LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ dlaed7()

subroutine dlaed7 ( integer  ICOMPQ,
integer  N,
integer  QSIZ,
integer  TLVLS,
integer  CURLVL,
integer  CURPBM,
double precision, dimension( * )  D,
double precision, dimension( ldq, * )  Q,
integer  LDQ,
integer, dimension( * )  INDXQ,
double precision  RHO,
integer  CUTPNT,
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,
double precision, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Download DLAED7 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DLAED7 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.  DLAED1 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 DLAED8.

       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 DLAED9).
       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 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 DOUBLE PRECISION 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 DOUBLE PRECISION
         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 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]WORK
          WORK is DOUBLE PRECISION 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 dlaed7.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 DOUBLE PRECISION RHO
269* ..
270* .. Array Arguments ..
271 INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
272 $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
273 DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
274 $ QSTORE( * ), WORK( * )
275* ..
276*
277* =====================================================================
278*
279* .. Parameters ..
280 DOUBLE PRECISION ONE, ZERO
281 parameter( one = 1.0d0, zero = 0.0d0 )
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 dgemm, dlaed8, dlaed9, dlaeda, dlamrg, 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( 'DLAED7', -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 DLAED8 and DLAED9.
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 dlaeda( 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 dlaed8( 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 dlaed9( 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 dgemm( '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 dlamrg( 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 DLAED7
403*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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
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
subroutine dlaed9(K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO)
DLAED9 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors....
Definition: dlaed9.f:156
subroutine dlaed8(ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL, GIVNUM, INDXP, INDX, INFO)
DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: dlaed8.f:243
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
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