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version 1.7, 2010/12/21 13:53:43 version 1.19, 2023/08/07 08:39:17
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   *> \brief \b ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZGEHD2 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgehd2.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgehd2.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgehd2.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            IHI, ILO, INFO, LDA, N
   *       ..
   *       .. Array Arguments ..
   *       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
   *> by a unitary similarity transformation:  Q**H * A * Q = H .
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] ILO
   *> \verbatim
   *>          ILO is INTEGER
   *> \endverbatim
   *>
   *> \param[in] IHI
   *> \verbatim
   *>          IHI is INTEGER
   *>
   *>          It is assumed that A is already upper triangular in rows
   *>          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
   *>          set by a previous call to ZGEBAL; otherwise they should be
   *>          set to 1 and N respectively. See Further Details.
   *>          1 <= ILO <= IHI <= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,N)
   *>          On entry, the n by n general matrix to be reduced.
   *>          On exit, the upper triangle and the first subdiagonal of A
   *>          are overwritten with the upper Hessenberg matrix H, and the
   *>          elements below the first subdiagonal, with the array TAU,
   *>          represent the unitary matrix Q as a product of elementary
   *>          reflectors. 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 (N-1)
   *>          The scalar factors of the elementary reflectors (see Further
   *>          Details).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup complex16GEcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The matrix Q is represented as a product of (ihi-ilo) elementary
   *>  reflectors
   *>
   *>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
   *>
   *>  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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
   *>  exit in A(i+2:ihi,i), and tau in TAU(i).
   *>
   *>  The contents of A are illustrated by the following example, with
   *>  n = 7, ilo = 2 and ihi = 6:
   *>
   *>  on entry,                        on exit,
   *>
   *>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
   *>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
   *>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
   *>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
   *>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
   *>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
   *>  (                         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 ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )        SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational 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..--
 *     November 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            IHI, ILO, INFO, LDA, N        INTEGER            IHI, ILO, INFO, LDA, N
Line 12 Line 158
       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )        COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H  
 *  by a unitary similarity transformation:  Q' * A * Q = H .  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A.  N >= 0.  
 *  
 *  ILO     (input) INTEGER  
 *  IHI     (input) INTEGER  
 *          It is assumed that A is already upper triangular in rows  
 *          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally  
 *          set by a previous call to ZGEBAL; otherwise they should be  
 *          set to 1 and N respectively. See Further Details.  
 *          1 <= ILO <= IHI <= max(1,N).  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)  
 *          On entry, the n by n general matrix to be reduced.  
 *          On exit, the upper triangle and the first subdiagonal of A  
 *          are overwritten with the upper Hessenberg matrix H, and the  
 *          elements below the first subdiagonal, with the array TAU,  
 *          represent the unitary matrix Q as a product of elementary  
 *          reflectors. See Further Details.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  TAU     (output) COMPLEX*16 array, dimension (N-1)  
 *          The scalar factors of the elementary reflectors (see Further  
 *          Details).  
 *  
 *  WORK    (workspace) COMPLEX*16 array, dimension (N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The matrix Q is represented as a product of (ihi-ilo) elementary  
 *  reflectors  
 *  
 *     Q = H(ilo) H(ilo+1) . . . H(ihi-1).  
 *  
 *  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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on  
 *  exit in A(i+2:ihi,i), and tau in TAU(i).  
 *  
 *  The contents of A are illustrated by the following example, with  
 *  n = 7, ilo = 2 and ihi = 6:  
 *  
 *  on entry,                        on exit,  
 *  
 *  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )  
 *  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )  
 *  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )  
 *  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )  
 *  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )  
 *  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )  
 *  (                         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 ..
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          CALL ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),           CALL ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
      $               A( 1, I+1 ), LDA, WORK )       $               A( 1, I+1 ), LDA, WORK )
 *  *
 *        Apply H(i)' to A(i+1:ihi,i+1:n) from the left  *        Apply H(i)**H to A(i+1:ihi,i+1:n) from the left
 *  *
          CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,           CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,
      $               DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )       $               DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )
Removed from v.1.7  
changed lines
  Added in v.1.19

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