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version 1.5, 2010/08/07 13:18:08 version 1.19, 2023/08/07 08:39:14
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   *> \brief \b DTZRQF
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DTZRQF + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtzrqf.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtzrqf.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtzrqf.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, M, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), TAU( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> This routine is deprecated and has been replaced by routine DTZRZF.
   *>
   *> DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
   *> to upper triangular form by means of orthogonal transformations.
   *>
   *> The upper trapezoidal matrix A is factored as
   *>
   *>    A = ( R  0 ) * Z,
   *>
   *> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
   *> triangular matrix.
   *> \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 >= M.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, the leading M-by-N upper trapezoidal part of the
   *>          array A must contain the matrix to be factorized.
   *>          On exit, the leading M-by-M upper triangular part of A
   *>          contains the upper triangular matrix R, and elements M+1 to
   *>          N of the first M rows of A, with the array TAU, represent the
   *>          orthogonal matrix Z as a product of M elementary reflectors.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[out] TAU
   *> \verbatim
   *>          TAU is DOUBLE PRECISION array, dimension (M)
   *>          The scalar factors of the elementary reflectors.
   *> \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.
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The factorization is obtained by Householder's method.  The kth
   *>  transformation matrix, Z( k ), which is used to introduce zeros into
   *>  the ( m - k + 1 )th row of A, is given in the form
   *>
   *>     Z( k ) = ( I     0   ),
   *>              ( 0  T( k ) )
   *>
   *>  where
   *>
   *>     T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
   *>                                                   (   0    )
   *>                                                   ( z( k ) )
   *>
   *>  tau is a scalar and z( k ) is an ( n - m ) element vector.
   *>  tau and z( k ) are chosen to annihilate the elements of the kth row
   *>  of X.
   *>
   *>  The scalar tau is returned in the kth element of TAU and the vector
   *>  u( k ) in the kth row of A, such that the elements of z( k ) are
   *>  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
   *>  the upper triangular part of A.
   *>
   *>  Z is given by
   *>
   *>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )        SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
 *  *
 *  -- LAPACK routine (version 3.2.2) --  *  -- LAPACK computational routine --
 *  -- 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  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, M, N        INTEGER            INFO, LDA, M, N
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       DOUBLE PRECISION   A( LDA, * ), TAU( * )        DOUBLE PRECISION   A( LDA, * ), TAU( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  This routine is deprecated and has been replaced by routine DTZRZF.  
 *  
 *  DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A  
 *  to upper triangular form by means of orthogonal transformations.  
 *  
 *  The upper trapezoidal matrix A is factored as  
 *  
 *     A = ( R  0 ) * Z,  
 *  
 *  where Z is an N-by-N orthogonal matrix and R is an M-by-M upper  
 *  triangular matrix.  
 *  
 *  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 >= M.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, the leading M-by-N upper trapezoidal part of the  
 *          array A must contain the matrix to be factorized.  
 *          On exit, the leading M-by-M upper triangular part of A  
 *          contains the upper triangular matrix R, and elements M+1 to  
 *          N of the first M rows of A, with the array TAU, represent the  
 *          orthogonal matrix Z as a product of M elementary reflectors.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,M).  
 *  
 *  TAU     (output) DOUBLE PRECISION array, dimension (M)  
 *          The scalar factors of the elementary reflectors.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The factorization is obtained by Householder's method.  The kth  
 *  transformation matrix, Z( k ), which is used to introduce zeros into  
 *  the ( m - k + 1 )th row of A, is given in the form  
 *  
 *     Z( k ) = ( I     0   ),  
 *              ( 0  T( k ) )  
 *  
 *  where  
 *  
 *     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),  
 *                                                 (   0    )  
 *                                                 ( z( k ) )  
 *  
 *  tau is a scalar and z( k ) is an ( n - m ) element vector.  
 *  tau and z( k ) are chosen to annihilate the elements of the kth row  
 *  of X.  
 *  
 *  The scalar tau is returned in the kth element of TAU and the vector  
 *  u( k ) in the kth row of A, such that the elements of z( k ) are  
 *  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in  
 *  the upper triangular part of A.  
 *  
 *  Z is given by  
 *  
 *     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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      $                     LDA, A( K, M1 ), LDA, ONE, TAU, 1 )       $                     LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
 *  *
 *              Now form  a( k ) := a( k ) - tau*w  *              Now form  a( k ) := a( k ) - tau*w
 *              and       B      := B      - tau*w*z( k )'.  *              and       B      := B      - tau*w*z( k )**T.
 *  *
                CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )                 CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
                CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA,                 CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA,
Removed from v.1.5  
changed lines
  Added in v.1.19

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