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version 1.8, 2011/07/22 07:38:06 version 1.9, 2011/11/21 20:42:55
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   *> \brief \b DLAHRD
   *
   *  =========== 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
   *>
   *> 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.
   *>
   *> This is an OBSOLETE auxiliary routine. 
   *> This routine will be 'deprecated' in a  future release.
   *> Please use the new routine DLAHR2 instead.
   *> \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 2011
   *
   *> \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 3.3.1) --  *  -- LAPACK auxiliary routine (version 3.4.0) --
 *  -- 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..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            K, LDA, LDT, LDY, N, NB        INTEGER            K, LDA, LDT, LDY, N, NB
Line 13 Line 182
      $                   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**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.  
 *  
 *  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**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).  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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