 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dlaed3()

 subroutine dlaed3 ( integer K, integer N, integer N1, double precision, dimension( * ) D, double precision, dimension( ldq, * ) Q, integer LDQ, double precision RHO, double precision, dimension( * ) DLAMDA, double precision, dimension( * ) Q2, integer, dimension( * ) INDX, integer, dimension( * ) CTOT, double precision, dimension( * ) W, double precision, dimension( * ) S, integer INFO )

DLAED3 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

Purpose:
``` DLAED3 finds the roots of the secular equation, as defined by the
values in D, W, and RHO, between 1 and K.  It makes the
appropriate calls to DLAED4 and then updates the eigenvectors by
multiplying the matrix of eigenvectors of the pair of eigensystems
being combined by the matrix of eigenvectors of the K-by-K system
which is solved here.

This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.```
Parameters
 [in] K ``` K is INTEGER The number of terms in the rational function to be solved by DLAED4. K >= 0.``` [in] N ``` N is INTEGER The number of rows and columns in the Q matrix. N >= K (deflation may result in N>K).``` [in] N1 ``` N1 is INTEGER The location of the last eigenvalue in the leading submatrix. min(1,N) <= N1 <= N/2.``` [out] D ``` D is DOUBLE PRECISION array, dimension (N) D(I) contains the updated eigenvalues for 1 <= I <= K.``` [out] Q ``` Q is DOUBLE PRECISION array, dimension (LDQ,N) Initially the first K columns are used as workspace. On output the columns 1 to K contain the updated eigenvectors.``` [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,out] 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. May be changed on output by having lowest order bit set to zero on Cray X-MP, Cray Y-MP, Cray-2, or Cray C-90, as described above.``` [in] Q2 ``` Q2 is DOUBLE PRECISION array, dimension (LDQ2*N) The first K columns of this matrix contain the non-deflated eigenvectors for the split problem.``` [in] INDX ``` INDX is INTEGER array, dimension (N) The permutation used to arrange the columns of the deflated Q matrix into three groups (see DLAED2). The rows of the eigenvectors found by DLAED4 must be likewise permuted before the matrix multiply can take place.``` [in] CTOT ``` CTOT is INTEGER array, dimension (4) A count of the total number of the various types of columns in Q, as described in INDX. The fourth column type is any column which has been deflated.``` [in,out] W ``` W is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.``` [out] S ``` S is DOUBLE PRECISION array, dimension (N1 + 1)*K Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.``` [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
Modified by Francoise Tisseur, University of Tennessee

Definition at line 183 of file dlaed3.f.

185 *
186 * -- LAPACK computational routine --
187 * -- LAPACK is a software package provided by Univ. of Tennessee, --
188 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189 *
190 * .. Scalar Arguments ..
191  INTEGER INFO, K, LDQ, N, N1
192  DOUBLE PRECISION RHO
193 * ..
194 * .. Array Arguments ..
195  INTEGER CTOT( * ), INDX( * )
196  DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
197  \$ S( * ), W( * )
198 * ..
199 *
200 * =====================================================================
201 *
202 * .. Parameters ..
203  DOUBLE PRECISION ONE, ZERO
204  parameter( one = 1.0d0, zero = 0.0d0 )
205 * ..
206 * .. Local Scalars ..
207  INTEGER I, II, IQ2, J, N12, N2, N23
208  DOUBLE PRECISION TEMP
209 * ..
210 * .. External Functions ..
211  DOUBLE PRECISION DLAMC3, DNRM2
212  EXTERNAL dlamc3, dnrm2
213 * ..
214 * .. External Subroutines ..
215  EXTERNAL dcopy, dgemm, dlacpy, dlaed4, dlaset, xerbla
216 * ..
217 * .. Intrinsic Functions ..
218  INTRINSIC max, sign, sqrt
219 * ..
220 * .. Executable Statements ..
221 *
222 * Test the input parameters.
223 *
224  info = 0
225 *
226  IF( k.LT.0 ) THEN
227  info = -1
228  ELSE IF( n.LT.k ) THEN
229  info = -2
230  ELSE IF( ldq.LT.max( 1, n ) ) THEN
231  info = -6
232  END IF
233  IF( info.NE.0 ) THEN
234  CALL xerbla( 'DLAED3', -info )
235  RETURN
236  END IF
237 *
238 * Quick return if possible
239 *
240  IF( k.EQ.0 )
241  \$ RETURN
242 *
243 * Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
244 * be computed with high relative accuracy (barring over/underflow).
245 * This is a problem on machines without a guard digit in
246 * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
247 * The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
248 * which on any of these machines zeros out the bottommost
249 * bit of DLAMDA(I) if it is 1; this makes the subsequent
250 * subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
251 * occurs. On binary machines with a guard digit (almost all
252 * machines) it does not change DLAMDA(I) at all. On hexadecimal
253 * and decimal machines with a guard digit, it slightly
254 * changes the bottommost bits of DLAMDA(I). It does not account
255 * for hexadecimal or decimal machines without guard digits
256 * (we know of none). We use a subroutine call to compute
257 * 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
258 * this code.
259 *
260  DO 10 i = 1, k
261  dlamda( i ) = dlamc3( dlamda( i ), dlamda( i ) ) - dlamda( i )
262  10 CONTINUE
263 *
264  DO 20 j = 1, k
265  CALL dlaed4( k, j, dlamda, w, q( 1, j ), rho, d( j ), info )
266 *
267 * If the zero finder fails, the computation is terminated.
268 *
269  IF( info.NE.0 )
270  \$ GO TO 120
271  20 CONTINUE
272 *
273  IF( k.EQ.1 )
274  \$ GO TO 110
275  IF( k.EQ.2 ) THEN
276  DO 30 j = 1, k
277  w( 1 ) = q( 1, j )
278  w( 2 ) = q( 2, j )
279  ii = indx( 1 )
280  q( 1, j ) = w( ii )
281  ii = indx( 2 )
282  q( 2, j ) = w( ii )
283  30 CONTINUE
284  GO TO 110
285  END IF
286 *
287 * Compute updated W.
288 *
289  CALL dcopy( k, w, 1, s, 1 )
290 *
291 * Initialize W(I) = Q(I,I)
292 *
293  CALL dcopy( k, q, ldq+1, w, 1 )
294  DO 60 j = 1, k
295  DO 40 i = 1, j - 1
296  w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
297  40 CONTINUE
298  DO 50 i = j + 1, k
299  w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
300  50 CONTINUE
301  60 CONTINUE
302  DO 70 i = 1, k
303  w( i ) = sign( sqrt( -w( i ) ), s( i ) )
304  70 CONTINUE
305 *
306 * Compute eigenvectors of the modified rank-1 modification.
307 *
308  DO 100 j = 1, k
309  DO 80 i = 1, k
310  s( i ) = w( i ) / q( i, j )
311  80 CONTINUE
312  temp = dnrm2( k, s, 1 )
313  DO 90 i = 1, k
314  ii = indx( i )
315  q( i, j ) = s( ii ) / temp
316  90 CONTINUE
317  100 CONTINUE
318 *
319 * Compute the updated eigenvectors.
320 *
321  110 CONTINUE
322 *
323  n2 = n - n1
324  n12 = ctot( 1 ) + ctot( 2 )
325  n23 = ctot( 2 ) + ctot( 3 )
326 *
327  CALL dlacpy( 'A', n23, k, q( ctot( 1 )+1, 1 ), ldq, s, n23 )
328  iq2 = n1*n12 + 1
329  IF( n23.NE.0 ) THEN
330  CALL dgemm( 'N', 'N', n2, k, n23, one, q2( iq2 ), n2, s, n23,
331  \$ zero, q( n1+1, 1 ), ldq )
332  ELSE
333  CALL dlaset( 'A', n2, k, zero, zero, q( n1+1, 1 ), ldq )
334  END IF
335 *
336  CALL dlacpy( 'A', n12, k, q, ldq, s, n12 )
337  IF( n12.NE.0 ) THEN
338  CALL dgemm( 'N', 'N', n1, k, n12, one, q2, n1, s, n12, zero, q,
339  \$ ldq )
340  ELSE
341  CALL dlaset( 'A', n1, k, zero, zero, q( 1, 1 ), ldq )
342  END IF
343 *
344 *
345  120 CONTINUE
346  RETURN
347 *
348 * End of DLAED3
349 *
double precision function dlamc3(A, B)
DLAMC3
Definition: dlamch.f:169
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
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
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition: dnrm2.f90:89
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