LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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slaed5.f
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1*> \brief \b SLAED5 used by SSTEDC. Solves the 2-by-2 secular equation.
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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11*> [TGZ]</a>
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed5.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SLAED5( I, D, Z, DELTA, RHO, DLAM )
22*
23* .. Scalar Arguments ..
24* INTEGER I
25* REAL DLAM, RHO
26* ..
27* .. Array Arguments ..
28* REAL D( 2 ), DELTA( 2 ), Z( 2 )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> This subroutine computes the I-th eigenvalue of a symmetric rank-one
38*> modification of a 2-by-2 diagonal matrix
39*>
40*> diag( D ) + RHO * Z * transpose(Z) .
41*>
42*> The diagonal elements in the array D are assumed to satisfy
43*>
44*> D(i) < D(j) for i < j .
45*>
46*> We also assume RHO > 0 and that the Euclidean norm of the vector
47*> Z is one.
48*> \endverbatim
49*
50* Arguments:
51* ==========
52*
53*> \param[in] I
54*> \verbatim
55*> I is INTEGER
56*> The index of the eigenvalue to be computed. I = 1 or I = 2.
57*> \endverbatim
58*>
59*> \param[in] D
60*> \verbatim
61*> D is REAL array, dimension (2)
62*> The original eigenvalues. We assume D(1) < D(2).
63*> \endverbatim
64*>
65*> \param[in] Z
66*> \verbatim
67*> Z is REAL array, dimension (2)
68*> The components of the updating vector.
69*> \endverbatim
70*>
71*> \param[out] DELTA
72*> \verbatim
73*> DELTA is REAL array, dimension (2)
74*> The vector DELTA contains the information necessary
75*> to construct the eigenvectors.
76*> \endverbatim
77*>
78*> \param[in] RHO
79*> \verbatim
80*> RHO is REAL
81*> The scalar in the symmetric updating formula.
82*> \endverbatim
83*>
84*> \param[out] DLAM
85*> \verbatim
86*> DLAM is REAL
87*> The computed lambda_I, the I-th updated eigenvalue.
88*> \endverbatim
89*
90* Authors:
91* ========
92*
93*> \author Univ. of Tennessee
94*> \author Univ. of California Berkeley
95*> \author Univ. of Colorado Denver
96*> \author NAG Ltd.
97*
98*> \ingroup auxOTHERcomputational
99*
100*> \par Contributors:
101* ==================
102*>
103*> Ren-Cang Li, Computer Science Division, University of California
104*> at Berkeley, USA
105*>
106* =====================================================================
107 SUBROUTINE slaed5( I, D, Z, DELTA, RHO, DLAM )
108*
109* -- LAPACK computational routine --
110* -- LAPACK is a software package provided by Univ. of Tennessee, --
111* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
112*
113* .. Scalar Arguments ..
114 INTEGER I
115 REAL DLAM, RHO
116* ..
117* .. Array Arguments ..
118 REAL D( 2 ), DELTA( 2 ), Z( 2 )
119* ..
120*
121* =====================================================================
122*
123* .. Parameters ..
124 REAL ZERO, ONE, TWO, FOUR
125 parameter( zero = 0.0e0, one = 1.0e0, two = 2.0e0,
126 \$ four = 4.0e0 )
127* ..
128* .. Local Scalars ..
129 REAL B, C, DEL, TAU, TEMP, W
130* ..
131* .. Intrinsic Functions ..
132 INTRINSIC abs, sqrt
133* ..
134* .. Executable Statements ..
135*
136 del = d( 2 ) - d( 1 )
137 IF( i.EQ.1 ) THEN
138 w = one + two*rho*( z( 2 )*z( 2 )-z( 1 )*z( 1 ) ) / del
139 IF( w.GT.zero ) THEN
140 b = del + rho*( z( 1 )*z( 1 )+z( 2 )*z( 2 ) )
141 c = rho*z( 1 )*z( 1 )*del
142*
143* B > ZERO, always
144*
145 tau = two*c / ( b+sqrt( abs( b*b-four*c ) ) )
146 dlam = d( 1 ) + tau
147 delta( 1 ) = -z( 1 ) / tau
148 delta( 2 ) = z( 2 ) / ( del-tau )
149 ELSE
150 b = -del + rho*( z( 1 )*z( 1 )+z( 2 )*z( 2 ) )
151 c = rho*z( 2 )*z( 2 )*del
152 IF( b.GT.zero ) THEN
153 tau = -two*c / ( b+sqrt( b*b+four*c ) )
154 ELSE
155 tau = ( b-sqrt( b*b+four*c ) ) / two
156 END IF
157 dlam = d( 2 ) + tau
158 delta( 1 ) = -z( 1 ) / ( del+tau )
159 delta( 2 ) = -z( 2 ) / tau
160 END IF
161 temp = sqrt( delta( 1 )*delta( 1 )+delta( 2 )*delta( 2 ) )
162 delta( 1 ) = delta( 1 ) / temp
163 delta( 2 ) = delta( 2 ) / temp
164 ELSE
165*
166* Now I=2
167*
168 b = -del + rho*( z( 1 )*z( 1 )+z( 2 )*z( 2 ) )
169 c = rho*z( 2 )*z( 2 )*del
170 IF( b.GT.zero ) THEN
171 tau = ( b+sqrt( b*b+four*c ) ) / two
172 ELSE
173 tau = two*c / ( -b+sqrt( b*b+four*c ) )
174 END IF
175 dlam = d( 2 ) + tau
176 delta( 1 ) = -z( 1 ) / ( del+tau )
177 delta( 2 ) = -z( 2 ) / tau
178 temp = sqrt( delta( 1 )*delta( 1 )+delta( 2 )*delta( 2 ) )
179 delta( 1 ) = delta( 1 ) / temp
180 delta( 2 ) = delta( 2 ) / temp
181 END IF
182 RETURN
183*
184* End of SLAED5
185*
186 END
subroutine slaed5(I, D, Z, DELTA, RHO, DLAM)
SLAED5 used by SSTEDC. Solves the 2-by-2 secular equation.
Definition: slaed5.f:108