LAPACK  3.9.0 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.

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```
Date
June 2016
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 262 of file slaed7.f.

262 *
263 * -- LAPACK computational routine (version 3.7.0) --
264 * -- LAPACK is a software package provided by Univ. of Tennessee, --
265 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266 * June 2016
267 *
268 * .. Scalar Arguments ..
269  INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
270  \$ QSIZ, TLVLS
271  REAL RHO
272 * ..
273 * .. Array Arguments ..
274  INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
275  \$ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
276  REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
277  \$ QSTORE( * ), WORK( * )
278 * ..
279 *
280 * =====================================================================
281 *
282 * .. Parameters ..
283  REAL ONE, ZERO
284  parameter( one = 1.0e0, zero = 0.0e0 )
285 * ..
286 * .. Local Scalars ..
287  INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
288  \$ IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
289 * ..
290 * .. External Subroutines ..
291  EXTERNAL sgemm, slaed8, slaed9, slaeda, slamrg, xerbla
292 * ..
293 * .. Intrinsic Functions ..
294  INTRINSIC max, min
295 * ..
296 * .. Executable Statements ..
297 *
298 * Test the input parameters.
299 *
300  info = 0
301 *
302  IF( icompq.LT.0 .OR. icompq.GT.1 ) THEN
303  info = -1
304  ELSE IF( n.LT.0 ) THEN
305  info = -2
306  ELSE IF( icompq.EQ.1 .AND. qsiz.LT.n ) THEN
307  info = -3
308  ELSE IF( ldq.LT.max( 1, n ) ) THEN
309  info = -9
310  ELSE IF( min( 1, n ).GT.cutpnt .OR. n.LT.cutpnt ) THEN
311  info = -12
312  END IF
313  IF( info.NE.0 ) THEN
314  CALL xerbla( 'SLAED7', -info )
315  RETURN
316  END IF
317 *
318 * Quick return if possible
319 *
320  IF( n.EQ.0 )
321  \$ RETURN
322 *
323 * The following values are for bookkeeping purposes only. They are
324 * integer pointers which indicate the portion of the workspace
325 * used by a particular array in SLAED8 and SLAED9.
326 *
327  IF( icompq.EQ.1 ) THEN
328  ldq2 = qsiz
329  ELSE
330  ldq2 = n
331  END IF
332 *
333  iz = 1
334  idlmda = iz + n
335  iw = idlmda + n
336  iq2 = iw + n
337  is = iq2 + n*ldq2
338 *
339  indx = 1
340  indxc = indx + n
341  coltyp = indxc + n
342  indxp = coltyp + n
343 *
344 * Form the z-vector which consists of the last row of Q_1 and the
345 * first row of Q_2.
346 *
347  ptr = 1 + 2**tlvls
348  DO 10 i = 1, curlvl - 1
349  ptr = ptr + 2**( tlvls-i )
350  10 CONTINUE
351  curr = ptr + curpbm
352  CALL slaeda( n, tlvls, curlvl, curpbm, prmptr, perm, givptr,
353  \$ givcol, givnum, qstore, qptr, work( iz ),
354  \$ work( iz+n ), info )
355 *
356 * When solving the final problem, we no longer need the stored data,
357 * so we will overwrite the data from this level onto the previously
358 * used storage space.
359 *
360  IF( curlvl.EQ.tlvls ) THEN
361  qptr( curr ) = 1
362  prmptr( curr ) = 1
363  givptr( curr ) = 1
364  END IF
365 *
366 * Sort and Deflate eigenvalues.
367 *
368  CALL slaed8( icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt,
369  \$ work( iz ), work( idlmda ), work( iq2 ), ldq2,
370  \$ work( iw ), perm( prmptr( curr ) ), givptr( curr+1 ),
371  \$ givcol( 1, givptr( curr ) ),
372  \$ givnum( 1, givptr( curr ) ), iwork( indxp ),
373  \$ iwork( indx ), info )
374  prmptr( curr+1 ) = prmptr( curr ) + n
375  givptr( curr+1 ) = givptr( curr+1 ) + givptr( curr )
376 *
377 * Solve Secular Equation.
378 *
379  IF( k.NE.0 ) THEN
380  CALL slaed9( k, 1, k, n, d, work( is ), k, rho, work( idlmda ),
381  \$ work( iw ), qstore( qptr( curr ) ), k, info )
382  IF( info.NE.0 )
383  \$ GO TO 30
384  IF( icompq.EQ.1 ) THEN
385  CALL sgemm( 'N', 'N', qsiz, k, k, one, work( iq2 ), ldq2,
386  \$ qstore( qptr( curr ) ), k, zero, q, ldq )
387  END IF
388  qptr( curr+1 ) = qptr( curr ) + k**2
389 *
390 * Prepare the INDXQ sorting permutation.
391 *
392  n1 = k
393  n2 = n - k
394  CALL slamrg( n1, n2, d, 1, -1, indxq )
395  ELSE
396  qptr( curr+1 ) = qptr( curr )
397  DO 20 i = 1, n
398  indxq( i ) = i
399  20 CONTINUE
400  END IF
401 *
402  30 CONTINUE
403  RETURN
404 *
405 * End of SLAED7
406 *
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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....
Definition: slaed9.f:158
slaed8
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:245
sgemm
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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
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