zlahrd.f

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*> \brief \b ZLAHRD 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 ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLAHRD( 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
*>
*> ZLAHRD 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 a 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 OBSOLETE auxiliary routine. 
*> This routine will be 'deprecated' in a  future release.
*> Please use the new routine ZLAHR2 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 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 >= max(1,N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \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   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 zlahrd( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*  -- LAPACK auxiliary routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER            k, lda, ldt, ldy, n, nb
*     ..
*     .. Array Arguments ..
      COMPLEX*16         a( lda, * ), t( ldt, nb ), tau( nb ),
     $                   y( ldy, nb )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         zero, one
      parameter( zero = ( 0.0d+0, 0.0d+0 ),
     $                   one = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            i
      COMPLEX*16         ei
*     ..
*     .. External Subroutines ..
      EXTERNAL           zaxpy, zcopy, zgemv, zlacgv, zlarfg, zscal,
     $                   ztrmv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          min
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( n.LE.1 )
     $   return
*
      DO 10 i = 1, nb
         IF( i.GT.1 ) THEN
*
*           Update A(1:n,i)
*
*           Compute i-th column of A - Y * V**H
*
            CALL zlacgv( i-1, a( k+i-1, 1 ), lda )
            CALL zgemv( 'No transpose', n, i-1, -one, y, ldy,
     $                  a( k+i-1, 1 ), lda, one, a( 1, i ), 1 )
            CALL zlacgv( i-1, a( k+i-1, 1 ), lda )
*
*           Apply I - V * T**H * V**H to this column (call it b) from the
*           left, using the last column of T as workspace
*
*           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
*                    ( V2 )             ( b2 )
*
*           where V1 is unit lower triangular
*
*           w := V1**H * b1
*
            CALL zcopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
            CALL ztrmv( 'Lower', 'Conjugate transpose', 'Unit', i-1,
     $                  a( k+1, 1 ), lda, t( 1, nb ), 1 )
*
*           w := w + V2**H *b2
*
            CALL zgemv( 'Conjugate transpose', n-k-i+1, i-1, one,
     $                  a( k+i, 1 ), lda, a( k+i, i ), 1, one,
     $                  t( 1, nb ), 1 )
*
*           w := T**H *w
*
            CALL ztrmv( 'Upper', 'Conjugate transpose', 'Non-unit', i-1,
     $                  t, ldt, t( 1, nb ), 1 )
*
*           b2 := b2 - V2*w
*
            CALL zgemv( 'No transpose', n-k-i+1, i-1, -one, a( k+i, 1 ),
     $                  lda, t( 1, nb ), 1, one, a( k+i, i ), 1 )
*
*           b1 := b1 - V1*w
*
            CALL ztrmv( 'Lower', 'No transpose', 'Unit', i-1,
     $                  a( k+1, 1 ), lda, t( 1, nb ), 1 )
            CALL zaxpy( i-1, -one, t( 1, nb ), 1, a( k+1, i ), 1 )
*
            a( k+i-1, i-1 ) = ei
         END IF
*
*        Generate the elementary reflector H(i) to annihilate
*        A(k+i+1:n,i)
*
         ei = a( k+i, i )
         CALL zlarfg( n-k-i+1, ei, a( min( k+i+1, n ), i ), 1,
     $                tau( i ) )
         a( k+i, i ) = one
*
*        Compute  Y(1:n,i)
*
         CALL zgemv( 'No transpose', n, n-k-i+1, one, a( 1, i+1 ), lda,
     $               a( k+i, i ), 1, zero, y( 1, i ), 1 )
         CALL zgemv( 'Conjugate transpose', n-k-i+1, i-1, one,
     $               a( k+i, 1 ), lda, a( k+i, i ), 1, zero, t( 1, i ),
     $               1 )
         CALL zgemv( 'No transpose', n, i-1, -one, y, ldy, t( 1, i ), 1,
     $               one, y( 1, i ), 1 )
         CALL zscal( n, tau( i ), y( 1, i ), 1 )
*
*        Compute T(1:i,i)
*
         CALL zscal( i-1, -tau( i ), t( 1, i ), 1 )
         CALL ztrmv( 'Upper', 'No transpose', 'Non-unit', i-1, t, ldt,
     $               t( 1, i ), 1 )
         t( i, i ) = tau( i )
*
   10 continue
      a( k+nb, nb ) = ei
*
      return
*
*     End of ZLAHRD
*
      END