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version 1.5, 2010/08/07 13:18:06 version 1.19, 2023/08/07 08:38:49
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   *> \brief \b DGEQPF
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DGEQPF + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeqpf.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeqpf.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeqpf.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, M, N
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            JPVT( * )
   *       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> This routine is deprecated and has been replaced by routine DGEQP3.
   *>
   *> DGEQPF computes a QR factorization with column pivoting of a
   *> real M-by-N matrix A: A*P = Q*R.
   *> \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,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          On entry, the M-by-N matrix A.
   *>          On exit, the upper triangle of the array contains the
   *>          min(M,N)-by-N upper triangular matrix R; the elements
   *>          below the diagonal, together with the array TAU,
   *>          represent the orthogonal matrix Q as a product of
   *>          min(m,n) elementary reflectors.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A. LDA >= max(1,M).
   *> \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 A*P (a leading column); if JPVT(i) = 0,
   *>          the i-th column of A 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[out] TAU
   *> \verbatim
   *>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
   *>          The scalar factors of the elementary reflectors.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (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.
   *
   *> \ingroup doubleGEcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The matrix Q is represented as a product of elementary reflectors
   *>
   *>     Q = H(1) H(2) . . . H(n)
   *>
   *>  Each H(i) has the form
   *>
   *>     H = I - tau * v * v**T
   *>
   *>  where tau is a real scalar, and v is a real vector with
   *>  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
   *>
   *>  The matrix P is represented in jpvt as follows: If
   *>     jpvt(j) = i
   *>  then the jth column of P is the ith canonical unit vector.
   *>
   *>  Partial column norm updating strategy modified by
   *>    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
   *>    University of Zagreb, Croatia.
   *>  -- April 2011                                                      --
   *>  For more details see LAPACK Working Note 176.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )        SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
 *  *
 *  -- LAPACK deprecated computational 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( * ), WORK( * )        DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  This routine is deprecated and has been replaced by routine DGEQP3.  
 *  
 *  DGEQPF computes a QR factorization with column pivoting of a  
 *  real M-by-N matrix A: A*P = Q*R.  
 *  
 *  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  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, the M-by-N matrix A.  
 *          On exit, the upper triangle of the array contains the  
 *          min(M,N)-by-N upper triangular matrix R; the elements  
 *          below the diagonal, together with the array TAU,  
 *          represent the orthogonal matrix Q as a product of  
 *          min(m,n) elementary reflectors.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1,M).  
 *  
 *  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 A*P (a leading column); if JPVT(i) = 0,  
 *          the i-th column of A 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.  
 *  
 *  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))  
 *          The scalar factors of the elementary reflectors.  
 *  
 *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The matrix Q is represented as a product of elementary reflectors  
 *  
 *     Q = H(1) H(2) . . . H(n)  
 *  
 *  Each H(i) has the form  
 *  
 *     H = I - tau * v * v'  
 *  
 *  where tau is a real scalar, and v is a real vector with  
 *  v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).  
 *  
 *  The matrix P is represented in jpvt as follows: If  
 *     jpvt(j) = i  
 *  then the jth column of P is the ith canonical unit vector.  
 *  
 *  Partial column norm updating strategy modified by  
 *    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,  
 *    University of Zagreb, Croatia.  
 *     June 2010  
 *  For more details see LAPACK Working Note 176.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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 *  *
 *                 NOTE: The following 4 lines follow from the analysis in  *                 NOTE: The following 4 lines follow from the analysis in
 *                 Lapack Working Note 176.  *                 Lapack Working Note 176.
 *                   *
                   TEMP = ABS( A( I, J ) ) / WORK( J )                    TEMP = ABS( A( I, J ) ) / WORK( J )
                   TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )                    TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
                   TEMP2 = TEMP*( WORK( J ) / WORK( N+J ) )**2                    TEMP2 = TEMP*( WORK( J ) / WORK( N+J ) )**2
                   IF( TEMP2 .LE. TOL3Z ) THEN                     IF( TEMP2 .LE. TOL3Z ) THEN
                      IF( M-I.GT.0 ) THEN                       IF( M-I.GT.0 ) THEN
                         WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )                          WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )
                         WORK( N+J ) = WORK( J )                          WORK( N+J ) = WORK( J )
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  Added in v.1.19

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