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    1: *> \brief \b DLAED5
    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: *> \date November 2011
   99: *
  100: *> \ingroup auxOTHERcomputational
  101: *
  102: *> \par Contributors:
  103: *  ==================
  104: *>
  105: *>     Ren-Cang Li, Computer Science Division, University of California
  106: *>     at Berkeley, USA
  107: *>
  108: *  =====================================================================
  109:       SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
  110: *
  111: *  -- LAPACK computational routine (version 3.4.0) --
  112: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  113: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  114: *     November 2011
  115: *
  116: *     .. Scalar Arguments ..
  117:       INTEGER            I
  118:       DOUBLE PRECISION   DLAM, RHO
  119: *     ..
  120: *     .. Array Arguments ..
  121:       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
  122: *     ..
  123: *
  124: *  =====================================================================
  125: *
  126: *     .. Parameters ..
  127:       DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
  128:       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
  129:      $                   FOUR = 4.0D0 )
  130: *     ..
  131: *     .. Local Scalars ..
  132:       DOUBLE PRECISION   B, C, DEL, TAU, TEMP, W
  133: *     ..
  134: *     .. Intrinsic Functions ..
  135:       INTRINSIC          ABS, SQRT
  136: *     ..
  137: *     .. Executable Statements ..
  138: *
  139:       DEL = D( 2 ) - D( 1 )
  140:       IF( I.EQ.1 ) THEN
  141:          W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
  142:          IF( W.GT.ZERO ) THEN
  143:             B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  144:             C = RHO*Z( 1 )*Z( 1 )*DEL
  145: *
  146: *           B > ZERO, always
  147: *
  148:             TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
  149:             DLAM = D( 1 ) + TAU
  150:             DELTA( 1 ) = -Z( 1 ) / TAU
  151:             DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
  152:          ELSE
  153:             B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  154:             C = RHO*Z( 2 )*Z( 2 )*DEL
  155:             IF( B.GT.ZERO ) THEN
  156:                TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
  157:             ELSE
  158:                TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
  159:             END IF
  160:             DLAM = D( 2 ) + TAU
  161:             DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
  162:             DELTA( 2 ) = -Z( 2 ) / TAU
  163:          END IF
  164:          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
  165:          DELTA( 1 ) = DELTA( 1 ) / TEMP
  166:          DELTA( 2 ) = DELTA( 2 ) / TEMP
  167:       ELSE
  168: *
  169: *     Now I=2
  170: *
  171:          B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  172:          C = RHO*Z( 2 )*Z( 2 )*DEL
  173:          IF( B.GT.ZERO ) THEN
  174:             TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
  175:          ELSE
  176:             TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
  177:          END IF
  178:          DLAM = D( 2 ) + TAU
  179:          DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
  180:          DELTA( 2 ) = -Z( 2 ) / TAU
  181:          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
  182:          DELTA( 1 ) = DELTA( 1 ) / TEMP
  183:          DELTA( 2 ) = DELTA( 2 ) / TEMP
  184:       END IF
  185:       RETURN
  186: *
  187: *     End OF DLAED5
  188: *
  189:       END