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    1: *> \brief \b ZLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download ZLAQR1 + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqr1.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr1.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr1.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZLAQR1( N, H, LDH, S1, S2, V )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       COMPLEX*16         S1, S2
   25: *       INTEGER            LDH, N
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       COMPLEX*16         H( LDH, * ), V( * )
   29: *       ..
   30: *
   31: *
   32: *> \par Purpose:
   33: *  =============
   34: *>
   35: *> \verbatim
   36: *>
   37: *>      Given a 2-by-2 or 3-by-3 matrix H, ZLAQR1 sets v to a
   38: *>      scalar multiple of the first column of the product
   39: *>
   40: *>      (*)  K = (H - s1*I)*(H - s2*I)
   41: *>
   42: *>      scaling to avoid overflows and most underflows.
   43: *>
   44: *>      This is useful for starting double implicit shift bulges
   45: *>      in the QR algorithm.
   46: *> \endverbatim
   47: *
   48: *  Arguments:
   49: *  ==========
   50: *
   51: *> \param[in] N
   52: *> \verbatim
   53: *>          N is integer
   54: *>              Order of the matrix H. N must be either 2 or 3.
   55: *> \endverbatim
   56: *>
   57: *> \param[in] H
   58: *> \verbatim
   59: *>          H is COMPLEX*16 array of dimension (LDH,N)
   60: *>              The 2-by-2 or 3-by-3 matrix H in (*).
   61: *> \endverbatim
   62: *>
   63: *> \param[in] LDH
   64: *> \verbatim
   65: *>          LDH is integer
   66: *>              The leading dimension of H as declared in
   67: *>              the calling procedure.  LDH.GE.N
   68: *> \endverbatim
   69: *>
   70: *> \param[in] S1
   71: *> \verbatim
   72: *>          S1 is COMPLEX*16
   73: *> \endverbatim
   74: *>
   75: *> \param[in] S2
   76: *> \verbatim
   77: *>          S2 is COMPLEX*16
   78: *>
   79: *>          S1 and S2 are the shifts defining K in (*) above.
   80: *> \endverbatim
   81: *>
   82: *> \param[out] V
   83: *> \verbatim
   84: *>          V is COMPLEX*16 array of dimension N
   85: *>              A scalar multiple of the first column of the
   86: *>              matrix K in (*).
   87: *> \endverbatim
   88: *
   89: *  Authors:
   90: *  ========
   91: *
   92: *> \author Univ. of Tennessee
   93: *> \author Univ. of California Berkeley
   94: *> \author Univ. of Colorado Denver
   95: *> \author NAG Ltd.
   96: *
   97: *> \date December 2016
   98: *
   99: *> \ingroup complex16OTHERauxiliary
  100: *
  101: *> \par Contributors:
  102: *  ==================
  103: *>
  104: *>       Karen Braman and Ralph Byers, Department of Mathematics,
  105: *>       University of Kansas, USA
  106: *>
  107: *  =====================================================================
  108:       SUBROUTINE ZLAQR1( N, H, LDH, S1, S2, V )
  109: *
  110: *  -- LAPACK auxiliary routine (version 3.7.0) --
  111: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  112: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  113: *     December 2016
  114: *
  115: *     .. Scalar Arguments ..
  116:       COMPLEX*16         S1, S2
  117:       INTEGER            LDH, N
  118: *     ..
  119: *     .. Array Arguments ..
  120:       COMPLEX*16         H( LDH, * ), V( * )
  121: *     ..
  122: *
  123: *  ================================================================
  124: *
  125: *     .. Parameters ..
  126:       COMPLEX*16         ZERO
  127:       PARAMETER          ( ZERO = ( 0.0d0, 0.0d0 ) )
  128:       DOUBLE PRECISION   RZERO
  129:       PARAMETER          ( RZERO = 0.0d0 )
  130: *     ..
  131: *     .. Local Scalars ..
  132:       COMPLEX*16         CDUM, H21S, H31S
  133:       DOUBLE PRECISION   S
  134: *     ..
  135: *     .. Intrinsic Functions ..
  136:       INTRINSIC          ABS, DBLE, DIMAG
  137: *     ..
  138: *     .. Statement Functions ..
  139:       DOUBLE PRECISION   CABS1
  140: *     ..
  141: *     .. Statement Function definitions ..
  142:       CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
  143: *     ..
  144: *     .. Executable Statements ..
  145:       IF( N.EQ.2 ) THEN
  146:          S = CABS1( H( 1, 1 )-S2 ) + CABS1( H( 2, 1 ) )
  147:          IF( S.EQ.RZERO ) THEN
  148:             V( 1 ) = ZERO
  149:             V( 2 ) = ZERO
  150:          ELSE
  151:             H21S = H( 2, 1 ) / S
  152:             V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-S1 )*
  153:      $               ( ( H( 1, 1 )-S2 ) / S )
  154:             V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-S1-S2 )
  155:          END IF
  156:       ELSE
  157:          S = CABS1( H( 1, 1 )-S2 ) + CABS1( H( 2, 1 ) ) +
  158:      $       CABS1( H( 3, 1 ) )
  159:          IF( S.EQ.ZERO ) THEN
  160:             V( 1 ) = ZERO
  161:             V( 2 ) = ZERO
  162:             V( 3 ) = ZERO
  163:          ELSE
  164:             H21S = H( 2, 1 ) / S
  165:             H31S = H( 3, 1 ) / S
  166:             V( 1 ) = ( H( 1, 1 )-S1 )*( ( H( 1, 1 )-S2 ) / S ) +
  167:      $               H( 1, 2 )*H21S + H( 1, 3 )*H31S
  168:             V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-S1-S2 ) + H( 2, 3 )*H31S
  169:             V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-S1-S2 ) + H21S*H( 3, 2 )
  170:          END IF
  171:       END IF
  172:       END