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    1: *> \brief \b ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download ZGEHD2 + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgehd2.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgehd2.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgehd2.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            IHI, ILO, INFO, LDA, N
   25: *       ..
   26: *       .. Array Arguments ..
   27: *       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
   28: *       ..
   29: *
   30: *
   31: *> \par Purpose:
   32: *  =============
   33: *>
   34: *> \verbatim
   35: *>
   36: *> ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
   37: *> by a unitary similarity transformation:  Q**H * A * Q = H .
   38: *> \endverbatim
   39: *
   40: *  Arguments:
   41: *  ==========
   42: *
   43: *> \param[in] N
   44: *> \verbatim
   45: *>          N is INTEGER
   46: *>          The order of the matrix A.  N >= 0.
   47: *> \endverbatim
   48: *>
   49: *> \param[in] ILO
   50: *> \verbatim
   51: *>          ILO is INTEGER
   52: *> \endverbatim
   53: *>
   54: *> \param[in] IHI
   55: *> \verbatim
   56: *>          IHI is INTEGER
   57: *>
   58: *>          It is assumed that A is already upper triangular in rows
   59: *>          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
   60: *>          set by a previous call to ZGEBAL; otherwise they should be
   61: *>          set to 1 and N respectively. See Further Details.
   62: *>          1 <= ILO <= IHI <= max(1,N).
   63: *> \endverbatim
   64: *>
   65: *> \param[in,out] A
   66: *> \verbatim
   67: *>          A is COMPLEX*16 array, dimension (LDA,N)
   68: *>          On entry, the n by n general matrix to be reduced.
   69: *>          On exit, the upper triangle and the first subdiagonal of A
   70: *>          are overwritten with the upper Hessenberg matrix H, and the
   71: *>          elements below the first subdiagonal, with the array TAU,
   72: *>          represent the unitary matrix Q as a product of elementary
   73: *>          reflectors. See Further Details.
   74: *> \endverbatim
   75: *>
   76: *> \param[in] LDA
   77: *> \verbatim
   78: *>          LDA is INTEGER
   79: *>          The leading dimension of the array A.  LDA >= max(1,N).
   80: *> \endverbatim
   81: *>
   82: *> \param[out] TAU
   83: *> \verbatim
   84: *>          TAU is COMPLEX*16 array, dimension (N-1)
   85: *>          The scalar factors of the elementary reflectors (see Further
   86: *>          Details).
   87: *> \endverbatim
   88: *>
   89: *> \param[out] WORK
   90: *> \verbatim
   91: *>          WORK is COMPLEX*16 array, dimension (N)
   92: *> \endverbatim
   93: *>
   94: *> \param[out] INFO
   95: *> \verbatim
   96: *>          INFO is INTEGER
   97: *>          = 0:  successful exit
   98: *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
   99: *> \endverbatim
  100: *
  101: *  Authors:
  102: *  ========
  103: *
  104: *> \author Univ. of Tennessee
  105: *> \author Univ. of California Berkeley
  106: *> \author Univ. of Colorado Denver
  107: *> \author NAG Ltd.
  108: *
  109: *> \date December 2016
  110: *
  111: *> \ingroup complex16GEcomputational
  112: *
  113: *> \par Further Details:
  114: *  =====================
  115: *>
  116: *> \verbatim
  117: *>
  118: *>  The matrix Q is represented as a product of (ihi-ilo) elementary
  119: *>  reflectors
  120: *>
  121: *>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
  122: *>
  123: *>  Each H(i) has the form
  124: *>
  125: *>     H(i) = I - tau * v * v**H
  126: *>
  127: *>  where tau is a complex scalar, and v is a complex vector with
  128: *>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
  129: *>  exit in A(i+2:ihi,i), and tau in TAU(i).
  130: *>
  131: *>  The contents of A are illustrated by the following example, with
  132: *>  n = 7, ilo = 2 and ihi = 6:
  133: *>
  134: *>  on entry,                        on exit,
  135: *>
  136: *>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
  137: *>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
  138: *>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
  139: *>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
  140: *>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
  141: *>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
  142: *>  (                         a )    (                          a )
  143: *>
  144: *>  where a denotes an element of the original matrix A, h denotes a
  145: *>  modified element of the upper Hessenberg matrix H, and vi denotes an
  146: *>  element of the vector defining H(i).
  147: *> \endverbatim
  148: *>
  149: *  =====================================================================
  150:       SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
  151: *
  152: *  -- LAPACK computational routine (version 3.7.0) --
  153: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  154: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  155: *     December 2016
  156: *
  157: *     .. Scalar Arguments ..
  158:       INTEGER            IHI, ILO, INFO, LDA, N
  159: *     ..
  160: *     .. Array Arguments ..
  161:       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
  162: *     ..
  163: *
  164: *  =====================================================================
  165: *
  166: *     .. Parameters ..
  167:       COMPLEX*16         ONE
  168:       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ) )
  169: *     ..
  170: *     .. Local Scalars ..
  171:       INTEGER            I
  172:       COMPLEX*16         ALPHA
  173: *     ..
  174: *     .. External Subroutines ..
  175:       EXTERNAL           XERBLA, ZLARF, ZLARFG
  176: *     ..
  177: *     .. Intrinsic Functions ..
  178:       INTRINSIC          DCONJG, MAX, MIN
  179: *     ..
  180: *     .. Executable Statements ..
  181: *
  182: *     Test the input parameters
  183: *
  184:       INFO = 0
  185:       IF( N.LT.0 ) THEN
  186:          INFO = -1
  187:       ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
  188:          INFO = -2
  189:       ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
  190:          INFO = -3
  191:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  192:          INFO = -5
  193:       END IF
  194:       IF( INFO.NE.0 ) THEN
  195:          CALL XERBLA( 'ZGEHD2', -INFO )
  196:          RETURN
  197:       END IF
  198: *
  199:       DO 10 I = ILO, IHI - 1
  200: *
  201: *        Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
  202: *
  203:          ALPHA = A( I+1, I )
  204:          CALL ZLARFG( IHI-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAU( I ) )
  205:          A( I+1, I ) = ONE
  206: *
  207: *        Apply H(i) to A(1:ihi,i+1:ihi) from the right
  208: *
  209:          CALL ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
  210:      $               A( 1, I+1 ), LDA, WORK )
  211: *
  212: *        Apply H(i)**H to A(i+1:ihi,i+1:n) from the left
  213: *
  214:          CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,
  215:      $               DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )
  216: *
  217:          A( I+1, I ) = ALPHA
  218:    10 CONTINUE
  219: *
  220:       RETURN
  221: *
  222: *     End of ZGEHD2
  223: *
  224:       END