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version 1.4, 2010/08/06 15:32:38 version 1.17, 2017/06/17 11:06:42
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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 good performance, LWORK should generally be larger.
   *>
   *>          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 December 2016
   *
   *> \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 3.2.1)                                  --  *  -- LAPACK computational routine (version 3.7.0) --
 *  -- 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                                                      --  *     December 2016
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            IHI, ILO, INFO, LDA, LWORK, N        INTEGER            IHI, ILO, INFO, LDA, LWORK, N
Line 12 Line 179
       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' * 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'  
 *  
 *  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 ..
       INTEGER            NBMAX, LDT        INTEGER            NBMAX, LDT, TSIZE
       PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )        PARAMETER          ( NBMAX = 64, LDT = NBMAX+1,
        $                     TSIZE = LDT*NBMAX )
       COMPLEX*16        ZERO, ONE        COMPLEX*16        ZERO, ONE
       PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),         PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),
      $                     ONE = ( 1.0D+0, 0.0D+0 ) )       $                     ONE = ( 1.0D+0, 0.0D+0 ) )
 *     ..  *     ..
 *     .. Local Scalars ..  *     .. Local Scalars ..
       LOGICAL            LQUERY        LOGICAL            LQUERY
       INTEGER            I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,        INTEGER            I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB,
      $                   NBMIN, NH, NX       $                   NBMIN, NH, NX
       COMPLEX*16        EI        COMPLEX*16        EI
 *     ..  *     ..
 *     .. Local Arrays ..  
       COMPLEX*16        T( LDT, NBMAX )  
 *     ..  
 *     .. External Subroutines ..  *     .. External Subroutines ..
       EXTERNAL           ZAXPY, ZGEHD2, ZGEMM, ZLAHR2, ZLARFB, ZTRMM,        EXTERNAL           ZAXPY, ZGEHD2, ZGEMM, ZLAHR2, ZLARFB, ZTRMM,
      $                   XERBLA       $                   XERBLA
Line 136 Line 211
 *     Test the input parameters  *     Test the input parameters
 *  *
       INFO = 0        INFO = 0
       NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )  
       LWKOPT = N*NB  
       WORK( 1 ) = LWKOPT  
       LQUERY = ( LWORK.EQ.-1 )        LQUERY = ( LWORK.EQ.-1 )
       IF( N.LT.0 ) THEN        IF( N.LT.0 ) THEN
          INFO = -1           INFO = -1
Line 151 Line 223
       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN        ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
          INFO = -8           INFO = -8
       END IF        END IF
   *
         IF( INFO.EQ.0 ) THEN
   *
   *        Compute the workspace requirements
   *
            NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
            LWKOPT = N*NB + TSIZE
            WORK( 1 ) = LWKOPT
         ENDIF
   *
       IF( INFO.NE.0 ) THEN        IF( INFO.NE.0 ) THEN
          CALL XERBLA( 'ZGEHRD', -INFO )           CALL XERBLA( 'ZGEHRD', -INFO )
          RETURN           RETURN
Line 179 Line 261
 *  *
       NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )        NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
       NBMIN = 2        NBMIN = 2
       IWS = 1  
       IF( NB.GT.1 .AND. NB.LT.NH ) THEN        IF( NB.GT.1 .AND. NB.LT.NH ) THEN
 *  *
 *        Determine when to cross over from blocked to unblocked code  *        Determine when to cross over from blocked to unblocked code
Line 190 Line 271
 *  *
 *           Determine if workspace is large enough for blocked code  *           Determine if workspace is large enough for blocked code
 *  *
             IWS = N*NB              IF( LWORK.LT.N*NB+TSIZE ) THEN
             IF( LWORK.LT.IWS ) THEN  
 *  *
 *              Not enough workspace to use optimal NB:  determine the  *              Not enough workspace to use optimal NB:  determine the
 *              minimum value of NB, and reduce NB or force use of  *              minimum value of NB, and reduce NB or force use of
Line 199 Line 279
 *  *
                NBMIN = MAX( 2, ILAENV( 2, 'ZGEHRD', ' ', N, ILO, IHI,                 NBMIN = MAX( 2, ILAENV( 2, 'ZGEHRD', ' ', N, ILO, IHI,
      $                 -1 ) )       $                 -1 ) )
                IF( LWORK.GE.N*NBMIN ) THEN                 IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN
                   NB = LWORK / N                    NB = (LWORK-TSIZE) / N
                ELSE                 ELSE
                   NB = 1                    NB = 1
                END IF                 END IF
Line 219 Line 299
 *  *
 *        Use blocked code  *        Use blocked code
 *  *
            IWT = 1 + N*NB
          DO 40 I = ILO, IHI - 1 - NX, NB           DO 40 I = ILO, IHI - 1 - NX, NB
             IB = MIN( NB, IHI-I )              IB = MIN( NB, IHI-I )
 *  *
 *           Reduce columns i:i+ib-1 to Hessenberg form, returning the  *           Reduce columns i:i+ib-1 to Hessenberg form, returning the
 *           matrices V and T of the block reflector H = I - V*T*V'  *           matrices V and T of the block reflector H = I - V*T*V**H
 *           which performs the reduction, and also the matrix Y = A*V*T  *           which performs the reduction, and also the matrix Y = A*V*T
 *  *
             CALL ZLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,              CALL ZLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ),
      $                   WORK, LDWORK )       $                   WORK( IWT ), LDT, WORK, LDWORK )
 *  *
 *           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the  *           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
 *           right, computing  A := A - Y * V'. V(i+ib,ib-1) must be set  *           right, computing  A := A - Y * V**H. V(i+ib,ib-1) must be set
 *           to 1  *           to 1
 *  *
             EI = A( I+IB, I+IB-1 )              EI = A( I+IB, I+IB-1 )
             A( I+IB, I+IB-1 ) = ONE              A( I+IB, I+IB-1 ) = ONE
             CALL ZGEMM( 'No transpose', 'Conjugate transpose',               CALL ZGEMM( 'No transpose', 'Conjugate transpose',
      $                  IHI, IHI-I-IB+1,       $                  IHI, IHI-I-IB+1,
      $                  IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,       $                  IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
      $                  A( 1, I+IB ), LDA )       $                  A( 1, I+IB ), LDA )
Line 257 Line 338
 *  *
             CALL ZLARFB( 'Left', 'Conjugate transpose', 'Forward',              CALL ZLARFB( 'Left', 'Conjugate transpose', 'Forward',
      $                   'Columnwise',       $                   'Columnwise',
      $                   IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,       $                   IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA,
      $                   A( I+1, I+IB ), LDA, WORK, LDWORK )       $                   WORK( IWT ), LDT, A( I+1, I+IB ), LDA,
        $                   WORK, LDWORK )
    40    CONTINUE     40    CONTINUE
       END IF        END IF
 *  *
 *     Use unblocked code to reduce the rest of the matrix  *     Use unblocked code to reduce the rest of the matrix
 *  *
       CALL ZGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )        CALL ZGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
       WORK( 1 ) = IWS        WORK( 1 ) = LWKOPT
 *  *
       RETURN        RETURN
 *  *
Removed from v.1.4  
changed lines
  Added in v.1.17

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