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version 1.8, 2011/07/22 07:38:14 version 1.9, 2011/11/21 20:43:09
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   *> \brief <b> ZGELSY solves overdetermined or underdetermined systems for GE matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGELSY + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgelsy.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgelsy.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgelsy.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
   *                          WORK, LWORK, RWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            JPVT( * )
   *       DOUBLE PRECISION   RWORK( * )
   *       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGELSY computes the minimum-norm solution to a complex linear least
   *> squares problem:
   *>     minimize || A * X - B ||
   *> using a complete orthogonal factorization of A.  A is an M-by-N
   *> matrix which may be rank-deficient.
   *>
   *> Several right hand side vectors b and solution vectors x can be
   *> handled in a single call; they are stored as the columns of the
   *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
   *> matrix X.
   *>
   *> The routine first computes a QR factorization with column pivoting:
   *>     A * P = Q * [ R11 R12 ]
   *>                 [  0  R22 ]
   *> with R11 defined as the largest leading submatrix whose estimated
   *> condition number is less than 1/RCOND.  The order of R11, RANK,
   *> is the effective rank of A.
   *>
   *> Then, R22 is considered to be negligible, and R12 is annihilated
   *> by unitary transformations from the right, arriving at the
   *> complete orthogonal factorization:
   *>    A * P = Q * [ T11 0 ] * Z
   *>                [  0  0 ]
   *> The minimum-norm solution is then
   *>    X = P * Z**H [ inv(T11)*Q1**H*B ]
   *>                 [        0         ]
   *> where Q1 consists of the first RANK columns of Q.
   *>
   *> This routine is basically identical to the original xGELSX except
   *> three differences:
   *>   o The permutation of matrix B (the right hand side) is faster and
   *>     more simple.
   *>   o The call to the subroutine xGEQPF has been substituted by the
   *>     the call to the subroutine xGEQP3. This subroutine is a Blas-3
   *>     version of the QR factorization with column pivoting.
   *>   o Matrix B (the right hand side) is updated with Blas-3.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows of the matrix A.  M >= 0.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] NRHS
   *> \verbatim
   *>          NRHS is INTEGER
   *>          The number of right hand sides, i.e., the number of
   *>          columns of matrices B and X. NRHS >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,N)
   *>          On entry, the M-by-N matrix A.
   *>          On exit, A has been overwritten by details of its
   *>          complete orthogonal factorization.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
   *>          On entry, the M-by-NRHS right hand side matrix B.
   *>          On exit, the N-by-NRHS solution matrix X.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,M,N).
   *> \endverbatim
   *>
   *> \param[in,out] JPVT
   *> \verbatim
   *>          JPVT is INTEGER array, dimension (N)
   *>          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
   *>          to the front of AP, otherwise column i is a free column.
   *>          On exit, if JPVT(i) = k, then the i-th column of A*P
   *>          was the k-th column of A.
   *> \endverbatim
   *>
   *> \param[in] RCOND
   *> \verbatim
   *>          RCOND is DOUBLE PRECISION
   *>          RCOND is used to determine the effective rank of A, which
   *>          is defined as the order of the largest leading triangular
   *>          submatrix R11 in the QR factorization with pivoting of A,
   *>          whose estimated condition number < 1/RCOND.
   *> \endverbatim
   *>
   *> \param[out] RANK
   *> \verbatim
   *>          RANK is INTEGER
   *>          The effective rank of A, i.e., the order of the submatrix
   *>          R11.  This is the same as the order of the submatrix T11
   *>          in the complete orthogonal factorization of A.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          The dimension of the array WORK.
   *>          The unblocked strategy requires that:
   *>            LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
   *>          where MN = min(M,N).
   *>          The block algorithm requires that:
   *>            LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
   *>          where NB is an upper bound on the blocksize returned
   *>          by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR,
   *>          and ZUNMRZ.
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal size of the WORK array, returns
   *>          this value as the first entry of the WORK array, and no error
   *>          message related to LWORK is issued by XERBLA.
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension (2*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 complex16GEsolve
   *
   *> \par Contributors:
   *  ==================
   *>
   *>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n 
   *>    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
   *>    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
   *>
   *  =====================================================================
       SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,        SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
      $                   WORK, LWORK, RWORK, INFO )       $                   WORK, LWORK, RWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.3.1) --  *  -- LAPACK driver 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..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK        INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
Line 16 Line 225
       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )        COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGELSY computes the minimum-norm solution to a complex linear least  
 *  squares problem:  
 *      minimize || A * X - B ||  
 *  using a complete orthogonal factorization of A.  A is an M-by-N  
 *  matrix which may be rank-deficient.  
 *  
 *  Several right hand side vectors b and solution vectors x can be  
 *  handled in a single call; they are stored as the columns of the  
 *  M-by-NRHS right hand side matrix B and the N-by-NRHS solution  
 *  matrix X.  
 *  
 *  The routine first computes a QR factorization with column pivoting:  
 *      A * P = Q * [ R11 R12 ]  
 *                  [  0  R22 ]  
 *  with R11 defined as the largest leading submatrix whose estimated  
 *  condition number is less than 1/RCOND.  The order of R11, RANK,  
 *  is the effective rank of A.  
 *  
 *  Then, R22 is considered to be negligible, and R12 is annihilated  
 *  by unitary transformations from the right, arriving at the  
 *  complete orthogonal factorization:  
 *     A * P = Q * [ T11 0 ] * Z  
 *                 [  0  0 ]  
 *  The minimum-norm solution is then  
 *     X = P * Z**H [ inv(T11)*Q1**H*B ]  
 *                  [        0         ]  
 *  where Q1 consists of the first RANK columns of Q.  
 *  
 *  This routine is basically identical to the original xGELSX except  
 *  three differences:  
 *    o The permutation of matrix B (the right hand side) is faster and  
 *      more simple.  
 *    o The call to the subroutine xGEQPF has been substituted by the  
 *      the call to the subroutine xGEQP3. This subroutine is a Blas-3  
 *      version of the QR factorization with column pivoting.  
 *    o Matrix B (the right hand side) is updated with Blas-3.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  M       (input) INTEGER  
 *          The number of rows of the matrix A.  M >= 0.  
 *  
 *  N       (input) INTEGER  
 *          The number of columns of the matrix A.  N >= 0.  
 *  
 *  NRHS    (input) INTEGER  
 *          The number of right hand sides, i.e., the number of  
 *          columns of matrices B and X. NRHS >= 0.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)  
 *          On entry, the M-by-N matrix A.  
 *          On exit, A has been overwritten by details of its  
 *          complete orthogonal factorization.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,M).  
 *  
 *  B       (input/output) COMPLEX*16 array, dimension (LDB,NRHS)  
 *          On entry, the M-by-NRHS right hand side matrix B.  
 *          On exit, the N-by-NRHS solution matrix X.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B. LDB >= max(1,M,N).  
 *  
 *  JPVT    (input/output) INTEGER array, dimension (N)  
 *          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted  
 *          to the front of AP, otherwise column i is a free column.  
 *          On exit, if JPVT(i) = k, then the i-th column of A*P  
 *          was the k-th column of A.  
 *  
 *  RCOND   (input) DOUBLE PRECISION  
 *          RCOND is used to determine the effective rank of A, which  
 *          is defined as the order of the largest leading triangular  
 *          submatrix R11 in the QR factorization with pivoting of A,  
 *          whose estimated condition number < 1/RCOND.  
 *  
 *  RANK    (output) INTEGER  
 *          The effective rank of A, i.e., the order of the submatrix  
 *          R11.  This is the same as the order of the submatrix T11  
 *          in the complete orthogonal factorization of A.  
 *  
 *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The dimension of the array WORK.  
 *          The unblocked strategy requires that:  
 *            LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )  
 *          where MN = min(M,N).  
 *          The block algorithm requires that:  
 *            LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )  
 *          where NB is an upper bound on the blocksize returned  
 *          by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR,  
 *          and ZUNMRZ.  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal size of the WORK array, returns  
 *          this value as the first entry of the WORK array, and no error  
 *          message related to LWORK is issued by XERBLA.  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (2*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  
 *    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA  
 *    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain  
 *    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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