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    1: *> \brief \b DLAED5 used by DSTEDC. Solves the 2-by-2 secular equation.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DLAED5 + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed5.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed5.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed5.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            I
   25: *       DOUBLE PRECISION   DLAM, RHO
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
   29: *       ..
   30: *
   31: *
   32: *> \par Purpose:
   33: *  =============
   34: *>
   35: *> \verbatim
   36: *>
   37: *> This subroutine computes the I-th eigenvalue of a symmetric rank-one
   38: *> modification of a 2-by-2 diagonal matrix
   39: *>
   40: *>            diag( D )  +  RHO * Z * transpose(Z) .
   41: *>
   42: *> The diagonal elements in the array D are assumed to satisfy
   43: *>
   44: *>            D(i) < D(j)  for  i < j .
   45: *>
   46: *> We also assume RHO > 0 and that the Euclidean norm of the vector
   47: *> Z is one.
   48: *> \endverbatim
   49: *
   50: *  Arguments:
   51: *  ==========
   52: *
   53: *> \param[in] I
   54: *> \verbatim
   55: *>          I is INTEGER
   56: *>         The index of the eigenvalue to be computed.  I = 1 or I = 2.
   57: *> \endverbatim
   58: *>
   59: *> \param[in] D
   60: *> \verbatim
   61: *>          D is DOUBLE PRECISION array, dimension (2)
   62: *>         The original eigenvalues.  We assume D(1) < D(2).
   63: *> \endverbatim
   64: *>
   65: *> \param[in] Z
   66: *> \verbatim
   67: *>          Z is DOUBLE PRECISION array, dimension (2)
   68: *>         The components of the updating vector.
   69: *> \endverbatim
   70: *>
   71: *> \param[out] DELTA
   72: *> \verbatim
   73: *>          DELTA is DOUBLE PRECISION array, dimension (2)
   74: *>         The vector DELTA contains the information necessary
   75: *>         to construct the eigenvectors.
   76: *> \endverbatim
   77: *>
   78: *> \param[in] RHO
   79: *> \verbatim
   80: *>          RHO is DOUBLE PRECISION
   81: *>         The scalar in the symmetric updating formula.
   82: *> \endverbatim
   83: *>
   84: *> \param[out] DLAM
   85: *> \verbatim
   86: *>          DLAM is DOUBLE PRECISION
   87: *>         The computed lambda_I, the I-th updated eigenvalue.
   88: *> \endverbatim
   89: *
   90: *  Authors:
   91: *  ========
   92: *
   93: *> \author Univ. of Tennessee
   94: *> \author Univ. of California Berkeley
   95: *> \author Univ. of Colorado Denver
   96: *> \author NAG Ltd.
   97: *
   98: *> \ingroup auxOTHERcomputational
   99: *
  100: *> \par Contributors:
  101: *  ==================
  102: *>
  103: *>     Ren-Cang Li, Computer Science Division, University of California
  104: *>     at Berkeley, USA
  105: *>
  106: *  =====================================================================
  107:       SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
  108: *
  109: *  -- LAPACK computational routine --
  110: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  111: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  112: *
  113: *     .. Scalar Arguments ..
  114:       INTEGER            I
  115:       DOUBLE PRECISION   DLAM, RHO
  116: *     ..
  117: *     .. Array Arguments ..
  118:       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
  119: *     ..
  120: *
  121: *  =====================================================================
  122: *
  123: *     .. Parameters ..
  124:       DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
  125:       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
  126:      $                   FOUR = 4.0D0 )
  127: *     ..
  128: *     .. Local Scalars ..
  129:       DOUBLE PRECISION   B, C, DEL, TAU, TEMP, W
  130: *     ..
  131: *     .. Intrinsic Functions ..
  132:       INTRINSIC          ABS, SQRT
  133: *     ..
  134: *     .. Executable Statements ..
  135: *
  136:       DEL = D( 2 ) - D( 1 )
  137:       IF( I.EQ.1 ) THEN
  138:          W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
  139:          IF( W.GT.ZERO ) THEN
  140:             B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  141:             C = RHO*Z( 1 )*Z( 1 )*DEL
  142: *
  143: *           B > ZERO, always
  144: *
  145:             TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
  146:             DLAM = D( 1 ) + TAU
  147:             DELTA( 1 ) = -Z( 1 ) / TAU
  148:             DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
  149:          ELSE
  150:             B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  151:             C = RHO*Z( 2 )*Z( 2 )*DEL
  152:             IF( B.GT.ZERO ) THEN
  153:                TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
  154:             ELSE
  155:                TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
  156:             END IF
  157:             DLAM = D( 2 ) + TAU
  158:             DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
  159:             DELTA( 2 ) = -Z( 2 ) / TAU
  160:          END IF
  161:          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
  162:          DELTA( 1 ) = DELTA( 1 ) / TEMP
  163:          DELTA( 2 ) = DELTA( 2 ) / TEMP
  164:       ELSE
  165: *
  166: *     Now I=2
  167: *
  168:          B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  169:          C = RHO*Z( 2 )*Z( 2 )*DEL
  170:          IF( B.GT.ZERO ) THEN
  171:             TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
  172:          ELSE
  173:             TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
  174:          END IF
  175:          DLAM = D( 2 ) + TAU
  176:          DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
  177:          DELTA( 2 ) = -Z( 2 ) / TAU
  178:          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
  179:          DELTA( 1 ) = DELTA( 1 ) / TEMP
  180:          DELTA( 2 ) = DELTA( 2 ) / TEMP
  181:       END IF
  182:       RETURN
  183: *
  184: *     End of DLAED5
  185: *
  186:       END