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version 1.5, 2010/08/07 13:18:09 version 1.18, 2017/06/17 11:06:55
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   *> \brief \b ZLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZLAQPS + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqps.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqps.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqps.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
   *                          VN2, AUXV, F, LDF )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            KB, LDA, LDF, M, N, NB, OFFSET
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            JPVT( * )
   *       DOUBLE PRECISION   VN1( * ), VN2( * )
   *       COMPLEX*16         A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZLAQPS computes a step of QR factorization with column pivoting
   *> of a complex M-by-N matrix A by using Blas-3.  It tries to factorize
   *> NB columns from A starting from the row OFFSET+1, and updates all
   *> of the matrix with Blas-3 xGEMM.
   *>
   *> In some cases, due to catastrophic cancellations, it cannot
   *> factorize NB columns.  Hence, the actual number of factorized
   *> columns is returned in KB.
   *>
   *> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
   *> \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] OFFSET
   *> \verbatim
   *>          OFFSET is INTEGER
   *>          The number of rows of A that have been factorized in
   *>          previous steps.
   *> \endverbatim
   *>
   *> \param[in] NB
   *> \verbatim
   *>          NB is INTEGER
   *>          The number of columns to factorize.
   *> \endverbatim
   *>
   *> \param[out] KB
   *> \verbatim
   *>          KB is INTEGER
   *>          The number of columns actually factorized.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,N)
   *>          On entry, the M-by-N matrix A.
   *>          On exit, block A(OFFSET+1:M,1:KB) is the triangular
   *>          factor obtained and block A(1:OFFSET,1:N) has been
   *>          accordingly pivoted, but no factorized.
   *>          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
   *>          been updated.
   *> \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)
   *>          JPVT(I) = K <==> Column K of the full matrix A has been
   *>          permuted into position I in AP.
   *> \endverbatim
   *>
   *> \param[out] TAU
   *> \verbatim
   *>          TAU is COMPLEX*16 array, dimension (KB)
   *>          The scalar factors of the elementary reflectors.
   *> \endverbatim
   *>
   *> \param[in,out] VN1
   *> \verbatim
   *>          VN1 is DOUBLE PRECISION array, dimension (N)
   *>          The vector with the partial column norms.
   *> \endverbatim
   *>
   *> \param[in,out] VN2
   *> \verbatim
   *>          VN2 is DOUBLE PRECISION array, dimension (N)
   *>          The vector with the exact column norms.
   *> \endverbatim
   *>
   *> \param[in,out] AUXV
   *> \verbatim
   *>          AUXV is COMPLEX*16 array, dimension (NB)
   *>          Auxiliar vector.
   *> \endverbatim
   *>
   *> \param[in,out] F
   *> \verbatim
   *>          F is COMPLEX*16 array, dimension (LDF,NB)
   *>          Matrix F**H = L * Y**H * A.
   *> \endverbatim
   *>
   *> \param[in] LDF
   *> \verbatim
   *>          LDF is INTEGER
   *>          The leading dimension of the array F. LDF >= max(1,N).
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \date December 2016
   *
   *> \ingroup complex16OTHERauxiliary
   *
   *> \par Contributors:
   *  ==================
   *>
   *>    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
   *>    X. Sun, Computer Science Dept., Duke University, USA
   *> \n
   *>  Partial column norm updating strategy modified on April 2011
   *>    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
   *>    University of Zagreb, Croatia.
   *
   *> \par References:
   *  ================
   *>
   *> LAPACK Working Note 176
   *
   *> \htmlonly
   *> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
   *> \endhtmlonly
   *
   *  =====================================================================
       SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,        SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
      $                   VN2, AUXV, F, LDF )       $                   VN2, AUXV, F, LDF )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2.2) --  *  -- LAPACK auxiliary routine (version 3.7.0) --
 *  -- 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..--
 *     June 2010  *     December 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            KB, LDA, LDF, M, N, NB, OFFSET        INTEGER            KB, LDA, LDF, M, N, NB, OFFSET
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       COMPLEX*16         A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )        COMPLEX*16         A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZLAQPS computes a step of QR factorization with column pivoting  
 *  of a complex M-by-N matrix A by using Blas-3.  It tries to factorize  
 *  NB columns from A starting from the row OFFSET+1, and updates all  
 *  of the matrix with Blas-3 xGEMM.  
 *  
 *  In some cases, due to catastrophic cancellations, it cannot  
 *  factorize NB columns.  Hence, the actual number of factorized  
 *  columns is returned in KB.  
 *  
 *  Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.  
 *  
 *  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  
 *  
 *  OFFSET  (input) INTEGER  
 *          The number of rows of A that have been factorized in  
 *          previous steps.  
 *  
 *  NB      (input) INTEGER  
 *          The number of columns to factorize.  
 *  
 *  KB      (output) INTEGER  
 *          The number of columns actually factorized.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)  
 *          On entry, the M-by-N matrix A.  
 *          On exit, block A(OFFSET+1:M,1:KB) is the triangular  
 *          factor obtained and block A(1:OFFSET,1:N) has been  
 *          accordingly pivoted, but no factorized.  
 *          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has  
 *          been updated.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1,M).  
 *  
 *  JPVT    (input/output) INTEGER array, dimension (N)  
 *          JPVT(I) = K <==> Column K of the full matrix A has been  
 *          permuted into position I in AP.  
 *  
 *  TAU     (output) COMPLEX*16 array, dimension (KB)  
 *          The scalar factors of the elementary reflectors.  
 *  
 *  VN1     (input/output) DOUBLE PRECISION array, dimension (N)  
 *          The vector with the partial column norms.  
 *  
 *  VN2     (input/output) DOUBLE PRECISION array, dimension (N)  
 *          The vector with the exact column norms.  
 *  
 *  AUXV    (input/output) COMPLEX*16 array, dimension (NB)  
 *          Auxiliar vector.  
 *  
 *  F       (input/output) COMPLEX*16 array, dimension (LDF,NB)  
 *          Matrix F' = L*Y'*A.  
 *  
 *  LDF     (input) INTEGER  
 *          The leading dimension of the array F. LDF >= max(1,N).  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain  
 *    X. Sun, Computer Science Dept., Duke University, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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          END IF           END IF
 *  *
 *        Apply previous Householder reflectors to column K:  *        Apply previous Householder reflectors to column K:
 *        A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'.  *        A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**H.
 *  *
          IF( K.GT.1 ) THEN           IF( K.GT.1 ) THEN
             DO 20 J = 1, K - 1              DO 20 J = 1, K - 1
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 *  *
 *        Compute Kth column of F:  *        Compute Kth column of F:
 *  *
 *        Compute  F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K).  *        Compute  F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**H*A(RK:M,K).
 *  *
          IF( K.LT.N ) THEN           IF( K.LT.N ) THEN
             CALL ZGEMV( 'Conjugate transpose', M-RK+1, N-K, TAU( K ),              CALL ZGEMV( 'Conjugate transpose', M-RK+1, N-K, TAU( K ),
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    40    CONTINUE     40    CONTINUE
 *  *
 *        Incremental updating of F:  *        Incremental updating of F:
 *        F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)'  *        F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**H
 *                    *A(RK:M,K).  *                    *A(RK:M,K).
 *  *
          IF( K.GT.1 ) THEN           IF( K.GT.1 ) THEN
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          END IF           END IF
 *  *
 *        Update the current row of A:  *        Update the current row of A:
 *        A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'.  *        A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**H.
 *  *
          IF( K.LT.N ) THEN           IF( K.LT.N ) THEN
             CALL ZGEMM( 'No transpose', 'Conjugate transpose', 1, N-K,              CALL ZGEMM( 'No transpose', 'Conjugate transpose', 1, N-K,
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 *  *
 *     Apply the block reflector to the rest of the matrix:  *     Apply the block reflector to the rest of the matrix:
 *     A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -  *     A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
 *                         A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)'.  *                         A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**H.
 *  *
       IF( KB.LT.MIN( N, M-OFFSET ) ) THEN        IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
          CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-RK, N-KB,           CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-RK, N-KB,
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          ITEMP = NINT( VN2( LSTICC ) )           ITEMP = NINT( VN2( LSTICC ) )
          VN1( LSTICC ) = DZNRM2( M-RK, A( RK+1, LSTICC ), 1 )           VN1( LSTICC ) = DZNRM2( M-RK, A( RK+1, LSTICC ), 1 )
 *  *
 *        NOTE: The computation of VN1( LSTICC ) relies on the fact that   *        NOTE: The computation of VN1( LSTICC ) relies on the fact that
 *        SNRM2 does not fail on vectors with norm below the value of  *        SNRM2 does not fail on vectors with norm below the value of
 *        SQRT(DLAMCH('S'))   *        SQRT(DLAMCH('S'))
 *  *
          VN2( LSTICC ) = VN1( LSTICC )           VN2( LSTICC ) = VN1( LSTICC )
          LSTICC = ITEMP           LSTICC = ITEMP
Removed from v.1.5  
changed lines
  Added in v.1.18

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