)  
 *         On entry D contains the main diagonal of the bidiagonal  
 *         matrix. On exit, if INFO = 0, D contains its singular values.  
 *  
 *  E      (input/output) DOUBLE PRECISION array, dimension (N-1)  
 *         Contains the super-diagonal entries of the bidiagonal matrix.  
 *         On exit, E has been destroyed.  
 *  
 *  B      (input/output) COMPLEX*16 array, dimension (LDB,NRHS)  
 *         On input, B contains the right hand sides of the least  
 *         squares problem. On output, B contains the solution X.  
 *  
 *  LDB    (input) INTEGER  
 *         The leading dimension of B in the calling subprogram.  
 *         LDB must be at least max(1,N).  
 *  
 *  RCOND  (input) DOUBLE PRECISION  
 *         The singular values of A less than or equal to RCOND times  
 *         the largest singular value are treated as zero in solving  
 *         the least squares problem. If RCOND is negative,  
 *         machine precision is used instead.  
 *         For example, if diag(S)*X=B were the least squares problem,  
 *         where diag(S) is a diagonal matrix of singular values, the  
 *         solution would be X(i) = B(i) / S(i) if S(i) is greater than  
 *         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to  
 *         RCOND*max(S).  
 *  
 *  RANK   (output) INTEGER  
 *         The number of singular values of A greater than RCOND times  
 *         the largest singular value.  
 *  
 *  WORK   (workspace) COMPLEX*16 array, dimension at least  
 *         (N * NRHS).  
 *  
 *  RWORK  (workspace) DOUBLE PRECISION array, dimension at least  
 *         (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +  
 *         MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),  
 *         where  
 *         NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )  
 *  
 *  IWORK  (workspace) INTEGER array, dimension at least  
 *         (3*N*NLVL + 11*N).  
 *  
 *  INFO   (output) INTEGER  
 *         = 0:  successful exit.  
 *         < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *         > 0:  The algorithm failed to compute a singular value while  
 *               working on the submatrix lying in rows and columns  
 *               INFO/(N+1) through MOD(INFO,N+1).  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Ming Gu and Ren-Cang Li, Computer Science Division, University of  
 *       California at Berkeley, USA  
 *     Osni Marques, LBNL/NERSC, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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          END IF           END IF
 *  *
 *        In the real version, B is passed to DLASDQ and multiplied  *        In the real version, B is passed to DLASDQ and multiplied
 *        internally by Q'. Here B is complex and that product is  *        internally by Q**H. Here B is complex and that product is
 *        computed below in two steps (real and imaginary parts).  *        computed below in two steps (real and imaginary parts).
 *  *
          J = IRWB - 1           J = IRWB - 1
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 *  *
 *        Since B is complex, the following call to DGEMM is performed  *        Since B is complex, the following call to DGEMM is performed
 *        in two steps (real and imaginary parts). That is for V * B  *        in two steps (real and imaginary parts). That is for V * B
 *        (in the real version of the code V' is stored in WORK).  *        (in the real version of the code V**H is stored in WORK).
 *  *
 *        CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO,  *        CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO,
 *    $               WORK( NWORK ), N )  *    $               WORK( NWORK ), N )
Line 431 Line 523
                END IF                 END IF
 *  *
 *              In the real version, B is passed to DLASDQ and multiplied  *              In the real version, B is passed to DLASDQ and multiplied
 *              internally by Q'. Here B is complex and that product is  *              internally by Q**H. Here B is complex and that product is
 *              computed below in two steps (real and imaginary parts).  *              computed below in two steps (real and imaginary parts).
 *  *
                J = IRWB - 1                 J = IRWB - 1
Removed from v.1.8  
changed lines
  Added in v.1.14

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