LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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slasd5.f
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1*> \brief \b SLASD5 computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slasd5.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
22*
23* .. Scalar Arguments ..
24* INTEGER I
25* REAL DSIGMA, RHO
26* ..
27* .. Array Arguments ..
28* REAL D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> This subroutine computes the square root of the I-th eigenvalue
38*> of a positive symmetric rank-one modification of a 2-by-2 diagonal
39*> matrix
40*>
41*> diag( D ) * diag( D ) + RHO * Z * transpose(Z) .
42*>
43*> The diagonal entries in the array D are assumed to satisfy
44*>
45*> 0 <= D(i) < D(j) for i < j .
46*>
47*> We also assume RHO > 0 and that the Euclidean norm of the vector
48*> Z is one.
49*> \endverbatim
50*
51* Arguments:
52* ==========
53*
54*> \param[in] I
55*> \verbatim
56*> I is INTEGER
57*> The index of the eigenvalue to be computed. I = 1 or I = 2.
58*> \endverbatim
59*>
60*> \param[in] D
61*> \verbatim
62*> D is REAL array, dimension (2)
63*> The original eigenvalues. We assume 0 <= D(1) < D(2).
64*> \endverbatim
65*>
66*> \param[in] Z
67*> \verbatim
68*> Z is REAL array, dimension (2)
69*> The components of the updating vector.
70*> \endverbatim
71*>
72*> \param[out] DELTA
73*> \verbatim
74*> DELTA is REAL array, dimension (2)
75*> Contains (D(j) - sigma_I) in its j-th component.
76*> The vector DELTA contains the information necessary
77*> to construct the eigenvectors.
78*> \endverbatim
79*>
80*> \param[in] RHO
81*> \verbatim
82*> RHO is REAL
83*> The scalar in the symmetric updating formula.
84*> \endverbatim
85*>
86*> \param[out] DSIGMA
87*> \verbatim
88*> DSIGMA is REAL
89*> The computed sigma_I, the I-th updated eigenvalue.
90*> \endverbatim
91*>
92*> \param[out] WORK
93*> \verbatim
94*> WORK is REAL array, dimension (2)
95*> WORK contains (D(j) + sigma_I) in its j-th component.
96*> \endverbatim
97*
98* Authors:
99* ========
100*
101*> \author Univ. of Tennessee
102*> \author Univ. of California Berkeley
103*> \author Univ. of Colorado Denver
104*> \author NAG Ltd.
105*
106*> \ingroup OTHERauxiliary
107*
108*> \par Contributors:
109* ==================
110*>
111*> Ren-Cang Li, Computer Science Division, University of California
112*> at Berkeley, USA
113*>
114* =====================================================================
115 SUBROUTINE slasd5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
116*
117* -- LAPACK auxiliary routine --
118* -- LAPACK is a software package provided by Univ. of Tennessee, --
119* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120*
121* .. Scalar Arguments ..
122 INTEGER I
123 REAL DSIGMA, RHO
124* ..
125* .. Array Arguments ..
126 REAL D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
127* ..
128*
129* =====================================================================
130*
131* .. Parameters ..
132 REAL ZERO, ONE, TWO, THREE, FOUR
133 parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0,
134 \$ three = 3.0e+0, four = 4.0e+0 )
135* ..
136* .. Local Scalars ..
137 REAL B, C, DEL, DELSQ, TAU, W
138* ..
139* .. Intrinsic Functions ..
140 INTRINSIC abs, sqrt
141* ..
142* .. Executable Statements ..
143*
144 del = d( 2 ) - d( 1 )
145 delsq = del*( d( 2 )+d( 1 ) )
146 IF( i.EQ.1 ) THEN
147 w = one + four*rho*( z( 2 )*z( 2 ) / ( d( 1 )+three*d( 2 ) )-
148 \$ z( 1 )*z( 1 ) / ( three*d( 1 )+d( 2 ) ) ) / del
149 IF( w.GT.zero ) THEN
150 b = delsq + rho*( z( 1 )*z( 1 )+z( 2 )*z( 2 ) )
151 c = rho*z( 1 )*z( 1 )*delsq
152*
153* B > ZERO, always
154*
155* The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
156*
157 tau = two*c / ( b+sqrt( abs( b*b-four*c ) ) )
158*
159* The following TAU is DSIGMA - D( 1 )
160*
161 tau = tau / ( d( 1 )+sqrt( d( 1 )*d( 1 )+tau ) )
162 dsigma = d( 1 ) + tau
163 delta( 1 ) = -tau
164 delta( 2 ) = del - tau
165 work( 1 ) = two*d( 1 ) + tau
166 work( 2 ) = ( d( 1 )+tau ) + d( 2 )
167* DELTA( 1 ) = -Z( 1 ) / TAU
168* DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
169 ELSE
170 b = -delsq + rho*( z( 1 )*z( 1 )+z( 2 )*z( 2 ) )
171 c = rho*z( 2 )*z( 2 )*delsq
172*
173* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
174*
175 IF( b.GT.zero ) THEN
176 tau = -two*c / ( b+sqrt( b*b+four*c ) )
177 ELSE
178 tau = ( b-sqrt( b*b+four*c ) ) / two
179 END IF
180*
181* The following TAU is DSIGMA - D( 2 )
182*
183 tau = tau / ( d( 2 )+sqrt( abs( d( 2 )*d( 2 )+tau ) ) )
184 dsigma = d( 2 ) + tau
185 delta( 1 ) = -( del+tau )
186 delta( 2 ) = -tau
187 work( 1 ) = d( 1 ) + tau + d( 2 )
188 work( 2 ) = two*d( 2 ) + tau
189* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
190* DELTA( 2 ) = -Z( 2 ) / TAU
191 END IF
192* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
193* DELTA( 1 ) = DELTA( 1 ) / TEMP
194* DELTA( 2 ) = DELTA( 2 ) / TEMP
195 ELSE
196*
197* Now I=2
198*
199 b = -delsq + rho*( z( 1 )*z( 1 )+z( 2 )*z( 2 ) )
200 c = rho*z( 2 )*z( 2 )*delsq
201*
202* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
203*
204 IF( b.GT.zero ) THEN
205 tau = ( b+sqrt( b*b+four*c ) ) / two
206 ELSE
207 tau = two*c / ( -b+sqrt( b*b+four*c ) )
208 END IF
209*
210* The following TAU is DSIGMA - D( 2 )
211*
212 tau = tau / ( d( 2 )+sqrt( d( 2 )*d( 2 )+tau ) )
213 dsigma = d( 2 ) + tau
214 delta( 1 ) = -( del+tau )
215 delta( 2 ) = -tau
216 work( 1 ) = d( 1 ) + tau + d( 2 )
217 work( 2 ) = two*d( 2 ) + tau
218* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
219* DELTA( 2 ) = -Z( 2 ) / TAU
220* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
221* DELTA( 1 ) = DELTA( 1 ) / TEMP
222* DELTA( 2 ) = DELTA( 2 ) / TEMP
223 END IF
224 RETURN
225*
226* End of SLASD5
227*
228 END
subroutine slasd5(I, D, Z, DELTA, RHO, DSIGMA, WORK)
SLASD5 computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification ...
Definition: slasd5.f:116