LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dlaed5.f
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00001 *> \brief \b DLAED5
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DLAED5 + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed5.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       INTEGER            I
00025 *       DOUBLE PRECISION   DLAM, RHO
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> This subroutine computes the I-th eigenvalue of a symmetric rank-one
00038 *> modification of a 2-by-2 diagonal matrix
00039 *>
00040 *>            diag( D )  +  RHO * Z * transpose(Z) .
00041 *>
00042 *> The diagonal elements in the array D are assumed to satisfy
00043 *>
00044 *>            D(i) < D(j)  for  i < j .
00045 *>
00046 *> We also assume RHO > 0 and that the Euclidean norm of the vector
00047 *> Z is one.
00048 *> \endverbatim
00049 *
00050 *  Arguments:
00051 *  ==========
00052 *
00053 *> \param[in] I
00054 *> \verbatim
00055 *>          I is INTEGER
00056 *>         The index of the eigenvalue to be computed.  I = 1 or I = 2.
00057 *> \endverbatim
00058 *>
00059 *> \param[in] D
00060 *> \verbatim
00061 *>          D is DOUBLE PRECISION array, dimension (2)
00062 *>         The original eigenvalues.  We assume D(1) < D(2).
00063 *> \endverbatim
00064 *>
00065 *> \param[in] Z
00066 *> \verbatim
00067 *>          Z is DOUBLE PRECISION array, dimension (2)
00068 *>         The components of the updating vector.
00069 *> \endverbatim
00070 *>
00071 *> \param[out] DELTA
00072 *> \verbatim
00073 *>          DELTA is DOUBLE PRECISION array, dimension (2)
00074 *>         The vector DELTA contains the information necessary
00075 *>         to construct the eigenvectors.
00076 *> \endverbatim
00077 *>
00078 *> \param[in] RHO
00079 *> \verbatim
00080 *>          RHO is DOUBLE PRECISION
00081 *>         The scalar in the symmetric updating formula.
00082 *> \endverbatim
00083 *>
00084 *> \param[out] DLAM
00085 *> \verbatim
00086 *>          DLAM is DOUBLE PRECISION
00087 *>         The computed lambda_I, the I-th updated eigenvalue.
00088 *> \endverbatim
00089 *
00090 *  Authors:
00091 *  ========
00092 *
00093 *> \author Univ. of Tennessee 
00094 *> \author Univ. of California Berkeley 
00095 *> \author Univ. of Colorado Denver 
00096 *> \author NAG Ltd. 
00097 *
00098 *> \date November 2011
00099 *
00100 *> \ingroup auxOTHERcomputational
00101 *
00102 *> \par Contributors:
00103 *  ==================
00104 *>
00105 *>     Ren-Cang Li, Computer Science Division, University of California
00106 *>     at Berkeley, USA
00107 *>
00108 *  =====================================================================
00109       SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
00110 *
00111 *  -- LAPACK computational routine (version 3.4.0) --
00112 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00113 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00114 *     November 2011
00115 *
00116 *     .. Scalar Arguments ..
00117       INTEGER            I
00118       DOUBLE PRECISION   DLAM, RHO
00119 *     ..
00120 *     .. Array Arguments ..
00121       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
00122 *     ..
00123 *
00124 *  =====================================================================
00125 *
00126 *     .. Parameters ..
00127       DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
00128       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
00129      $                   FOUR = 4.0D0 )
00130 *     ..
00131 *     .. Local Scalars ..
00132       DOUBLE PRECISION   B, C, DEL, TAU, TEMP, W
00133 *     ..
00134 *     .. Intrinsic Functions ..
00135       INTRINSIC          ABS, SQRT
00136 *     ..
00137 *     .. Executable Statements ..
00138 *
00139       DEL = D( 2 ) - D( 1 )
00140       IF( I.EQ.1 ) THEN
00141          W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
00142          IF( W.GT.ZERO ) THEN
00143             B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00144             C = RHO*Z( 1 )*Z( 1 )*DEL
00145 *
00146 *           B > ZERO, always
00147 *
00148             TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
00149             DLAM = D( 1 ) + TAU
00150             DELTA( 1 ) = -Z( 1 ) / TAU
00151             DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
00152          ELSE
00153             B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00154             C = RHO*Z( 2 )*Z( 2 )*DEL
00155             IF( B.GT.ZERO ) THEN
00156                TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
00157             ELSE
00158                TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
00159             END IF
00160             DLAM = D( 2 ) + TAU
00161             DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
00162             DELTA( 2 ) = -Z( 2 ) / TAU
00163          END IF
00164          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
00165          DELTA( 1 ) = DELTA( 1 ) / TEMP
00166          DELTA( 2 ) = DELTA( 2 ) / TEMP
00167       ELSE
00168 *
00169 *     Now I=2
00170 *
00171          B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00172          C = RHO*Z( 2 )*Z( 2 )*DEL
00173          IF( B.GT.ZERO ) THEN
00174             TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
00175          ELSE
00176             TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
00177          END IF
00178          DLAM = D( 2 ) + TAU
00179          DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
00180          DELTA( 2 ) = -Z( 2 ) / TAU
00181          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
00182          DELTA( 1 ) = DELTA( 1 ) / TEMP
00183          DELTA( 2 ) = DELTA( 2 ) / TEMP
00184       END IF
00185       RETURN
00186 *
00187 *     End OF DLAED5
00188 *
00189       END
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