LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
subroutine claed7 ( integer  N,
integer  CUTPNT,
integer  QSIZ,
integer  TLVLS,
integer  CURLVL,
integer  CURPBM,
real, dimension( * )  D,
complex, dimension( ldq, * )  Q,
integer  LDQ,
real  RHO,
integer, dimension( * )  INDXQ,
real, dimension( * )  QSTORE,
integer, dimension( * )  QPTR,
integer, dimension( * )  PRMPTR,
integer, dimension( * )  PERM,
integer, dimension( * )  GIVPTR,
integer, dimension( 2, * )  GIVCOL,
real, dimension( 2, * )  GIVNUM,
complex, dimension( * )  WORK,
real, dimension( * )  RWORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

CLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Download CLAED7 + dependencies [TGZ] [ZIP] [TXT]

Purpose: 
 CLAED7 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 SLAED2.

       The second stage consists of calculating the updated
       eigenvalues. This is done by finding the roots of the secular
       equation via the routine SLAED4 (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 REAL 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 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 REAL
         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 REAL array,
                                 dimension (3*N+2*QSIZ*N)
[out]WORK
          WORK is COMPLEX array, dimension (QSIZ*N)
[in,out]QSTORE
          QSTORE is REAL 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 REAL 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.
Date
June 2016

Definition at line 251 of file claed7.f.

251 *
252 * -- LAPACK computational routine (version 3.6.1) --
253 * -- LAPACK is a software package provided by Univ. of Tennessee, --
254 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
255 * June 2016
256 *
257 * .. Scalar Arguments ..
258  INTEGER curlvl, curpbm, cutpnt, info, ldq, n, qsiz,
259  $ tlvls
260  REAL rho
261 * ..
262 * .. Array Arguments ..
263  INTEGER givcol( 2, * ), givptr( * ), indxq( * ),
264  $ iwork( * ), perm( * ), prmptr( * ), qptr( * )
265  REAL d( * ), givnum( 2, * ), qstore( * ), rwork( * )
266  COMPLEX q( ldq, * ), work( * )
267 * ..
268 *
269 * =====================================================================
270 *
271 * .. Local Scalars ..
272  INTEGER coltyp, curr, i, idlmda, indx,
273  $ indxc, indxp, iq, iw, iz, k, n1, n2, ptr
274 * ..
275 * .. External Subroutines ..
276  EXTERNAL clacrm, claed8, slaed9, slaeda, slamrg, xerbla
277 * ..
278 * .. Intrinsic Functions ..
279  INTRINSIC max, min
280 * ..
281 * .. Executable Statements ..
282 *
283 * Test the input parameters.
284 *
285  info = 0
286 *
287 * IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
288 * INFO = -1
289 * ELSE IF( N.LT.0 ) THEN
290  IF( n.LT.0 ) THEN
291  info = -1
292  ELSE IF( min( 1, n ).GT.cutpnt .OR. n.LT.cutpnt ) THEN
293  info = -2
294  ELSE IF( qsiz.LT.n ) THEN
295  info = -3
296  ELSE IF( ldq.LT.max( 1, n ) ) THEN
297  info = -9
298  END IF
299  IF( info.NE.0 ) THEN
300  CALL xerbla( 'CLAED7', -info )
301  RETURN
302  END IF
303 *
304 * Quick return if possible
305 *
306  IF( n.EQ.0 )
307  $ RETURN
308 *
309 * The following values are for bookkeeping purposes only. They are
310 * integer pointers which indicate the portion of the workspace
311 * used by a particular array in SLAED2 and SLAED3.
312 *
313  iz = 1
314  idlmda = iz + n
315  iw = idlmda + n
316  iq = iw + n
317 *
318  indx = 1
319  indxc = indx + n
320  coltyp = indxc + n
321  indxp = coltyp + n
322 *
323 * Form the z-vector which consists of the last row of Q_1 and the
324 * first row of Q_2.
325 *
326  ptr = 1 + 2**tlvls
327  DO 10 i = 1, curlvl - 1
328  ptr = ptr + 2**( tlvls-i )
329  10 CONTINUE
330  curr = ptr + curpbm
331  CALL slaeda( n, tlvls, curlvl, curpbm, prmptr, perm, givptr,
332  $ givcol, givnum, qstore, qptr, rwork( iz ),
333  $ rwork( iz+n ), info )
334 *
335 * When solving the final problem, we no longer need the stored data,
336 * so we will overwrite the data from this level onto the previously
337 * used storage space.
338 *
339  IF( curlvl.EQ.tlvls ) THEN
340  qptr( curr ) = 1
341  prmptr( curr ) = 1
342  givptr( curr ) = 1
343  END IF
344 *
345 * Sort and Deflate eigenvalues.
346 *
347  CALL claed8( k, n, qsiz, q, ldq, d, rho, cutpnt, rwork( iz ),
348  $ rwork( idlmda ), work, qsiz, rwork( iw ),
349  $ iwork( indxp ), iwork( indx ), indxq,
350  $ perm( prmptr( curr ) ), givptr( curr+1 ),
351  $ givcol( 1, givptr( curr ) ),
352  $ givnum( 1, givptr( curr ) ), info )
353  prmptr( curr+1 ) = prmptr( curr ) + n
354  givptr( curr+1 ) = givptr( curr+1 ) + givptr( curr )
355 *
356 * Solve Secular Equation.
357 *
358  IF( k.NE.0 ) THEN
359  CALL slaed9( k, 1, k, n, d, rwork( iq ), k, rho,
360  $ rwork( idlmda ), rwork( iw ),
361  $ qstore( qptr( curr ) ), k, info )
362  CALL clacrm( qsiz, k, work, qsiz, qstore( qptr( curr ) ), k, q,
363  $ ldq, rwork( iq ) )
364  qptr( curr+1 ) = qptr( curr ) + k**2
365  IF( info.NE.0 ) THEN
366  RETURN
367  END IF
368 *
369 * Prepare the INDXQ sorting premutation.
370 *
371  n1 = k
372  n2 = n - k
373  CALL slamrg( n1, n2, d, 1, -1, indxq )
374  ELSE
375  qptr( curr+1 ) = qptr( curr )
376  DO 20 i = 1, n
377  indxq( i ) = i
378  20 CONTINUE
379  END IF
380 *
381  RETURN
382 *
383 * End of CLAED7
384 *
slaeda
subroutine slaeda(N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO)
SLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal ma...
Definition: slaeda.f:168
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
claed8
subroutine claed8(K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO)
CLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: claed8.f:230
slaed9
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. Used when the original matrix is dense.
Definition: slaed9.f:158
slamrg
subroutine slamrg(N1, N2, A, STRD1, STRD2, INDEX)
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition: slamrg.f:101
clacrm
subroutine clacrm(M, N, A, LDA, B, LDB, C, LDC, RWORK)
CLACRM multiplies a complex matrix by a square real matrix.
Definition: clacrm.f:116

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