1      SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
2*
3*  -- LAPACK routine (version 3.0) --
4*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
5*     Courant Institute, NAG Ltd., and Rice University
6*     September 30, 1994
7*
8*     .. Scalar Arguments ..
9      INTEGER            I
10      DOUBLE PRECISION   DLAM, RHO
11*     ..
12*     .. Array Arguments ..
13      DOUBLE PRECISION   D( 2 ), DELTA( 2 ), Z( 2 )
14*     ..
15*
16*  Purpose
17*  =======
18*
19*  This subroutine computes the I-th eigenvalue of a symmetric rank-one
20*  modification of a 2-by-2 diagonal matrix
21*
22*             diag( D )  +  RHO *  Z * transpose(Z) .
23*
24*  The diagonal elements in the array D are assumed to satisfy
25*
26*             D(i) < D(j)  for  i < j .
27*
28*  We also assume RHO > 0 and that the Euclidean norm of the vector
29*  Z is one.
30*
31*  Arguments
32*  =========
33*
34*  I      (input) INTEGER
35*         The index of the eigenvalue to be computed.  I = 1 or I = 2.
36*
37*  D      (input) DOUBLE PRECISION array, dimension (2)
38*         The original eigenvalues.  We assume D(1) < D(2).
39*
40*  Z      (input) DOUBLE PRECISION array, dimension (2)
41*         The components of the updating vector.
42*
43*  DELTA  (output) DOUBLE PRECISION array, dimension (2)
44*         The vector DELTA contains the information necessary
45*         to construct the eigenvectors.
46*
47*  RHO    (input) DOUBLE PRECISION
48*         The scalar in the symmetric updating formula.
49*
50*  DLAM   (output) DOUBLE PRECISION
51*         The computed lambda_I, the I-th updated eigenvalue.
52*
53*  Further Details
54*  ===============
55*
56*  Based on contributions by
57*     Ren-Cang Li, Computer Science Division, University of California
58*     at Berkeley, USA
59*
60*  =====================================================================
61*
62*     .. Parameters ..
63      DOUBLE PRECISION   ZERO, ONE, TWO, FOUR
64      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
65     $                   FOUR = 4.0D0 )
66*     ..
67*     .. Local Scalars ..
68      DOUBLE PRECISION   B, C, DEL, TAU, TEMP, W
69*     ..
70*     .. Intrinsic Functions ..
71      INTRINSIC          ABS, SQRT
72*     ..
73*     .. Executable Statements ..
74*
75      DEL = D( 2 ) - D( 1 )
76      IF( I.EQ.1 ) THEN
77         W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
78         IF( W.GT.ZERO ) THEN
79            B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
80            C = RHO*Z( 1 )*Z( 1 )*DEL
81*
82*           B > ZERO, always
83*
84            TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
85            DLAM = D( 1 ) + TAU
86            DELTA( 1 ) = -Z( 1 ) / TAU
87            DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
88         ELSE
89            B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
90            C = RHO*Z( 2 )*Z( 2 )*DEL
91            IF( B.GT.ZERO ) THEN
92               TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
93            ELSE
94               TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
95            END IF
96            DLAM = D( 2 ) + TAU
97            DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
98            DELTA( 2 ) = -Z( 2 ) / TAU
99         END IF
100         TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
101         DELTA( 1 ) = DELTA( 1 ) / TEMP
102         DELTA( 2 ) = DELTA( 2 ) / TEMP
103      ELSE
104*
105*     Now I=2
106*
107         B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
108         C = RHO*Z( 2 )*Z( 2 )*DEL
109         IF( B.GT.ZERO ) THEN
110            TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
111         ELSE
112            TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
113         END IF
114         DLAM = D( 2 ) + TAU
115         DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
116         DELTA( 2 ) = -Z( 2 ) / TAU
117         TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
118         DELTA( 1 ) = DELTA( 1 ) / TEMP
119         DELTA( 2 ) = DELTA( 2 ) / TEMP
120      END IF
121      RETURN
122*
123*     End OF DLAED5
124*
125      END
126