LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
slasd5.f
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00001 *> \brief \b SLASD5
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SLASD5 + 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/slasd5.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       INTEGER            I
00025 *       REAL               DSIGMA, RHO
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       REAL               D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> This subroutine computes the square root of the I-th eigenvalue
00038 *> of a positive symmetric rank-one modification of a 2-by-2 diagonal
00039 *> matrix
00040 *>
00041 *>            diag( D ) * diag( D ) +  RHO * Z * transpose(Z) .
00042 *>
00043 *> The diagonal entries in the array D are assumed to satisfy
00044 *>
00045 *>            0 <= D(i) < D(j)  for  i < j .
00046 *>
00047 *> We also assume RHO > 0 and that the Euclidean norm of the vector
00048 *> Z is one.
00049 *> \endverbatim
00050 *
00051 *  Arguments:
00052 *  ==========
00053 *
00054 *> \param[in] I
00055 *> \verbatim
00056 *>          I is INTEGER
00057 *>         The index of the eigenvalue to be computed.  I = 1 or I = 2.
00058 *> \endverbatim
00059 *>
00060 *> \param[in] D
00061 *> \verbatim
00062 *>          D is REAL array, dimension (2)
00063 *>         The original eigenvalues.  We assume 0 <= D(1) < D(2).
00064 *> \endverbatim
00065 *>
00066 *> \param[in] Z
00067 *> \verbatim
00068 *>          Z is REAL array, dimension (2)
00069 *>         The components of the updating vector.
00070 *> \endverbatim
00071 *>
00072 *> \param[out] DELTA
00073 *> \verbatim
00074 *>          DELTA is REAL array, dimension (2)
00075 *>         Contains (D(j) - sigma_I) in its  j-th component.
00076 *>         The vector DELTA contains the information necessary
00077 *>         to construct the eigenvectors.
00078 *> \endverbatim
00079 *>
00080 *> \param[in] RHO
00081 *> \verbatim
00082 *>          RHO is REAL
00083 *>         The scalar in the symmetric updating formula.
00084 *> \endverbatim
00085 *>
00086 *> \param[out] DSIGMA
00087 *> \verbatim
00088 *>          DSIGMA is REAL
00089 *>         The computed sigma_I, the I-th updated eigenvalue.
00090 *> \endverbatim
00091 *>
00092 *> \param[out] WORK
00093 *> \verbatim
00094 *>          WORK is REAL array, dimension (2)
00095 *>         WORK contains (D(j) + sigma_I) in its  j-th component.
00096 *> \endverbatim
00097 *
00098 *  Authors:
00099 *  ========
00100 *
00101 *> \author Univ. of Tennessee 
00102 *> \author Univ. of California Berkeley 
00103 *> \author Univ. of Colorado Denver 
00104 *> \author NAG Ltd. 
00105 *
00106 *> \date November 2011
00107 *
00108 *> \ingroup auxOTHERauxiliary
00109 *
00110 *> \par Contributors:
00111 *  ==================
00112 *>
00113 *>     Ren-Cang Li, Computer Science Division, University of California
00114 *>     at Berkeley, USA
00115 *>
00116 *  =====================================================================
00117       SUBROUTINE SLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
00118 *
00119 *  -- LAPACK auxiliary routine (version 3.4.0) --
00120 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00121 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00122 *     November 2011
00123 *
00124 *     .. Scalar Arguments ..
00125       INTEGER            I
00126       REAL               DSIGMA, RHO
00127 *     ..
00128 *     .. Array Arguments ..
00129       REAL               D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
00130 *     ..
00131 *
00132 *  =====================================================================
00133 *
00134 *     .. Parameters ..
00135       REAL               ZERO, ONE, TWO, THREE, FOUR
00136       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0,
00137      $                   THREE = 3.0E+0, FOUR = 4.0E+0 )
00138 *     ..
00139 *     .. Local Scalars ..
00140       REAL               B, C, DEL, DELSQ, TAU, W
00141 *     ..
00142 *     .. Intrinsic Functions ..
00143       INTRINSIC          ABS, SQRT
00144 *     ..
00145 *     .. Executable Statements ..
00146 *
00147       DEL = D( 2 ) - D( 1 )
00148       DELSQ = DEL*( D( 2 )+D( 1 ) )
00149       IF( I.EQ.1 ) THEN
00150          W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )-
00151      $       Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL
00152          IF( W.GT.ZERO ) THEN
00153             B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00154             C = RHO*Z( 1 )*Z( 1 )*DELSQ
00155 *
00156 *           B > ZERO, always
00157 *
00158 *           The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
00159 *
00160             TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
00161 *
00162 *           The following TAU is DSIGMA - D( 1 )
00163 *
00164             TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) )
00165             DSIGMA = D( 1 ) + TAU
00166             DELTA( 1 ) = -TAU
00167             DELTA( 2 ) = DEL - TAU
00168             WORK( 1 ) = TWO*D( 1 ) + TAU
00169             WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 )
00170 *           DELTA( 1 ) = -Z( 1 ) / TAU
00171 *           DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
00172          ELSE
00173             B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00174             C = RHO*Z( 2 )*Z( 2 )*DELSQ
00175 *
00176 *           The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
00177 *
00178             IF( B.GT.ZERO ) THEN
00179                TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
00180             ELSE
00181                TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
00182             END IF
00183 *
00184 *           The following TAU is DSIGMA - D( 2 )
00185 *
00186             TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) )
00187             DSIGMA = D( 2 ) + TAU
00188             DELTA( 1 ) = -( DEL+TAU )
00189             DELTA( 2 ) = -TAU
00190             WORK( 1 ) = D( 1 ) + TAU + D( 2 )
00191             WORK( 2 ) = TWO*D( 2 ) + TAU
00192 *           DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
00193 *           DELTA( 2 ) = -Z( 2 ) / TAU
00194          END IF
00195 *        TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
00196 *        DELTA( 1 ) = DELTA( 1 ) / TEMP
00197 *        DELTA( 2 ) = DELTA( 2 ) / TEMP
00198       ELSE
00199 *
00200 *        Now I=2
00201 *
00202          B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00203          C = RHO*Z( 2 )*Z( 2 )*DELSQ
00204 *
00205 *        The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
00206 *
00207          IF( B.GT.ZERO ) THEN
00208             TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
00209          ELSE
00210             TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
00211          END IF
00212 *
00213 *        The following TAU is DSIGMA - D( 2 )
00214 *
00215          TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) )
00216          DSIGMA = D( 2 ) + TAU
00217          DELTA( 1 ) = -( DEL+TAU )
00218          DELTA( 2 ) = -TAU
00219          WORK( 1 ) = D( 1 ) + TAU + D( 2 )
00220          WORK( 2 ) = TWO*D( 2 ) + TAU
00221 *        DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
00222 *        DELTA( 2 ) = -Z( 2 ) / TAU
00223 *        TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
00224 *        DELTA( 1 ) = DELTA( 1 ) / TEMP
00225 *        DELTA( 2 ) = DELTA( 2 ) / TEMP
00226       END IF
00227       RETURN
00228 *
00229 *     End of SLASD5
00230 *
00231       END
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