LAPACK: SRC/dlaed5.f Source File

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00001       SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
00002 *
00003 *  -- LAPACK routine (version 3.2) --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       INTEGER            I
00010       DOUBLE PRECISION   DLAM, RHO
00011 *     ..
00012 *     .. Array Arguments ..
00013       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
00014 *     ..
00015 *
00016 *  Purpose
00017 *  =======
00018 *
00019 *  This subroutine computes the I-th eigenvalue of a symmetric rank-one
00020 *  modification of a 2-by-2 diagonal matrix
00021 *
00022 *             diag( D )  +  RHO *  Z * transpose(Z) .
00023 *
00024 *  The diagonal elements in the array D are assumed to satisfy
00025 *
00026 *             D(i) < D(j)  for  i < j .
00027 *
00028 *  We also assume RHO > 0 and that the Euclidean norm of the vector
00029 *  Z is one.
00030 *
00031 *  Arguments
00032 *  =========
00033 *
00034 *  I      (input) INTEGER
00035 *         The index of the eigenvalue to be computed.  I = 1 or I = 2.
00036 *
00037 *  D      (input) DOUBLE PRECISION array, dimension (2)
00038 *         The original eigenvalues.  We assume D(1) < D(2).
00039 *
00040 *  Z      (input) DOUBLE PRECISION array, dimension (2)
00041 *         The components of the updating vector.
00042 *
00043 *  DELTA  (output) DOUBLE PRECISION array, dimension (2)
00044 *         The vector DELTA contains the information necessary
00045 *         to construct the eigenvectors.
00046 *
00047 *  RHO    (input) DOUBLE PRECISION
00048 *         The scalar in the symmetric updating formula.
00049 *
00050 *  DLAM   (output) DOUBLE PRECISION
00051 *         The computed lambda_I, the I-th updated eigenvalue.
00052 *
00053 *  Further Details
00054 *  ===============
00055 *
00056 *  Based on contributions by
00057 *     Ren-Cang Li, Computer Science Division, University of California
00058 *     at Berkeley, USA
00059 *
00060 *  =====================================================================
00061 *
00062 *     .. Parameters ..
00063       DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
00064       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
00065      $                   FOUR = 4.0D0 )
00066 *     ..
00067 *     .. Local Scalars ..
00068       DOUBLE PRECISION   B, C, DEL, TAU, TEMP, W
00069 *     ..
00070 *     .. Intrinsic Functions ..
00071       INTRINSIC          ABS, SQRT
00072 *     ..
00073 *     .. Executable Statements ..
00074 *
00075       DEL = D( 2 ) - D( 1 )
00076       IF( I.EQ.1 ) THEN
00077          W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
00078          IF( W.GT.ZERO ) THEN
00079             B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00080             C = RHO*Z( 1 )*Z( 1 )*DEL
00081 *
00082 *           B > ZERO, always
00083 *
00084             TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
00085             DLAM = D( 1 ) + TAU
00086             DELTA( 1 ) = -Z( 1 ) / TAU
00087             DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
00088          ELSE
00089             B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00090             C = RHO*Z( 2 )*Z( 2 )*DEL
00091             IF( B.GT.ZERO ) THEN
00092                TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
00093             ELSE
00094                TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
00095             END IF
00096             DLAM = D( 2 ) + TAU
00097             DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
00098             DELTA( 2 ) = -Z( 2 ) / TAU
00099          END IF
00100          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
00101          DELTA( 1 ) = DELTA( 1 ) / TEMP
00102          DELTA( 2 ) = DELTA( 2 ) / TEMP
00103       ELSE
00104 *
00105 *     Now I=2
00106 *
00107          B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
00108          C = RHO*Z( 2 )*Z( 2 )*DEL
00109          IF( B.GT.ZERO ) THEN
00110             TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
00111          ELSE
00112             TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
00113          END IF
00114          DLAM = D( 2 ) + TAU
00115          DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
00116          DELTA( 2 ) = -Z( 2 ) / TAU
00117          TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
00118          DELTA( 1 ) = DELTA( 1 ) / TEMP
00119          DELTA( 2 ) = DELTA( 2 ) / TEMP
00120       END IF
00121       RETURN
00122 *
00123 *     End OF DLAED5
00124 *
00125       END