1 SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
2 *
3 * -- LAPACK routine (version 3.2) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * November 2006
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
2 *
3 * -- LAPACK routine (version 3.2) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * November 2006
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