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version 1.8, 2011/07/22 07:38:06 version 1.9, 2011/11/21 20:42:55
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   *> \brief \b DLALSA
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLALSA + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlalsa.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlalsa.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlalsa.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
   *                          LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
   *                          GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
   *                          IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
   *      $                   SMLSIZ
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
   *      $                   K( * ), PERM( LDGCOL, * )
   *       DOUBLE PRECISION   B( LDB, * ), BX( LDBX, * ), C( * ),
   *      $                   DIFL( LDU, * ), DIFR( LDU, * ),
   *      $                   GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
   *      $                   U( LDU, * ), VT( LDU, * ), WORK( * ),
   *      $                   Z( LDU, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLALSA is an itermediate step in solving the least squares problem
   *> by computing the SVD of the coefficient matrix in compact form (The
   *> singular vectors are computed as products of simple orthorgonal
   *> matrices.).
   *>
   *> If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
   *> matrix of an upper bidiagonal matrix to the right hand side; and if
   *> ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
   *> right hand side. The singular vector matrices were generated in
   *> compact form by DLALSA.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] ICOMPQ
   *> \verbatim
   *>          ICOMPQ is INTEGER
   *>         Specifies whether the left or the right singular vector
   *>         matrix is involved.
   *>         = 0: Left singular vector matrix
   *>         = 1: Right singular vector 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 row and column dimensions of the upper bidiagonal matrix.
   *> \endverbatim
   *>
   *> \param[in] NRHS
   *> \verbatim
   *>          NRHS is INTEGER
   *>         The number of columns of B and BX. NRHS must be at least 1.
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
   *>         On input, B contains the right hand sides of the least
   *>         squares problem in rows 1 through M.
   *>         On output, B contains the solution X in rows 1 through N.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>         The leading dimension of B in the calling subprogram.
   *>         LDB must be at least max(1,MAX( M, N ) ).
   *> \endverbatim
   *>
   *> \param[out] BX
   *> \verbatim
   *>          BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
   *>         On exit, the result of applying the left or right singular
   *>         vector matrix to B.
   *> \endverbatim
   *>
   *> \param[in] LDBX
   *> \verbatim
   *>          LDBX is INTEGER
   *>         The leading dimension of BX.
   *> \endverbatim
   *>
   *> \param[in] U
   *> \verbatim
   *>          U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
   *>         On entry, 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[in] VT
   *> \verbatim
   *>          VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
   *>         On entry, VT**T contains the right singular vector matrices of
   *>         all subproblems at the bottom level.
   *> \endverbatim
   *>
   *> \param[in] K
   *> \verbatim
   *>          K is INTEGER array, dimension ( N ).
   *> \endverbatim
   *>
   *> \param[in] DIFL
   *> \verbatim
   *>          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
   *>         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
   *> \endverbatim
   *>
   *> \param[in] DIFR
   *> \verbatim
   *>          DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
   *>         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
   *>         distances between singular values on the I-th level and
   *>         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
   *>         record the normalizing factors of the right singular vectors
   *>         matrices of subproblems on I-th level.
   *> \endverbatim
   *>
   *> \param[in] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
   *>         On entry, Z(1, I) contains the components of the deflation-
   *>         adjusted updating row vector for subproblems on the I-th
   *>         level.
   *> \endverbatim
   *>
   *> \param[in] POLES
   *> \verbatim
   *>          POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
   *>         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
   *>         singular values involved in the secular equations on the I-th
   *>         level.
   *> \endverbatim
   *>
   *> \param[in] GIVPTR
   *> \verbatim
   *>          GIVPTR is INTEGER array, dimension ( N ).
   *>         On entry, GIVPTR( I ) records the number of Givens
   *>         rotations performed on the I-th problem on the computation
   *>         tree.
   *> \endverbatim
   *>
   *> \param[in] GIVCOL
   *> \verbatim
   *>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
   *>         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records 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[in] PERM
   *> \verbatim
   *>          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
   *>         On entry, PERM(*, I) records permutations done on the I-th
   *>         level of the computation tree.
   *> \endverbatim
   *>
   *> \param[in] GIVNUM
   *> \verbatim
   *>          GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
   *>         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
   *>         values of Givens rotations performed on the I-th level on the
   *>         computation tree.
   *> \endverbatim
   *>
   *> \param[in] C
   *> \verbatim
   *>          C is DOUBLE PRECISION array, dimension ( N ).
   *>         On entry, if the I-th subproblem is not square,
   *>         C( I ) contains the C-value of a Givens rotation related to
   *>         the right null space of the I-th subproblem.
   *> \endverbatim
   *>
   *> \param[in] S
   *> \verbatim
   *>          S is DOUBLE PRECISION array, dimension ( N ).
   *>         On entry, if the I-th subproblem is not square,
   *>         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.
   *>         The dimension must be at least N.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array.
   *>         The dimension must be at least 3 * N
   *> \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 doubleOTHERcomputational
   *
   *> \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 DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,        SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
      $                   LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,       $                   LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
      $                   GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,       $                   GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
      $                   IWORK, INFO )       $                   IWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational 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            ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,        INTEGER            ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
Line 22 Line 288
      $                   Z( LDU, * )       $                   Z( LDU, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLALSA is an itermediate step in solving the least squares problem  
 *  by computing the SVD of the coefficient matrix in compact form (The  
 *  singular vectors are computed as products of simple orthorgonal  
 *  matrices.).  
 *  
 *  If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector  
 *  matrix of an upper bidiagonal matrix to the right hand side; and if  
 *  ICOMPQ = 1, DLALSA applies the right singular vector matrix to the  
 *  right hand side. The singular vector matrices were generated in  
 *  compact form by DLALSA.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  
 *  ICOMPQ (input) INTEGER  
 *         Specifies whether the left or the right singular vector  
 *         matrix is involved.  
 *         = 0: Left singular vector matrix  
 *         = 1: Right singular vector matrix  
 *  
 *  SMLSIZ (input) INTEGER  
 *         The maximum size of the subproblems at the bottom of the  
 *         computation tree.  
 *  
 *  N      (input) INTEGER  
 *         The row and column dimensions of the upper bidiagonal matrix.  
 *  
 *  NRHS   (input) INTEGER  
 *         The number of columns of B and BX. NRHS must be at least 1.  
 *  
 *  B      (input/output) DOUBLE PRECISION array, dimension ( LDB, NRHS )  
 *         On input, B contains the right hand sides of the least  
 *         squares problem in rows 1 through M.  
 *         On output, B contains the solution X in rows 1 through N.  
 *  
 *  LDB    (input) INTEGER  
 *         The leading dimension of B in the calling subprogram.  
 *         LDB must be at least max(1,MAX( M, N ) ).  
 *  
 *  BX     (output) DOUBLE PRECISION array, dimension ( LDBX, NRHS )  
 *         On exit, the result of applying the left or right singular  
 *         vector matrix to B.  
 *  
 *  LDBX   (input) INTEGER  
 *         The leading dimension of BX.  
 *  
 *  U      (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).  
 *         On entry, 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     (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).  
 *         On entry, VT**T contains the right singular vector matrices of  
 *         all subproblems at the bottom level.  
 *  
 *  K      (input) INTEGER array, dimension ( N ).  
 *  
 *  DIFL   (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).  
 *         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.  
 *  
 *  DIFR   (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).  
 *         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record  
 *         distances between singular values on the I-th level and  
 *         singular values on the (I -1)-th level, and DIFR(*, 2 * I)  
 *         record the normalizing factors of the right singular vectors  
 *         matrices of subproblems on I-th level.  
 *  
 *  Z      (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).  
 *         On entry, Z(1, I) contains the components of the deflation-  
 *         adjusted updating row vector for subproblems on the I-th  
 *         level.  
 *  
 *  POLES  (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).  
 *         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old  
 *         singular values involved in the secular equations on the I-th  
 *         level.  
 *  
 *  GIVPTR (input) INTEGER array, dimension ( N ).  
 *         On entry, GIVPTR( I ) records the number of Givens  
 *         rotations performed on the I-th problem on the computation  
 *         tree.  
 *  
 *  GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).  
 *         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records 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   (input) INTEGER array, dimension ( LDGCOL, NLVL ).  
 *         On entry, PERM(*, I) records permutations done on the I-th  
 *         level of the computation tree.  
 *  
 *  GIVNUM (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).  
 *         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-  
 *         values of Givens rotations performed on the I-th level on the  
 *         computation tree.  
 *  
 *  C      (input) DOUBLE PRECISION array, dimension ( N ).  
 *         On entry, if the I-th subproblem is not square,  
 *         C( I ) contains the C-value of a Givens rotation related to  
 *         the right null space of the I-th subproblem.  
 *  
 *  S      (input) DOUBLE PRECISION array, dimension ( N ).  
 *         On entry, if the I-th subproblem is not square,  
 *         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.  
 *         The dimension must be at least N.  
 *  
 *  IWORK  (workspace) INTEGER array.  
 *         The dimension must be at least 3 * N  
 *  
 *  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 Ren-Cang Li, Computer Science Division, University of  
 *       California at Berkeley, USA  
 *     Osni Marques, LBNL/NERSC, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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