LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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slaed9.f
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1*> \brief \b SLAED9 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaed9.f">
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed9.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
22* S, LDS, INFO )
23*
24* .. Scalar Arguments ..
25* INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
26* REAL RHO
27* ..
28* .. Array Arguments ..
29* REAL D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
30* \$ W( * )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*>
39*> SLAED9 finds the roots of the secular equation, as defined by the
40*> values in D, Z, and RHO, between KSTART and KSTOP. It makes the
41*> appropriate calls to SLAED4 and then stores the new matrix of
42*> eigenvectors for use in calculating the next level of Z vectors.
43*> \endverbatim
44*
45* Arguments:
46* ==========
47*
48*> \param[in] K
49*> \verbatim
50*> K is INTEGER
51*> The number of terms in the rational function to be solved by
52*> SLAED4. K >= 0.
53*> \endverbatim
54*>
55*> \param[in] KSTART
56*> \verbatim
57*> KSTART is INTEGER
58*> \endverbatim
59*>
60*> \param[in] KSTOP
61*> \verbatim
62*> KSTOP is INTEGER
63*> The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
64*> are to be computed. 1 <= KSTART <= KSTOP <= K.
65*> \endverbatim
66*>
67*> \param[in] N
68*> \verbatim
69*> N is INTEGER
70*> The number of rows and columns in the Q matrix.
71*> N >= K (delation may result in N > K).
72*> \endverbatim
73*>
74*> \param[out] D
75*> \verbatim
76*> D is REAL array, dimension (N)
77*> D(I) contains the updated eigenvalues
78*> for KSTART <= I <= KSTOP.
79*> \endverbatim
80*>
81*> \param[out] Q
82*> \verbatim
83*> Q is REAL array, dimension (LDQ,N)
84*> \endverbatim
85*>
86*> \param[in] LDQ
87*> \verbatim
88*> LDQ is INTEGER
89*> The leading dimension of the array Q. LDQ >= max( 1, N ).
90*> \endverbatim
91*>
92*> \param[in] RHO
93*> \verbatim
94*> RHO is REAL
95*> The value of the parameter in the rank one update equation.
96*> RHO >= 0 required.
97*> \endverbatim
98*>
99*> \param[in] DLAMDA
100*> \verbatim
101*> DLAMDA is REAL array, dimension (K)
102*> The first K elements of this array contain the old roots
103*> of the deflated updating problem. These are the poles
104*> of the secular equation.
105*> \endverbatim
106*>
107*> \param[in] W
108*> \verbatim
109*> W is REAL array, dimension (K)
110*> The first K elements of this array contain the components
111*> of the deflation-adjusted updating vector.
112*> \endverbatim
113*>
114*> \param[out] S
115*> \verbatim
116*> S is REAL array, dimension (LDS, K)
117*> Will contain the eigenvectors of the repaired matrix which
118*> will be stored for subsequent Z vector calculation and
119*> multiplied by the previously accumulated eigenvectors
120*> to update the system.
121*> \endverbatim
122*>
123*> \param[in] LDS
124*> \verbatim
125*> LDS is INTEGER
126*> The leading dimension of S. LDS >= max( 1, K ).
127*> \endverbatim
128*>
129*> \param[out] INFO
130*> \verbatim
131*> INFO is INTEGER
132*> = 0: successful exit.
133*> < 0: if INFO = -i, the i-th argument had an illegal value.
134*> > 0: if INFO = 1, an eigenvalue did not converge
135*> \endverbatim
136*
137* Authors:
138* ========
139*
140*> \author Univ. of Tennessee
141*> \author Univ. of California Berkeley
142*> \author Univ. of Colorado Denver
143*> \author NAG Ltd.
144*
145*> \ingroup auxOTHERcomputational
146*
147*> \par Contributors:
148* ==================
149*>
150*> Jeff Rutter, Computer Science Division, University of California
151*> at Berkeley, USA
152*
153* =====================================================================
154 SUBROUTINE slaed9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
155 \$ S, LDS, INFO )
156*
157* -- LAPACK computational routine --
158* -- LAPACK is a software package provided by Univ. of Tennessee, --
159* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
160*
161* .. Scalar Arguments ..
162 INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
163 REAL RHO
164* ..
165* .. Array Arguments ..
166 REAL D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
167 \$ w( * )
168* ..
169*
170* =====================================================================
171*
172* .. Local Scalars ..
173 INTEGER I, J
174 REAL TEMP
175* ..
176* .. External Functions ..
177 REAL SLAMC3, SNRM2
178 EXTERNAL slamc3, snrm2
179* ..
180* .. External Subroutines ..
181 EXTERNAL scopy, slaed4, xerbla
182* ..
183* .. Intrinsic Functions ..
184 INTRINSIC max, sign, sqrt
185* ..
186* .. Executable Statements ..
187*
188* Test the input parameters.
189*
190 info = 0
191*
192 IF( k.LT.0 ) THEN
193 info = -1
194 ELSE IF( kstart.LT.1 .OR. kstart.GT.max( 1, k ) ) THEN
195 info = -2
196 ELSE IF( max( 1, kstop ).LT.kstart .OR. kstop.GT.max( 1, k ) )
197 \$ THEN
198 info = -3
199 ELSE IF( n.LT.k ) THEN
200 info = -4
201 ELSE IF( ldq.LT.max( 1, k ) ) THEN
202 info = -7
203 ELSE IF( lds.LT.max( 1, k ) ) THEN
204 info = -12
205 END IF
206 IF( info.NE.0 ) THEN
207 CALL xerbla( 'SLAED9', -info )
208 RETURN
209 END IF
210*
211* Quick return if possible
212*
213 IF( k.EQ.0 )
214 \$ RETURN
215*
216* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
217* be computed with high relative accuracy (barring over/underflow).
218* This is a problem on machines without a guard digit in
219* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
220* The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
221* which on any of these machines zeros out the bottommost
222* bit of DLAMDA(I) if it is 1; this makes the subsequent
223* subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
224* occurs. On binary machines with a guard digit (almost all
225* machines) it does not change DLAMDA(I) at all. On hexadecimal
226* and decimal machines with a guard digit, it slightly
227* changes the bottommost bits of DLAMDA(I). It does not account
228* for hexadecimal or decimal machines without guard digits
229* (we know of none). We use a subroutine call to compute
230* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
231* this code.
232*
233 DO 10 i = 1, n
234 dlamda( i ) = slamc3( dlamda( i ), dlamda( i ) ) - dlamda( i )
235 10 CONTINUE
236*
237 DO 20 j = kstart, kstop
238 CALL slaed4( k, j, dlamda, w, q( 1, j ), rho, d( j ), info )
239*
240* If the zero finder fails, the computation is terminated.
241*
242 IF( info.NE.0 )
243 \$ GO TO 120
244 20 CONTINUE
245*
246 IF( k.EQ.1 .OR. k.EQ.2 ) THEN
247 DO 40 i = 1, k
248 DO 30 j = 1, k
249 s( j, i ) = q( j, i )
250 30 CONTINUE
251 40 CONTINUE
252 GO TO 120
253 END IF
254*
255* Compute updated W.
256*
257 CALL scopy( k, w, 1, s, 1 )
258*
259* Initialize W(I) = Q(I,I)
260*
261 CALL scopy( k, q, ldq+1, w, 1 )
262 DO 70 j = 1, k
263 DO 50 i = 1, j - 1
264 w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
265 50 CONTINUE
266 DO 60 i = j + 1, k
267 w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
268 60 CONTINUE
269 70 CONTINUE
270 DO 80 i = 1, k
271 w( i ) = sign( sqrt( -w( i ) ), s( i, 1 ) )
272 80 CONTINUE
273*
274* Compute eigenvectors of the modified rank-1 modification.
275*
276 DO 110 j = 1, k
277 DO 90 i = 1, k
278 q( i, j ) = w( i ) / q( i, j )
279 90 CONTINUE
280 temp = snrm2( k, q( 1, j ), 1 )
281 DO 100 i = 1, k
282 s( i, j ) = q( i, j ) / temp
283 100 CONTINUE
284 110 CONTINUE
285*
286 120 CONTINUE
287 RETURN
288*
289* End of SLAED9
290*
291 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slaed9(K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO)
SLAED9 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors....
Definition: slaed9.f:156
subroutine slaed4(N, I, D, Z, DELTA, RHO, DLAM, INFO)
SLAED4 used by SSTEDC. Finds a single root of the secular equation.
Definition: slaed4.f:145
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82