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version 1.1, 2010/01/26 15:22:45 version 1.19, 2023/08/07 08:39:29
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   *> \brief \b ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZLAHR2 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahr2.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahr2.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahr2.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            K, LDA, LDT, LDY, N, NB
   *       ..
   *       .. Array Arguments ..
   *       COMPLEX*16        A( LDA, * ), T( LDT, NB ), TAU( NB ),
   *      $                   Y( LDY, NB )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
   *> matrix A so that elements below the k-th subdiagonal are zero. The
   *> reduction is performed by an unitary similarity transformation
   *> Q**H * A * Q. The routine returns the matrices V and T which determine
   *> Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
   *>
   *> This is an auxiliary routine called by ZGEHRD.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.
   *> \endverbatim
   *>
   *> \param[in] K
   *> \verbatim
   *>          K is INTEGER
   *>          The offset for the reduction. Elements below the k-th
   *>          subdiagonal in the first NB columns are reduced to zero.
   *>          K < N.
   *> \endverbatim
   *>
   *> \param[in] NB
   *> \verbatim
   *>          NB is INTEGER
   *>          The number of columns to be reduced.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,N-K+1)
   *>          On entry, the n-by-(n-k+1) general matrix A.
   *>          On exit, the elements on and above the k-th subdiagonal in
   *>          the first NB columns are overwritten with the corresponding
   *>          elements of the reduced matrix; the elements below the k-th
   *>          subdiagonal, with the array TAU, represent the matrix Q as a
   *>          product of elementary reflectors. The other columns of A are
   *>          unchanged. See Further Details.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] TAU
   *> \verbatim
   *>          TAU is COMPLEX*16 array, dimension (NB)
   *>          The scalar factors of the elementary reflectors. See Further
   *>          Details.
   *> \endverbatim
   *>
   *> \param[out] T
   *> \verbatim
   *>          T is COMPLEX*16 array, dimension (LDT,NB)
   *>          The upper triangular matrix T.
   *> \endverbatim
   *>
   *> \param[in] LDT
   *> \verbatim
   *>          LDT is INTEGER
   *>          The leading dimension of the array T.  LDT >= NB.
   *> \endverbatim
   *>
   *> \param[out] Y
   *> \verbatim
   *>          Y is COMPLEX*16 array, dimension (LDY,NB)
   *>          The n-by-nb matrix Y.
   *> \endverbatim
   *>
   *> \param[in] LDY
   *> \verbatim
   *>          LDY is INTEGER
   *>          The leading dimension of the array Y. LDY >= N.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup complex16OTHERauxiliary
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The matrix Q is represented as a product of nb elementary reflectors
   *>
   *>     Q = H(1) H(2) . . . H(nb).
   *>
   *>  Each H(i) has the form
   *>
   *>     H(i) = I - tau * v * v**H
   *>
   *>  where tau is a complex scalar, and v is a complex vector with
   *>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
   *>  A(i+k+1:n,i), and tau in TAU(i).
   *>
   *>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
   *>  V which is needed, with T and Y, to apply the transformation to the
   *>  unreduced part of the matrix, using an update of the form:
   *>  A := (I - V*T*V**H) * (A - Y*V**H).
   *>
   *>  The contents of A on exit are illustrated by the following example
   *>  with n = 7, k = 3 and nb = 2:
   *>
   *>     ( a   a   a   a   a )
   *>     ( a   a   a   a   a )
   *>     ( a   a   a   a   a )
   *>     ( h   h   a   a   a )
   *>     ( v1  h   a   a   a )
   *>     ( v1  v2  a   a   a )
   *>     ( v1  v2  a   a   a )
   *>
   *>  where a denotes an element of the original matrix A, h denotes a
   *>  modified element of the upper Hessenberg matrix H, and vi denotes an
   *>  element of the vector defining H(i).
   *>
   *>  This subroutine is a slight modification of LAPACK-3.0's ZLAHRD
   *>  incorporating improvements proposed by Quintana-Orti and Van de
   *>  Gejin. Note that the entries of A(1:K,2:NB) differ from those
   *>  returned by the original LAPACK-3.0's ZLAHRD routine. (This
   *>  subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)
   *> \endverbatim
   *
   *> \par References:
   *  ================
   *>
   *>  Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
   *>  performance of reduction to Hessenberg form," ACM Transactions on
   *>  Mathematical Software, 32(2):180-194, June 2006.
   *>
   *  =====================================================================
       SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )        SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2.1)                        --  *  -- LAPACK auxiliary 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..--
 *  -- April 2009                                                      --  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            K, LDA, LDT, LDY, N, NB        INTEGER            K, LDA, LDT, LDY, N, NB
Line 13 Line 191
      $                   Y( LDY, NB )       $                   Y( LDY, NB )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)  
 *  matrix A so that elements below the k-th subdiagonal are zero. The  
 *  reduction is performed by an unitary similarity transformation  
 *  Q' * A * Q. The routine returns the matrices V and T which determine  
 *  Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.  
 *  
 *  This is an auxiliary routine called by ZGEHRD.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A.  
 *  
 *  K       (input) INTEGER  
 *          The offset for the reduction. Elements below the k-th  
 *          subdiagonal in the first NB columns are reduced to zero.  
 *          K < N.  
 *  
 *  NB      (input) INTEGER  
 *          The number of columns to be reduced.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)  
 *          On entry, the n-by-(n-k+1) general matrix A.  
 *          On exit, the elements on and above the k-th subdiagonal in  
 *          the first NB columns are overwritten with the corresponding  
 *          elements of the reduced matrix; the elements below the k-th  
 *          subdiagonal, with the array TAU, represent the matrix Q as a  
 *          product of elementary reflectors. The other columns of A are  
 *          unchanged. See Further Details.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  TAU     (output) COMPLEX*16 array, dimension (NB)  
 *          The scalar factors of the elementary reflectors. See Further  
 *          Details.  
 *  
 *  T       (output) COMPLEX*16 array, dimension (LDT,NB)  
 *          The upper triangular matrix T.  
 *  
 *  LDT     (input) INTEGER  
 *          The leading dimension of the array T.  LDT >= NB.  
 *  
 *  Y       (output) COMPLEX*16 array, dimension (LDY,NB)  
 *          The n-by-nb matrix Y.  
 *  
 *  LDY     (input) INTEGER  
 *          The leading dimension of the array Y. LDY >= N.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The matrix Q is represented as a product of nb elementary reflectors  
 *  
 *     Q = H(1) H(2) . . . H(nb).  
 *  
 *  Each H(i) has the form  
 *  
 *     H(i) = I - tau * v * v'  
 *  
 *  where tau is a complex scalar, and v is a complex vector with  
 *  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in  
 *  A(i+k+1:n,i), and tau in TAU(i).  
 *  
 *  The elements of the vectors v together form the (n-k+1)-by-nb matrix  
 *  V which is needed, with T and Y, to apply the transformation to the  
 *  unreduced part of the matrix, using an update of the form:  
 *  A := (I - V*T*V') * (A - Y*V').  
 *  
 *  The contents of A on exit are illustrated by the following example  
 *  with n = 7, k = 3 and nb = 2:  
 *  
 *     ( a   a   a   a   a )  
 *     ( a   a   a   a   a )  
 *     ( a   a   a   a   a )  
 *     ( h   h   a   a   a )  
 *     ( v1  h   a   a   a )  
 *     ( v1  v2  a   a   a )  
 *     ( v1  v2  a   a   a )  
 *  
 *  where a denotes an element of the original matrix A, h denotes a  
 *  modified element of the upper Hessenberg matrix H, and vi denotes an  
 *  element of the vector defining H(i).  
 *  
 *  This subroutine is a slight modification of LAPACK-3.0's DLAHRD  
 *  incorporating improvements proposed by Quintana-Orti and Van de  
 *  Gejin. Note that the entries of A(1:K,2:NB) differ from those  
 *  returned by the original LAPACK-3.0's DLAHRD routine. (This  
 *  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)  
 *  
 *  References  
 *  ==========  
 *  
 *  Gregorio Quintana-Orti and Robert van de Geijn, "Improving the  
 *  performance of reduction to Hessenberg form," ACM Transactions on  
 *  Mathematical Software, 32(2):180-194, June 2006.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
       COMPLEX*16        ZERO, ONE        COMPLEX*16        ZERO, ONE
       PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),         PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),
      $                     ONE = ( 1.0D+0, 0.0D+0 ) )       $                     ONE = ( 1.0D+0, 0.0D+0 ) )
 *     ..  *     ..
 *     .. Local Scalars ..  *     .. Local Scalars ..
Line 144 Line 221
 *  *
 *           Update A(K+1:N,I)  *           Update A(K+1:N,I)
 *  *
 *           Update I-th column of A - Y * V'  *           Update I-th column of A - Y * V**H
 *  *
             CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )               CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
             CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,              CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
      $                  A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )       $                  A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
             CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )               CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
 *  *
 *           Apply I - V * T' * V' to this column (call it b) from the  *           Apply I - V * T**H * V**H to this column (call it b) from the
 *           left, using the last column of T as workspace  *           left, using the last column of T as workspace
 *  *
 *           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)  *           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
Line 159 Line 236
 *  *
 *           where V1 is unit lower triangular  *           where V1 is unit lower triangular
 *  *
 *           w := V1' * b1  *           w := V1**H * b1
 *  *
             CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )              CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
             CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT',               CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT',
      $                  I-1, A( K+1, 1 ),       $                  I-1, A( K+1, 1 ),
      $                  LDA, T( 1, NB ), 1 )       $                  LDA, T( 1, NB ), 1 )
 *  *
 *           w := w + V2'*b2  *           w := w + V2**H * b2
 *  *
             CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,               CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
      $                  ONE, A( K+I, 1 ),       $                  ONE, A( K+I, 1 ),
      $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )       $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
 *  *
 *           w := T'*w  *           w := T**H * w
 *  *
             CALL ZTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',               CALL ZTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',
      $                  I-1, T, LDT,       $                  I-1, T, LDT,
      $                  T( 1, NB ), 1 )       $                  T( 1, NB ), 1 )
 *  *
 *           b2 := b2 - V2*w  *           b2 := b2 - V2*w
 *  *
             CALL ZGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,               CALL ZGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
      $                  A( K+I, 1 ),       $                  A( K+I, 1 ),
      $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )       $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
 *  *
 *           b1 := b1 - V1*w  *           b1 := b1 - V1*w
 *  *
             CALL ZTRMV( 'Lower', 'NO TRANSPOSE',               CALL ZTRMV( 'Lower', 'NO TRANSPOSE',
      $                  'UNIT', I-1,       $                  'UNIT', I-1,
      $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )       $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
             CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )              CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
Line 204 Line 281
 *  *
 *        Compute  Y(K+1:N,I)  *        Compute  Y(K+1:N,I)
 *  *
          CALL ZGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,            CALL ZGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
      $               ONE, A( K+1, I+1 ),       $               ONE, A( K+1, I+1 ),
      $               LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )       $               LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
          CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,            CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
      $               ONE, A( K+I, 1 ), LDA,       $               ONE, A( K+I, 1 ), LDA,
      $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )       $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
          CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,            CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
      $               Y( K+1, 1 ), LDY,       $               Y( K+1, 1 ), LDY,
      $               T( 1, I ), 1, ONE, Y( K+1, I ), 1 )       $               T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
          CALL ZSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )           CALL ZSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
Line 218 Line 295
 *        Compute T(1:I,I)  *        Compute T(1:I,I)
 *  *
          CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )           CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
          CALL ZTRMV( 'Upper', 'No Transpose', 'NON-UNIT',            CALL ZTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
      $               I-1, T, LDT,       $               I-1, T, LDT,
      $               T( 1, I ), 1 )       $               T( 1, I ), 1 )
          T( I, I ) = TAU( I )           T( I, I ) = TAU( I )
Line 229 Line 306
 *     Compute Y(1:K,1:NB)  *     Compute Y(1:K,1:NB)
 *  *
       CALL ZLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )        CALL ZLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
       CALL ZTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',         CALL ZTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
      $            'UNIT', K, NB,       $            'UNIT', K, NB,
      $            ONE, A( K+1, 1 ), LDA, Y, LDY )       $            ONE, A( K+1, 1 ), LDA, Y, LDY )
       IF( N.GT.K+NB )        IF( N.GT.K+NB )
      $   CALL ZGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,        $   CALL ZGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
      $               NB, N-K-NB, ONE,       $               NB, N-K-NB, ONE,
      $               A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,       $               A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
      $               LDY )       $               LDY )
       CALL ZTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',         CALL ZTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
      $            'NON-UNIT', K, NB,       $            'NON-UNIT', K, NB,
      $            ONE, T, LDT, Y, LDY )       $            ONE, T, LDT, Y, LDY )
 *  *
Removed from v.1.1  
changed lines
  Added in v.1.19

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