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[project @ 2007-10-26 15:52:57 by jwe]
author jwe
date Fri, 26 Oct 2007 15:52:58 +0000
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7071:c3b479e753dd 7072:b48d486f641d
1 SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
2 *
3 * -- LAPACK auxiliary routine (version 3.1) --
4 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
5 * November 2006
6 *
7 * .. Scalar Arguments ..
8 INTEGER I
9 DOUBLE PRECISION DSIGMA, RHO
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
13 * ..
14 *
15 * Purpose
16 * =======
17 *
18 * This subroutine computes the square root of the I-th eigenvalue
19 * of a positive symmetric rank-one modification of a 2-by-2 diagonal
20 * matrix
21 *
22 * diag( D ) * diag( D ) + RHO * Z * transpose(Z) .
23 *
24 * The diagonal entries in the array D are assumed to satisfy
25 *
26 * 0 <= 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 0 <= 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 * Contains (D(j) - sigma_I) in its j-th component.
45 * The vector DELTA contains the information necessary
46 * to construct the eigenvectors.
47 *
48 * RHO (input) DOUBLE PRECISION
49 * The scalar in the symmetric updating formula.
50 *
51 * DSIGMA (output) DOUBLE PRECISION
52 * The computed sigma_I, the I-th updated eigenvalue.
53 *
54 * WORK (workspace) DOUBLE PRECISION array, dimension ( 2 )
55 * WORK contains (D(j) + sigma_I) in its j-th component.
56 *
57 * Further Details
58 * ===============
59 *
60 * Based on contributions by
61 * Ren-Cang Li, Computer Science Division, University of California
62 * at Berkeley, USA
63 *
64 * =====================================================================
65 *
66 * .. Parameters ..
67 DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR
68 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
69 $ THREE = 3.0D+0, FOUR = 4.0D+0 )
70 * ..
71 * .. Local Scalars ..
72 DOUBLE PRECISION B, C, DEL, DELSQ, TAU, W
73 * ..
74 * .. Intrinsic Functions ..
75 INTRINSIC ABS, SQRT
76 * ..
77 * .. Executable Statements ..
78 *
79 DEL = D( 2 ) - D( 1 )
80 DELSQ = DEL*( D( 2 )+D( 1 ) )
81 IF( I.EQ.1 ) THEN
82 W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )-
83 $ Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL
84 IF( W.GT.ZERO ) THEN
85 B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
86 C = RHO*Z( 1 )*Z( 1 )*DELSQ
87 *
88 * B > ZERO, always
89 *
90 * The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
91 *
92 TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
93 *
94 * The following TAU is DSIGMA - D( 1 )
95 *
96 TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) )
97 DSIGMA = D( 1 ) + TAU
98 DELTA( 1 ) = -TAU
99 DELTA( 2 ) = DEL - TAU
100 WORK( 1 ) = TWO*D( 1 ) + TAU
101 WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 )
102 * DELTA( 1 ) = -Z( 1 ) / TAU
103 * DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
104 ELSE
105 B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
106 C = RHO*Z( 2 )*Z( 2 )*DELSQ
107 *
108 * The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
109 *
110 IF( B.GT.ZERO ) THEN
111 TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
112 ELSE
113 TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
114 END IF
115 *
116 * The following TAU is DSIGMA - D( 2 )
117 *
118 TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) )
119 DSIGMA = D( 2 ) + TAU
120 DELTA( 1 ) = -( DEL+TAU )
121 DELTA( 2 ) = -TAU
122 WORK( 1 ) = D( 1 ) + TAU + D( 2 )
123 WORK( 2 ) = TWO*D( 2 ) + TAU
124 * DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
125 * DELTA( 2 ) = -Z( 2 ) / TAU
126 END IF
127 * TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
128 * DELTA( 1 ) = DELTA( 1 ) / TEMP
129 * DELTA( 2 ) = DELTA( 2 ) / TEMP
130 ELSE
131 *
132 * Now I=2
133 *
134 B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
135 C = RHO*Z( 2 )*Z( 2 )*DELSQ
136 *
137 * The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
138 *
139 IF( B.GT.ZERO ) THEN
140 TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
141 ELSE
142 TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
143 END IF
144 *
145 * The following TAU is DSIGMA - D( 2 )
146 *
147 TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) )
148 DSIGMA = D( 2 ) + TAU
149 DELTA( 1 ) = -( DEL+TAU )
150 DELTA( 2 ) = -TAU
151 WORK( 1 ) = D( 1 ) + TAU + D( 2 )
152 WORK( 2 ) = TWO*D( 2 ) + TAU
153 * DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
154 * DELTA( 2 ) = -Z( 2 ) / TAU
155 * TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
156 * DELTA( 1 ) = DELTA( 1 ) / TEMP
157 * DELTA( 2 ) = DELTA( 2 ) / TEMP
158 END IF
159 RETURN
160 *
161 * End of DLASD5
162 *
163 END