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version 1.7, 2010/12/21 13:53:29 version 1.16, 2016/08/27 15:34:27
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   *> \brief \b DLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th 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 DLAHRD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahrd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahrd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahrd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            K, LDA, LDT, LDY, N, NB
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), T( LDT, NB ), TAU( NB ),
   *      $                   Y( LDY, NB )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> This routine is deprecated and has been replaced by routine DLAHR2.
   *>
   *> DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
   *> matrix A so that elements below the k-th subdiagonal are zero. The
   *> reduction is performed by an orthogonal similarity transformation
   *> Q**T * A * Q. The routine returns the matrices V and T which determine
   *> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
   *> \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.
   *> \endverbatim
   *>
   *> \param[in] NB
   *> \verbatim
   *>          NB is INTEGER
   *>          The number of columns to be reduced.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)
   *>          The scalar factors of the elementary reflectors. See Further
   *>          Details.
   *> \endverbatim
   *>
   *> \param[out] T
   *> \verbatim
   *>          T is DOUBLE PRECISION 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 DOUBLE PRECISION 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. 
   *
   *> \date November 2015
   *
   *> \ingroup doubleOTHERauxiliary
   *
   *> \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**T
   *>
   *>  where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
   *>
   *>  The contents of A on exit are illustrated by the following example
   *>  with n = 7, k = 3 and nb = 2:
   *>
   *>     ( a   h   a   a   a )
   *>     ( a   h   a   a   a )
   *>     ( a   h   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).
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )        SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine (version 3.6.0) --
 *  -- 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..--
 *     November 2006  *     November 2015
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            K, LDA, LDT, LDY, N, NB        INTEGER            K, LDA, LDT, LDY, N, NB
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      $                   Y( LDY, NB )       $                   Y( LDY, NB )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLAHRD reduces the first NB columns of a real general n-by-(n-k+1)  
 *  matrix A so that elements below the k-th subdiagonal are zero. The  
 *  reduction is performed by an orthogonal 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 OBSOLETE auxiliary routine.   
 *  This routine will be 'deprecated' in a  future release.  
 *  Please use the new routine DLAHR2 instead.  
 *  
 *  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.  
 *  
 *  NB      (input) INTEGER  
 *          The number of columns to be reduced.  
 *  
 *  A       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)  
 *          The scalar factors of the elementary reflectors. See Further  
 *          Details.  
 *  
 *  T       (output) DOUBLE PRECISION array, dimension (LDT,NB)  
 *          The upper triangular matrix T.  
 *  
 *  LDT     (input) INTEGER  
 *          The leading dimension of the array T.  LDT >= NB.  
 *  
 *  Y       (output) DOUBLE PRECISION 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 real scalar, and v is a real 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   h   a   a   a )  
 *     ( a   h   a   a   a )  
 *     ( a   h   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).  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 130 Line 208
 *  *
 *           Update A(1:n,i)  *           Update A(1:n,i)
 *  *
 *           Compute i-th column of A - Y * V'  *           Compute i-th column of A - Y * V**T
 *  *
             CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,              CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
      $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )       $                  A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
 *  *
 *           Apply I - V * T' * V' to this column (call it b) from the  *           Apply I - V * T**T * V**T 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)
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 *  *
 *           where V1 is unit lower triangular  *           where V1 is unit lower triangular
 *  *
 *           w := V1' * b1  *           w := V1**T * b1
 *  *
             CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )              CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
             CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),              CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
      $                  LDA, T( 1, NB ), 1 )       $                  LDA, T( 1, NB ), 1 )
 *  *
 *           w := w + V2'*b2  *           w := w + V2**T *b2
 *  *
             CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),              CALL DGEMV( 'Transpose', N-K-I+1, 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**T *w
 *  *
             CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,              CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
      $                  T( 1, NB ), 1 )       $                  T( 1, NB ), 1 )
Removed from v.1.7  
changed lines
  Added in v.1.16

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