1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
|
SUBROUTINE SLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
*
* -- LAPACK auxiliary routine (version 3.2) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER I
REAL DSIGMA, RHO
* ..
* .. Array Arguments ..
REAL D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
* ..
*
* Purpose
* =======
*
* This subroutine computes the square root of the I-th eigenvalue
* of a positive symmetric rank-one modification of a 2-by-2 diagonal
* matrix
*
* diag( D ) * diag( D ) + RHO * Z * transpose(Z) .
*
* The diagonal entries in the array D are assumed to satisfy
*
* 0 <= D(i) < D(j) for i < j .
*
* We also assume RHO > 0 and that the Euclidean norm of the vector
* Z is one.
*
* Arguments
* =========
*
* I (input) INTEGER
* The index of the eigenvalue to be computed. I = 1 or I = 2.
*
* D (input) REAL array, dimension (2)
* The original eigenvalues. We assume 0 <= D(1) < D(2).
*
* Z (input) REAL array, dimension (2)
* The components of the updating vector.
*
* DELTA (output) REAL array, dimension (2)
* Contains (D(j) - sigma_I) in its j-th component.
* The vector DELTA contains the information necessary
* to construct the eigenvectors.
*
* RHO (input) REAL
* The scalar in the symmetric updating formula.
*
* DSIGMA (output) REAL
* The computed sigma_I, the I-th updated eigenvalue.
*
* WORK (workspace) REAL array, dimension (2)
* WORK contains (D(j) + sigma_I) in its j-th component.
*
* Further Details
* ===============
*
* Based on contributions by
* Ren-Cang Li, Computer Science Division, University of California
* at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO, THREE, FOUR
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0,
$ THREE = 3.0E+0, FOUR = 4.0E+0 )
* ..
* .. Local Scalars ..
REAL B, C, DEL, DELSQ, TAU, W
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
DEL = D( 2 ) - D( 1 )
DELSQ = DEL*( D( 2 )+D( 1 ) )
IF( I.EQ.1 ) THEN
W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )-
$ Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL
IF( W.GT.ZERO ) THEN
B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 1 )*Z( 1 )*DELSQ
*
* B > ZERO, always
*
* The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
*
TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
*
* The following TAU is DSIGMA - D( 1 )
*
TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) )
DSIGMA = D( 1 ) + TAU
DELTA( 1 ) = -TAU
DELTA( 2 ) = DEL - TAU
WORK( 1 ) = TWO*D( 1 ) + TAU
WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 )
* DELTA( 1 ) = -Z( 1 ) / TAU
* DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
ELSE
B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DELSQ
*
* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
*
IF( B.GT.ZERO ) THEN
TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
ELSE
TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
END IF
*
* The following TAU is DSIGMA - D( 2 )
*
TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) )
DSIGMA = D( 2 ) + TAU
DELTA( 1 ) = -( DEL+TAU )
DELTA( 2 ) = -TAU
WORK( 1 ) = D( 1 ) + TAU + D( 2 )
WORK( 2 ) = TWO*D( 2 ) + TAU
* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
* DELTA( 2 ) = -Z( 2 ) / TAU
END IF
* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
* DELTA( 1 ) = DELTA( 1 ) / TEMP
* DELTA( 2 ) = DELTA( 2 ) / TEMP
ELSE
*
* Now I=2
*
B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DELSQ
*
* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
*
IF( B.GT.ZERO ) THEN
TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
ELSE
TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
END IF
*
* The following TAU is DSIGMA - D( 2 )
*
TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) )
DSIGMA = D( 2 ) + TAU
DELTA( 1 ) = -( DEL+TAU )
DELTA( 2 ) = -TAU
WORK( 1 ) = D( 1 ) + TAU + D( 2 )
WORK( 2 ) = TWO*D( 2 ) + TAU
* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
* DELTA( 2 ) = -Z( 2 ) / TAU
* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
* DELTA( 1 ) = DELTA( 1 ) / TEMP
* DELTA( 2 ) = DELTA( 2 ) / TEMP
END IF
RETURN
*
* End of SLASD5
*
END
|