LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ slaed7()

subroutine slaed7 ( integer  ICOMPQ,
integer  N,
integer  QSIZ,
integer  TLVLS,
integer  CURLVL,
integer  CURPBM,
real, dimension( * )  D,
real, dimension( ldq, * )  Q,
integer  LDQ,
integer, dimension( * )  INDXQ,
real  RHO,
integer  CUTPNT,
real, dimension( * )  QSTORE,
integer, dimension( * )  QPTR,
integer, dimension( * )  PRMPTR,
integer, dimension( * )  PERM,
integer, dimension( * )  GIVPTR,
integer, dimension( 2, * )  GIVCOL,
real, dimension( 2, * )  GIVNUM,
real, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

SLAED7 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 SLAED7 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SLAED7 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 symmetric matrix
 that has been reduced to tridiagonal form.  SLAED1 handles
 the case in which all eigenvalues and eigenvectors of a symmetric
 tridiagonal matrix are desired.

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

    where Z = Q**Tu, 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 SLAED8.

       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 SLAED9).
       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]ICOMPQ
          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.
[in]N
          N is INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.
[in]QSIZ
          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
[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 REAL 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).
[out]INDXQ
          INDXQ is INTEGER array, dimension (N)
         The permutation which will reintegrate the subproblem just
         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
         will be in ascending order.
[in]RHO
          RHO is REAL
         The subdiagonal element used to create the rank-1
         modification.
[in]CUTPNT
          CUTPNT is INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= 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]WORK
          WORK is REAL array, dimension (3*N+2*QSIZ*N)
[out]IWORK
          IWORK is INTEGER array, dimension (4*N)
[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.
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 256 of file slaed7.f.

260 *
261 * -- LAPACK computational routine --
262 * -- LAPACK is a software package provided by Univ. of Tennessee, --
263 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
264 *
265 * .. Scalar Arguments ..
266  INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
267  $ QSIZ, TLVLS
268  REAL RHO
269 * ..
270 * .. Array Arguments ..
271  INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
272  $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
273  REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
274  $ QSTORE( * ), WORK( * )
275 * ..
276 *
277 * =====================================================================
278 *
279 * .. Parameters ..
280  REAL ONE, ZERO
281  parameter( one = 1.0e0, zero = 0.0e0 )
282 * ..
283 * .. Local Scalars ..
284  INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
285  $ IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
286 * ..
287 * .. External Subroutines ..
288  EXTERNAL sgemm, slaed8, slaed9, slaeda, slamrg, xerbla
289 * ..
290 * .. Intrinsic Functions ..
291  INTRINSIC max, min
292 * ..
293 * .. Executable Statements ..
294 *
295 * Test the input parameters.
296 *
297  info = 0
298 *
299  IF( icompq.LT.0 .OR. icompq.GT.1 ) THEN
300  info = -1
301  ELSE IF( n.LT.0 ) THEN
302  info = -2
303  ELSE IF( icompq.EQ.1 .AND. qsiz.LT.n ) THEN
304  info = -3
305  ELSE IF( ldq.LT.max( 1, n ) ) THEN
306  info = -9
307  ELSE IF( min( 1, n ).GT.cutpnt .OR. n.LT.cutpnt ) THEN
308  info = -12
309  END IF
310  IF( info.NE.0 ) THEN
311  CALL xerbla( 'SLAED7', -info )
312  RETURN
313  END IF
314 *
315 * Quick return if possible
316 *
317  IF( n.EQ.0 )
318  $ RETURN
319 *
320 * The following values are for bookkeeping purposes only. They are
321 * integer pointers which indicate the portion of the workspace
322 * used by a particular array in SLAED8 and SLAED9.
323 *
324  IF( icompq.EQ.1 ) THEN
325  ldq2 = qsiz
326  ELSE
327  ldq2 = n
328  END IF
329 *
330  iz = 1
331  idlmda = iz + n
332  iw = idlmda + n
333  iq2 = iw + n
334  is = iq2 + n*ldq2
335 *
336  indx = 1
337  indxc = indx + n
338  coltyp = indxc + n
339  indxp = coltyp + n
340 *
341 * Form the z-vector which consists of the last row of Q_1 and the
342 * first row of Q_2.
343 *
344  ptr = 1 + 2**tlvls
345  DO 10 i = 1, curlvl - 1
346  ptr = ptr + 2**( tlvls-i )
347  10 CONTINUE
348  curr = ptr + curpbm
349  CALL slaeda( n, tlvls, curlvl, curpbm, prmptr, perm, givptr,
350  $ givcol, givnum, qstore, qptr, work( iz ),
351  $ work( iz+n ), info )
352 *
353 * When solving the final problem, we no longer need the stored data,
354 * so we will overwrite the data from this level onto the previously
355 * used storage space.
356 *
357  IF( curlvl.EQ.tlvls ) THEN
358  qptr( curr ) = 1
359  prmptr( curr ) = 1
360  givptr( curr ) = 1
361  END IF
362 *
363 * Sort and Deflate eigenvalues.
364 *
365  CALL slaed8( icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt,
366  $ work( iz ), work( idlmda ), work( iq2 ), ldq2,
367  $ work( iw ), perm( prmptr( curr ) ), givptr( curr+1 ),
368  $ givcol( 1, givptr( curr ) ),
369  $ givnum( 1, givptr( curr ) ), iwork( indxp ),
370  $ iwork( indx ), info )
371  prmptr( curr+1 ) = prmptr( curr ) + n
372  givptr( curr+1 ) = givptr( curr+1 ) + givptr( curr )
373 *
374 * Solve Secular Equation.
375 *
376  IF( k.NE.0 ) THEN
377  CALL slaed9( k, 1, k, n, d, work( is ), k, rho, work( idlmda ),
378  $ work( iw ), qstore( qptr( curr ) ), k, info )
379  IF( info.NE.0 )
380  $ GO TO 30
381  IF( icompq.EQ.1 ) THEN
382  CALL sgemm( 'N', 'N', qsiz, k, k, one, work( iq2 ), ldq2,
383  $ qstore( qptr( curr ) ), k, zero, q, ldq )
384  END IF
385  qptr( curr+1 ) = qptr( curr ) + k**2
386 *
387 * Prepare the INDXQ sorting permutation.
388 *
389  n1 = k
390  n2 = n - k
391  CALL slamrg( n1, n2, d, 1, -1, indxq )
392  ELSE
393  qptr( curr+1 ) = qptr( curr )
394  DO 20 i = 1, n
395  indxq( i ) = i
396  20 CONTINUE
397  END IF
398 *
399  30 CONTINUE
400  RETURN
401 *
402 * End of SLAED7
403 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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:99
subroutine slaed8(ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL, GIVNUM, INDXP, INDX, INFO)
SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: slaed8.f:243
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....
Definition: slaed9.f:156
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:166
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
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