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version 1.8, 2011/07/22 07:38:14 version 1.9, 2011/11/21 20:43:09
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   *> \brief \b ZGEHRD
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGEHRD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgehrd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgehrd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgehrd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            IHI, ILO, INFO, LDA, LWORK, N
   *       ..
   *       .. Array Arguments ..
   *       COMPLEX*16        A( LDA, * ), TAU( * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
   *> an 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.
   *> \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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
   *>          zero.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (LWORK)
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          The length of the array WORK.  LWORK >= max(1,N).
   *>          For optimum performance LWORK >= N*NB, where NB is the
   *>          optimal blocksize.
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal size of the WORK array, returns
   *>          this value as the first entry of the WORK array, and no error
   *>          message related to LWORK is issued by XERBLA.
   *> \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. 
   *
   *> \date November 2011
   *
   *> \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).
   *>
   *>  This file is a slight modification of LAPACK-3.0's DGEHRD
   *>  subroutine incorporating improvements proposed by Quintana-Orti and
   *>  Van de Geijn (2006). (See DLAHR2.)
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )        SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.3.1)                                  --  *  -- LAPACK computational 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 2009                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            IHI, ILO, INFO, LDA, LWORK, N        INTEGER            IHI, ILO, INFO, LDA, LWORK, N
Line 12 Line 180
       COMPLEX*16        A( LDA, * ), TAU( * ), WORK( * )        COMPLEX*16        A( LDA, * ), TAU( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by  
 *  an unitary similarity transformation:  Q**H * 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.  
 *  
 *  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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to  
 *          zero.  
 *  
 *  WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The length of the array WORK.  LWORK >= max(1,N).  
 *          For optimum performance LWORK >= N*NB, where NB is the  
 *          optimal blocksize.  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal size of the WORK array, returns  
 *          this value as the first entry of the WORK array, and no error  
 *          message related to LWORK is issued by XERBLA.  
 *  
 *  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**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).  
 *  
 *  This file is a slight modification of LAPACK-3.0's DGEHRD  
 *  subroutine incorporating improvements proposed by Quintana-Orti and  
 *  Van de Geijn (2006). (See DLAHR2.)  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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