 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dlaed9()

 subroutine dlaed9 ( integer K, integer KSTART, integer KSTOP, integer N, double precision, dimension( * ) D, double precision, dimension( ldq, * ) Q, integer LDQ, double precision RHO, double precision, dimension( * ) DLAMDA, double precision, dimension( * ) W, double precision, dimension( lds, * ) S, integer LDS, integer INFO )

DLAED9 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.

Purpose:
``` DLAED9 finds the roots of the secular equation, as defined by the
values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
appropriate calls to DLAED4 and then stores the new matrix of
eigenvectors for use in calculating the next level of Z vectors.```
Parameters
 [in] K ``` K is INTEGER The number of terms in the rational function to be solved by DLAED4. K >= 0.``` [in] KSTART ` KSTART is INTEGER` [in] KSTOP ``` KSTOP is INTEGER The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP are to be computed. 1 <= KSTART <= KSTOP <= K.``` [in] N ``` N is INTEGER The number of rows and columns in the Q matrix. N >= K (delation may result in N > K).``` [out] D ``` D is DOUBLE PRECISION array, dimension (N) D(I) contains the updated eigenvalues for KSTART <= I <= KSTOP.``` [out] Q ` Q is DOUBLE PRECISION array, dimension (LDQ,N)` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. LDQ >= max( 1, N ).``` [in] RHO ``` RHO is DOUBLE PRECISION The value of the parameter in the rank one update equation. RHO >= 0 required.``` [in] DLAMDA ``` DLAMDA is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.``` [in] W ``` W is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector.``` [out] S ``` S is DOUBLE PRECISION array, dimension (LDS, K) Will contain the eigenvectors of the repaired matrix which will be stored for subsequent Z vector calculation and multiplied by the previously accumulated eigenvectors to update the system.``` [in] LDS ``` LDS is INTEGER The leading dimension of S. LDS >= max( 1, K ).``` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge```
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 154 of file dlaed9.f.

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 DOUBLE PRECISION RHO
164* ..
165* .. Array Arguments ..
166 DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
167 \$ W( * )
168* ..
169*
170* =====================================================================
171*
172* .. Local Scalars ..
173 INTEGER I, J
174 DOUBLE PRECISION TEMP
175* ..
176* .. External Functions ..
177 DOUBLE PRECISION DLAMC3, DNRM2
178 EXTERNAL dlamc3, dnrm2
179* ..
180* .. External Subroutines ..
181 EXTERNAL dcopy, dlaed4, 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( 'DLAED9', -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 ) = dlamc3( dlamda( i ), dlamda( i ) ) - dlamda( i )
235 10 CONTINUE
236*
237 DO 20 j = kstart, kstop
238 CALL dlaed4( 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 dcopy( k, w, 1, s, 1 )
258*
259* Initialize W(I) = Q(I,I)
260*
261 CALL dcopy( 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 = dnrm2( 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 DLAED9
290*
double precision function dlamc3(A, B)
DLAMC3
Definition: dlamch.f:169
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlaed4(N, I, D, Z, DELTA, RHO, DLAM, INFO)
DLAED4 used by DSTEDC. Finds a single root of the secular equation.
Definition: dlaed4.f:145
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition: dnrm2.f90:89
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