LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ zlaed7()

subroutine zlaed7 ( integer  n,
integer  cutpnt,
integer  qsiz,
integer  tlvls,
integer  curlvl,
integer  curpbm,
double precision, dimension( * )  d,
complex*16, dimension( ldq, * )  q,
integer  ldq,
double precision  rho,
integer, dimension( * )  indxq,
double precision, dimension( * )  qstore,
integer, dimension( * )  qptr,
integer, dimension( * )  prmptr,
integer, dimension( * )  perm,
integer, dimension( * )  givptr,
integer, dimension( 2, * )  givcol,
double precision, dimension( 2, * )  givnum,
complex*16, dimension( * )  work,
double precision, dimension( * )  rwork,
integer, dimension( * )  iwork,
integer  info 
)

ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Download ZLAED7 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 ZLAED7 computes the updated eigensystem of a diagonal
 matrix after modification by a rank-one symmetric matrix. This
 routine is used only for the eigenproblem which requires all
 eigenvalues and optionally eigenvectors of a dense or banded
 Hermitian matrix that has been reduced to tridiagonal form.

   T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)

   where Z = Q**Hu, u is a vector of length N with ones in the
   CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

    The eigenvectors of the original matrix are stored in Q, and the
    eigenvalues are in D.  The algorithm consists of three stages:

       The first stage consists of deflating the size of the problem
       when there are multiple eigenvalues or if there is a zero in
       the Z vector.  For each such occurrence the dimension of the
       secular equation problem is reduced by one.  This stage is
       performed by the routine DLAED2.

       The second stage consists of calculating the updated
       eigenvalues. This is done by finding the roots of the secular
       equation via the routine DLAED4 (as called by SLAED3).
       This routine also calculates the eigenvectors of the current
       problem.

       The final stage consists of computing the updated eigenvectors
       directly using the updated eigenvalues.  The eigenvectors for
       the current problem are multiplied with the eigenvectors from
       the overall problem.
Parameters
[in]N
          N is INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.
[in]CUTPNT
          CUTPNT is INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.
[in]QSIZ
          QSIZ is INTEGER
         The dimension of the unitary matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N.
[in]TLVLS
          TLVLS is INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.
[in]CURLVL
          CURLVL is INTEGER
         The current level in the overall merge routine,
         0 <= curlvl <= tlvls.
[in]CURPBM
          CURPBM is INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).
[in,out]D
          D is DOUBLE PRECISION array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.
[in,out]Q
          Q is COMPLEX*16 array, dimension (LDQ,N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.
[in]LDQ
          LDQ is INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).
[in]RHO
          RHO is DOUBLE PRECISION
         Contains the subdiagonal element used to create the rank-1
         modification.
[out]INDXQ
          INDXQ is INTEGER array, dimension (N)
         This contains the permutation which will reintegrate the
         subproblem just solved back into sorted order,
         ie. D( INDXQ( I = 1, N ) ) will be in ascending order.
[out]IWORK
          IWORK is INTEGER array, dimension (4*N)
[out]RWORK
          RWORK is DOUBLE PRECISION array,
                                 dimension (3*N+2*QSIZ*N)
[out]WORK
          WORK is COMPLEX*16 array, dimension (QSIZ*N)
[in,out]QSTORE
          QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
         Stores eigenvectors of submatrices encountered during
         divide and conquer, packed together. QPTR points to
         beginning of the submatrices.
[in,out]QPTR
          QPTR is INTEGER array, dimension (N+2)
         List of indices pointing to beginning of submatrices stored
         in QSTORE. The submatrices are numbered starting at the
         bottom left of the divide and conquer tree, from left to
         right and bottom to top.
[in]PRMPTR
          PRMPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and also the size of
         the full, non-deflated problem.
[in]PERM
          PERM is INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.
[in]GIVPTR
          GIVPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
         indicates the number of Givens rotations.
[in]GIVCOL
          GIVCOL is INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.
[in]GIVNUM
          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.
[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
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 245 of file zlaed7.f.

249*
250* -- LAPACK computational routine --
251* -- LAPACK is a software package provided by Univ. of Tennessee, --
252* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
253*
254* .. Scalar Arguments ..
255 INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ,
256 $ TLVLS
257 DOUBLE PRECISION RHO
258* ..
259* .. Array Arguments ..
260 INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
261 $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
262 DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
263 COMPLEX*16 Q( LDQ, * ), WORK( * )
264* ..
265*
266* =====================================================================
267*
268* .. Local Scalars ..
269 INTEGER COLTYP, CURR, I, IDLMDA, INDX,
270 $ INDXC, INDXP, IQ, IW, IZ, K, N1, N2, PTR
271* ..
272* .. External Subroutines ..
273 EXTERNAL dlaed9, dlaeda, dlamrg, xerbla, zlacrm, zlaed8
274* ..
275* .. Intrinsic Functions ..
276 INTRINSIC max, min
277* ..
278* .. Executable Statements ..
279*
280* Test the input parameters.
281*
282 info = 0
283*
284* IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
285* INFO = -1
286* ELSE IF( N.LT.0 ) THEN
287 IF( n.LT.0 ) THEN
288 info = -1
289 ELSE IF( min( 1, n ).GT.cutpnt .OR. n.LT.cutpnt ) THEN
290 info = -2
291 ELSE IF( qsiz.LT.n ) THEN
292 info = -3
293 ELSE IF( ldq.LT.max( 1, n ) ) THEN
294 info = -9
295 END IF
296 IF( info.NE.0 ) THEN
297 CALL xerbla( 'ZLAED7', -info )
298 RETURN
299 END IF
300*
301* Quick return if possible
302*
303 IF( n.EQ.0 )
304 $ RETURN
305*
306* The following values are for bookkeeping purposes only. They are
307* integer pointers which indicate the portion of the workspace
308* used by a particular array in DLAED2 and SLAED3.
309*
310 iz = 1
311 idlmda = iz + n
312 iw = idlmda + n
313 iq = iw + n
314*
315 indx = 1
316 indxc = indx + n
317 coltyp = indxc + n
318 indxp = coltyp + n
319*
320* Form the z-vector which consists of the last row of Q_1 and the
321* first row of Q_2.
322*
323 ptr = 1 + 2**tlvls
324 DO 10 i = 1, curlvl - 1
325 ptr = ptr + 2**( tlvls-i )
326 10 CONTINUE
327 curr = ptr + curpbm
328 CALL dlaeda( n, tlvls, curlvl, curpbm, prmptr, perm, givptr,
329 $ givcol, givnum, qstore, qptr, rwork( iz ),
330 $ rwork( iz+n ), info )
331*
332* When solving the final problem, we no longer need the stored data,
333* so we will overwrite the data from this level onto the previously
334* used storage space.
335*
336 IF( curlvl.EQ.tlvls ) THEN
337 qptr( curr ) = 1
338 prmptr( curr ) = 1
339 givptr( curr ) = 1
340 END IF
341*
342* Sort and Deflate eigenvalues.
343*
344 CALL zlaed8( k, n, qsiz, q, ldq, d, rho, cutpnt, rwork( iz ),
345 $ rwork( idlmda ), work, qsiz, rwork( iw ),
346 $ iwork( indxp ), iwork( indx ), indxq,
347 $ perm( prmptr( curr ) ), givptr( curr+1 ),
348 $ givcol( 1, givptr( curr ) ),
349 $ givnum( 1, givptr( curr ) ), info )
350 prmptr( curr+1 ) = prmptr( curr ) + n
351 givptr( curr+1 ) = givptr( curr+1 ) + givptr( curr )
352*
353* Solve Secular Equation.
354*
355 IF( k.NE.0 ) THEN
356 CALL dlaed9( k, 1, k, n, d, rwork( iq ), k, rho,
357 $ rwork( idlmda ), rwork( iw ),
358 $ qstore( qptr( curr ) ), k, info )
359 CALL zlacrm( qsiz, k, work, qsiz, qstore( qptr( curr ) ), k, q,
360 $ ldq, rwork( iq ) )
361 qptr( curr+1 ) = qptr( curr ) + k**2
362 IF( info.NE.0 ) THEN
363 RETURN
364 END IF
365*
366* Prepare the INDXQ sorting permutation.
367*
368 n1 = k
369 n2 = n - k
370 CALL dlamrg( n1, n2, d, 1, -1, indxq )
371 ELSE
372 qptr( curr+1 ) = qptr( curr )
373 DO 20 i = 1, n
374 indxq( i ) = i
375 20 CONTINUE
376 END IF
377*
378 RETURN
379*
380* End of ZLAED7
381*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
zlacrm
subroutine zlacrm(m, n, a, lda, b, ldb, c, ldc, rwork)
ZLACRM multiplies a complex matrix by a square real matrix.
Definition zlacrm.f:114
zlaed8
subroutine zlaed8(k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info)
ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition zlaed8.f:228
dlaed9
subroutine dlaed9(k, kstart, kstop, n, d, q, ldq, rho, dlambda, w, s, lds, info)
DLAED9 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors....
Definition dlaed9.f:156
dlaeda
subroutine dlaeda(n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info)
DLAEDA used by DSTEDC. Computes the Z vector determining the rank-one modification of the diagonal ma...
Definition dlaeda.f:166
dlamrg
subroutine dlamrg(n1, n2, a, dtrd1, dtrd2, index)
DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition dlamrg.f:99
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