 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dlaed8()

 subroutine dlaed8 ( integer ICOMPQ, integer K, integer N, integer QSIZ, double precision, dimension( * ) D, double precision, dimension( ldq, * ) Q, integer LDQ, integer, dimension( * ) INDXQ, double precision RHO, integer CUTPNT, double precision, dimension( * ) Z, double precision, dimension( * ) DLAMDA, double precision, dimension( ldq2, * ) Q2, integer LDQ2, double precision, dimension( * ) W, integer, dimension( * ) PERM, integer GIVPTR, integer, dimension( 2, * ) GIVCOL, double precision, dimension( 2, * ) GIVNUM, integer, dimension( * ) INDXP, integer, dimension( * ) INDX, integer INFO )

DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:
``` DLAED8 merges the two sets of eigenvalues together into a single
sorted set.  Then it tries to deflate the size of the problem.
There are two ways in which deflation can occur:  when two or more
eigenvalues are close together or if there is a tiny element in the
Z vector.  For each such occurrence the order of the related secular
equation problem is reduced by one.```
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.``` [out] K ``` K is INTEGER The number of non-deflated eigenvalues, and the order of the related secular equation.``` [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,out] D ``` D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.``` [in,out] Q ``` Q is DOUBLE PRECISION array, dimension (LDQ,N) If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.``` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).``` [in] INDXQ ``` INDXQ is INTEGER array, dimension (N) The permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate.``` [in,out] RHO ``` RHO is DOUBLE PRECISION On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by DLAED3.``` [in] CUTPNT ``` CUTPNT is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.``` [in] Z ``` Z is DOUBLE PRECISION array, dimension (N) On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z are destroyed by the updating process.``` [out] DLAMDA ``` DLAMDA is DOUBLE PRECISION array, dimension (N) A copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation.``` [out] Q2 ``` Q2 is DOUBLE PRECISION array, dimension (LDQ2,N) If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors.``` [in] LDQ2 ``` LDQ2 is INTEGER The leading dimension of the array Q2. LDQ2 >= max(1,N).``` [out] W ``` W is DOUBLE PRECISION array, dimension (N) The first k values of the final deflation-altered z-vector and will be passed to DLAED3.``` [out] PERM ``` PERM is INTEGER array, dimension (N) The permutations (from deflation and sorting) to be applied to each eigenblock.``` [out] GIVPTR ``` GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem.``` [out] GIVCOL ``` GIVCOL is INTEGER array, dimension (2, N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.``` [out] GIVNUM ``` GIVNUM is DOUBLE PRECISION array, dimension (2, N) Each number indicates the S value to be used in the corresponding Givens rotation.``` [out] INDXP ``` INDXP is INTEGER array, dimension (N) The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.``` [out] INDX ``` INDX is INTEGER array, dimension (N) The permutation used to sort the contents of D into ascending order.``` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 240 of file dlaed8.f.

243 *
244 * -- LAPACK computational routine --
245 * -- LAPACK is a software package provided by Univ. of Tennessee, --
246 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
247 *
248 * .. Scalar Arguments ..
249  INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
250  \$ QSIZ
251  DOUBLE PRECISION RHO
252 * ..
253 * .. Array Arguments ..
254  INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
255  \$ INDXQ( * ), PERM( * )
256  DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ),
257  \$ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
258 * ..
259 *
260 * =====================================================================
261 *
262 * .. Parameters ..
263  DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
264  parameter( mone = -1.0d0, zero = 0.0d0, one = 1.0d0,
265  \$ two = 2.0d0, eight = 8.0d0 )
266 * ..
267 * .. Local Scalars ..
268 *
269  INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
270  DOUBLE PRECISION C, EPS, S, T, TAU, TOL
271 * ..
272 * .. External Functions ..
273  INTEGER IDAMAX
274  DOUBLE PRECISION DLAMCH, DLAPY2
275  EXTERNAL idamax, dlamch, dlapy2
276 * ..
277 * .. External Subroutines ..
278  EXTERNAL dcopy, dlacpy, dlamrg, drot, dscal, xerbla
279 * ..
280 * .. Intrinsic Functions ..
281  INTRINSIC abs, max, min, sqrt
282 * ..
283 * .. Executable Statements ..
284 *
285 * Test the input parameters.
286 *
287  info = 0
288 *
289  IF( icompq.LT.0 .OR. icompq.GT.1 ) THEN
290  info = -1
291  ELSE IF( n.LT.0 ) THEN
292  info = -3
293  ELSE IF( icompq.EQ.1 .AND. qsiz.LT.n ) THEN
294  info = -4
295  ELSE IF( ldq.LT.max( 1, n ) ) THEN
296  info = -7
297  ELSE IF( cutpnt.LT.min( 1, n ) .OR. cutpnt.GT.n ) THEN
298  info = -10
299  ELSE IF( ldq2.LT.max( 1, n ) ) THEN
300  info = -14
301  END IF
302  IF( info.NE.0 ) THEN
303  CALL xerbla( 'DLAED8', -info )
304  RETURN
305  END IF
306 *
307 * Need to initialize GIVPTR to O here in case of quick exit
308 * to prevent an unspecified code behavior (usually sigfault)
309 * when IWORK array on entry to *stedc is not zeroed
310 * (or at least some IWORK entries which used in *laed7 for GIVPTR).
311 *
312  givptr = 0
313 *
314 * Quick return if possible
315 *
316  IF( n.EQ.0 )
317  \$ RETURN
318 *
319  n1 = cutpnt
320  n2 = n - n1
321  n1p1 = n1 + 1
322 *
323  IF( rho.LT.zero ) THEN
324  CALL dscal( n2, mone, z( n1p1 ), 1 )
325  END IF
326 *
327 * Normalize z so that norm(z) = 1
328 *
329  t = one / sqrt( two )
330  DO 10 j = 1, n
331  indx( j ) = j
332  10 CONTINUE
333  CALL dscal( n, t, z, 1 )
334  rho = abs( two*rho )
335 *
336 * Sort the eigenvalues into increasing order
337 *
338  DO 20 i = cutpnt + 1, n
339  indxq( i ) = indxq( i ) + cutpnt
340  20 CONTINUE
341  DO 30 i = 1, n
342  dlamda( i ) = d( indxq( i ) )
343  w( i ) = z( indxq( i ) )
344  30 CONTINUE
345  i = 1
346  j = cutpnt + 1
347  CALL dlamrg( n1, n2, dlamda, 1, 1, indx )
348  DO 40 i = 1, n
349  d( i ) = dlamda( indx( i ) )
350  z( i ) = w( indx( i ) )
351  40 CONTINUE
352 *
353 * Calculate the allowable deflation tolerance
354 *
355  imax = idamax( n, z, 1 )
356  jmax = idamax( n, d, 1 )
357  eps = dlamch( 'Epsilon' )
358  tol = eight*eps*abs( d( jmax ) )
359 *
360 * If the rank-1 modifier is small enough, no more needs to be done
361 * except to reorganize Q so that its columns correspond with the
362 * elements in D.
363 *
364  IF( rho*abs( z( imax ) ).LE.tol ) THEN
365  k = 0
366  IF( icompq.EQ.0 ) THEN
367  DO 50 j = 1, n
368  perm( j ) = indxq( indx( j ) )
369  50 CONTINUE
370  ELSE
371  DO 60 j = 1, n
372  perm( j ) = indxq( indx( j ) )
373  CALL dcopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
374  60 CONTINUE
375  CALL dlacpy( 'A', qsiz, n, q2( 1, 1 ), ldq2, q( 1, 1 ),
376  \$ ldq )
377  END IF
378  RETURN
379  END IF
380 *
381 * If there are multiple eigenvalues then the problem deflates. Here
382 * the number of equal eigenvalues are found. As each equal
383 * eigenvalue is found, an elementary reflector is computed to rotate
384 * the corresponding eigensubspace so that the corresponding
385 * components of Z are zero in this new basis.
386 *
387  k = 0
388  k2 = n + 1
389  DO 70 j = 1, n
390  IF( rho*abs( z( j ) ).LE.tol ) THEN
391 *
392 * Deflate due to small z component.
393 *
394  k2 = k2 - 1
395  indxp( k2 ) = j
396  IF( j.EQ.n )
397  \$ GO TO 110
398  ELSE
399  jlam = j
400  GO TO 80
401  END IF
402  70 CONTINUE
403  80 CONTINUE
404  j = j + 1
405  IF( j.GT.n )
406  \$ GO TO 100
407  IF( rho*abs( z( j ) ).LE.tol ) THEN
408 *
409 * Deflate due to small z component.
410 *
411  k2 = k2 - 1
412  indxp( k2 ) = j
413  ELSE
414 *
415 * Check if eigenvalues are close enough to allow deflation.
416 *
417  s = z( jlam )
418  c = z( j )
419 *
420 * Find sqrt(a**2+b**2) without overflow or
421 * destructive underflow.
422 *
423  tau = dlapy2( c, s )
424  t = d( j ) - d( jlam )
425  c = c / tau
426  s = -s / tau
427  IF( abs( t*c*s ).LE.tol ) THEN
428 *
429 * Deflation is possible.
430 *
431  z( j ) = tau
432  z( jlam ) = zero
433 *
434 * Record the appropriate Givens rotation
435 *
436  givptr = givptr + 1
437  givcol( 1, givptr ) = indxq( indx( jlam ) )
438  givcol( 2, givptr ) = indxq( indx( j ) )
439  givnum( 1, givptr ) = c
440  givnum( 2, givptr ) = s
441  IF( icompq.EQ.1 ) THEN
442  CALL drot( qsiz, q( 1, indxq( indx( jlam ) ) ), 1,
443  \$ q( 1, indxq( indx( j ) ) ), 1, c, s )
444  END IF
445  t = d( jlam )*c*c + d( j )*s*s
446  d( j ) = d( jlam )*s*s + d( j )*c*c
447  d( jlam ) = t
448  k2 = k2 - 1
449  i = 1
450  90 CONTINUE
451  IF( k2+i.LE.n ) THEN
452  IF( d( jlam ).LT.d( indxp( k2+i ) ) ) THEN
453  indxp( k2+i-1 ) = indxp( k2+i )
454  indxp( k2+i ) = jlam
455  i = i + 1
456  GO TO 90
457  ELSE
458  indxp( k2+i-1 ) = jlam
459  END IF
460  ELSE
461  indxp( k2+i-1 ) = jlam
462  END IF
463  jlam = j
464  ELSE
465  k = k + 1
466  w( k ) = z( jlam )
467  dlamda( k ) = d( jlam )
468  indxp( k ) = jlam
469  jlam = j
470  END IF
471  END IF
472  GO TO 80
473  100 CONTINUE
474 *
475 * Record the last eigenvalue.
476 *
477  k = k + 1
478  w( k ) = z( jlam )
479  dlamda( k ) = d( jlam )
480  indxp( k ) = jlam
481 *
482  110 CONTINUE
483 *
484 * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
485 * and Q2 respectively. The eigenvalues/vectors which were not
486 * deflated go into the first K slots of DLAMDA and Q2 respectively,
487 * while those which were deflated go into the last N - K slots.
488 *
489  IF( icompq.EQ.0 ) THEN
490  DO 120 j = 1, n
491  jp = indxp( j )
492  dlamda( j ) = d( jp )
493  perm( j ) = indxq( indx( jp ) )
494  120 CONTINUE
495  ELSE
496  DO 130 j = 1, n
497  jp = indxp( j )
498  dlamda( j ) = d( jp )
499  perm( j ) = indxq( indx( jp ) )
500  CALL dcopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
501  130 CONTINUE
502  END IF
503 *
504 * The deflated eigenvalues and their corresponding vectors go back
505 * into the last N - K slots of D and Q respectively.
506 *
507  IF( k.LT.n ) THEN
508  IF( icompq.EQ.0 ) THEN
509  CALL dcopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
510  ELSE
511  CALL dcopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
512  CALL dlacpy( 'A', qsiz, n-k, q2( 1, k+1 ), ldq2,
513  \$ q( 1, k+1 ), ldq )
514  END IF
515  END IF
516 *
517  RETURN
518 *
519 * End of DLAED8
520 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
double precision function dlapy2(X, Y)
DLAPY2 returns sqrt(x2+y2).
Definition: dlapy2.f:63
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
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 dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:92
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
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