LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

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

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
Date
June 2016

Definition at line 251 of file zlaed7.f.

251 *
252 * -- LAPACK computational routine (version 3.7.0) --
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  DOUBLE PRECISION rho
261 * ..
262 * .. Array Arguments ..
263  INTEGER givcol( 2, * ), givptr( * ), indxq( * ),
264  \$ iwork( * ), perm( * ), prmptr( * ), qptr( * )
265  DOUBLE PRECISION d( * ), givnum( 2, * ), qstore( * ), rwork( * )
266  COMPLEX*16 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 dlaed9, dlaeda, dlamrg, xerbla, zlacrm, zlaed8
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( 'ZLAED7', -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 DLAED2 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 dlaeda( 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 zlaed8( 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 dlaed9( k, 1, k, n, d, rwork( iq ), k, rho,
360  \$ rwork( idlmda ), rwork( iw ),
361  \$ qstore( qptr( curr ) ), k, info )
362  CALL zlacrm( 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 dlamrg( 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 ZLAED7
384 *
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:101
subroutine dlaed9(K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO)
DLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
Definition: dlaed9.f:158
subroutine dlaeda(N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO)
DLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal ma...
Definition: dlaeda.f:168
subroutine zlacrm(M, N, A, LDA, B, LDB, C, LDC, RWORK)
ZLACRM multiplies a complex matrix by a square real matrix.
Definition: zlacrm.f:116
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zlaed8(K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO)
ZLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: zlaed8.f:230
Here is the call graph for this function:
Here is the caller graph for this function: