LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dlasd5()

 subroutine dlasd5 ( integer i, double precision, dimension( 2 ) d, double precision, dimension( 2 ) z, double precision, dimension( 2 ) delta, double precision rho, double precision dsigma, double precision, dimension( 2 ) work )

DLASD5 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.

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.
Parameters
 [in] I I is INTEGER The index of the eigenvalue to be computed. I = 1 or I = 2. [in] D D is DOUBLE PRECISION array, dimension ( 2 ) The original eigenvalues. We assume 0 <= D(1) < D(2). [in] Z Z is DOUBLE PRECISION array, dimension ( 2 ) The components of the updating vector. [out] DELTA DELTA is DOUBLE PRECISION array, dimension ( 2 ) Contains (D(j) - sigma_I) in its j-th component. The vector DELTA contains the information necessary to construct the eigenvectors. [in] RHO RHO is DOUBLE PRECISION The scalar in the symmetric updating formula. [out] DSIGMA DSIGMA is DOUBLE PRECISION The computed sigma_I, the I-th updated eigenvalue. [out] WORK WORK is DOUBLE PRECISION array, dimension ( 2 ) WORK contains (D(j) + sigma_I) in its j-th component.
Contributors:
Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 115 of file dlasd5.f.

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 DOUBLE PRECISION DSIGMA, RHO
124* ..
125* .. Array Arguments ..
126 DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
127* ..
128*
129* =====================================================================
130*
131* .. Parameters ..
132 DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR
133 parameter( zero = 0.0d+0, one = 1.0d+0, two = 2.0d+0,
134 \$ three = 3.0d+0, four = 4.0d+0 )
135* ..
136* .. Local Scalars ..
137 DOUBLE PRECISION 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 DLASD5
227*
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