 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zlaed8()

 subroutine zlaed8 ( integer K, integer N, integer QSIZ, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) D, double precision RHO, integer CUTPNT, double precision, dimension( * ) Z, double precision, dimension( * ) DLAMDA, complex*16, dimension( ldq2, * ) Q2, integer LDQ2, double precision, dimension( * ) W, integer, dimension( * ) INDXP, integer, dimension( * ) INDX, integer, dimension( * ) INDXQ, integer, dimension( * ) PERM, integer GIVPTR, integer, dimension( 2, * ) GIVCOL, double precision, dimension( 2, * ) GIVNUM, integer INFO )

ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:
``` ZLAED8 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
 [out] K ``` K is INTEGER Contains the number of non-deflated eigenvalues. This is 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 unitary matrix used to reduce the dense or band matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.``` [in,out] Q ``` Q is COMPLEX*16 array, dimension (LDQ,N) 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,out] D ``` D is DOUBLE PRECISION array, dimension (N) On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.``` [in,out] RHO ``` RHO is DOUBLE PRECISION Contains the off diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. RHO is modified during the computation to the value required by DLAED3.``` [in] CUTPNT ``` CUTPNT is INTEGER Contains 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 input this vector contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). The contents of Z are destroyed during the updating process.``` [out] DLAMDA ``` DLAMDA is DOUBLE PRECISION array, dimension (N) Contains a copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation.``` [out] Q2 ``` Q2 is COMPLEX*16 array, dimension (LDQ2,N) If ICOMPQ = 0, Q2 is not referenced. Otherwise, Contains 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) This will hold the first k values of the final deflation-altered z-vector and will be passed to DLAED3.``` [out] INDXP ``` INDXP is INTEGER array, dimension (N) This will contain the permutation used to place deflated values of D at the end of the array. On output 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) This will contain the permutation used to sort the contents of D into ascending order.``` [in] INDXQ ``` INDXQ is INTEGER array, dimension (N) This contains 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.``` [out] PERM ``` PERM is INTEGER array, dimension (N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock.``` [out] GIVPTR ``` GIVPTR is INTEGER Contains 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] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```

Definition at line 225 of file zlaed8.f.

228 *
229 * -- LAPACK computational routine --
230 * -- LAPACK is a software package provided by Univ. of Tennessee, --
231 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232 *
233 * .. Scalar Arguments ..
234  INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
235  DOUBLE PRECISION RHO
236 * ..
237 * .. Array Arguments ..
238  INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
239  \$ INDXQ( * ), PERM( * )
240  DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
241  \$ Z( * )
242  COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * )
243 * ..
244 *
245 * =====================================================================
246 *
247 * .. Parameters ..
248  DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
249  parameter( mone = -1.0d0, zero = 0.0d0, one = 1.0d0,
250  \$ two = 2.0d0, eight = 8.0d0 )
251 * ..
252 * .. Local Scalars ..
253  INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
254  DOUBLE PRECISION C, EPS, S, T, TAU, TOL
255 * ..
256 * .. External Functions ..
257  INTEGER IDAMAX
258  DOUBLE PRECISION DLAMCH, DLAPY2
259  EXTERNAL idamax, dlamch, dlapy2
260 * ..
261 * .. External Subroutines ..
262  EXTERNAL dcopy, dlamrg, dscal, xerbla, zcopy, zdrot,
263  \$ zlacpy
264 * ..
265 * .. Intrinsic Functions ..
266  INTRINSIC abs, max, min, sqrt
267 * ..
268 * .. Executable Statements ..
269 *
270 * Test the input parameters.
271 *
272  info = 0
273 *
274  IF( n.LT.0 ) THEN
275  info = -2
276  ELSE IF( qsiz.LT.n ) THEN
277  info = -3
278  ELSE IF( ldq.LT.max( 1, n ) ) THEN
279  info = -5
280  ELSE IF( cutpnt.LT.min( 1, n ) .OR. cutpnt.GT.n ) THEN
281  info = -8
282  ELSE IF( ldq2.LT.max( 1, n ) ) THEN
283  info = -12
284  END IF
285  IF( info.NE.0 ) THEN
286  CALL xerbla( 'ZLAED8', -info )
287  RETURN
288  END IF
289 *
290 * Need to initialize GIVPTR to O here in case of quick exit
291 * to prevent an unspecified code behavior (usually sigfault)
292 * when IWORK array on entry to *stedc is not zeroed
293 * (or at least some IWORK entries which used in *laed7 for GIVPTR).
294 *
295  givptr = 0
296 *
297 * Quick return if possible
298 *
299  IF( n.EQ.0 )
300  \$ RETURN
301 *
302  n1 = cutpnt
303  n2 = n - n1
304  n1p1 = n1 + 1
305 *
306  IF( rho.LT.zero ) THEN
307  CALL dscal( n2, mone, z( n1p1 ), 1 )
308  END IF
309 *
310 * Normalize z so that norm(z) = 1
311 *
312  t = one / sqrt( two )
313  DO 10 j = 1, n
314  indx( j ) = j
315  10 CONTINUE
316  CALL dscal( n, t, z, 1 )
317  rho = abs( two*rho )
318 *
319 * Sort the eigenvalues into increasing order
320 *
321  DO 20 i = cutpnt + 1, n
322  indxq( i ) = indxq( i ) + cutpnt
323  20 CONTINUE
324  DO 30 i = 1, n
325  dlamda( i ) = d( indxq( i ) )
326  w( i ) = z( indxq( i ) )
327  30 CONTINUE
328  i = 1
329  j = cutpnt + 1
330  CALL dlamrg( n1, n2, dlamda, 1, 1, indx )
331  DO 40 i = 1, n
332  d( i ) = dlamda( indx( i ) )
333  z( i ) = w( indx( i ) )
334  40 CONTINUE
335 *
336 * Calculate the allowable deflation tolerance
337 *
338  imax = idamax( n, z, 1 )
339  jmax = idamax( n, d, 1 )
340  eps = dlamch( 'Epsilon' )
341  tol = eight*eps*abs( d( jmax ) )
342 *
343 * If the rank-1 modifier is small enough, no more needs to be done
344 * -- except to reorganize Q so that its columns correspond with the
345 * elements in D.
346 *
347  IF( rho*abs( z( imax ) ).LE.tol ) THEN
348  k = 0
349  DO 50 j = 1, n
350  perm( j ) = indxq( indx( j ) )
351  CALL zcopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
352  50 CONTINUE
353  CALL zlacpy( 'A', qsiz, n, q2( 1, 1 ), ldq2, q( 1, 1 ), ldq )
354  RETURN
355  END IF
356 *
357 * If there are multiple eigenvalues then the problem deflates. Here
358 * the number of equal eigenvalues are found. As each equal
359 * eigenvalue is found, an elementary reflector is computed to rotate
360 * the corresponding eigensubspace so that the corresponding
361 * components of Z are zero in this new basis.
362 *
363  k = 0
364  k2 = n + 1
365  DO 60 j = 1, n
366  IF( rho*abs( z( j ) ).LE.tol ) THEN
367 *
368 * Deflate due to small z component.
369 *
370  k2 = k2 - 1
371  indxp( k2 ) = j
372  IF( j.EQ.n )
373  \$ GO TO 100
374  ELSE
375  jlam = j
376  GO TO 70
377  END IF
378  60 CONTINUE
379  70 CONTINUE
380  j = j + 1
381  IF( j.GT.n )
382  \$ GO TO 90
383  IF( rho*abs( z( j ) ).LE.tol ) THEN
384 *
385 * Deflate due to small z component.
386 *
387  k2 = k2 - 1
388  indxp( k2 ) = j
389  ELSE
390 *
391 * Check if eigenvalues are close enough to allow deflation.
392 *
393  s = z( jlam )
394  c = z( j )
395 *
396 * Find sqrt(a**2+b**2) without overflow or
397 * destructive underflow.
398 *
399  tau = dlapy2( c, s )
400  t = d( j ) - d( jlam )
401  c = c / tau
402  s = -s / tau
403  IF( abs( t*c*s ).LE.tol ) THEN
404 *
405 * Deflation is possible.
406 *
407  z( j ) = tau
408  z( jlam ) = zero
409 *
410 * Record the appropriate Givens rotation
411 *
412  givptr = givptr + 1
413  givcol( 1, givptr ) = indxq( indx( jlam ) )
414  givcol( 2, givptr ) = indxq( indx( j ) )
415  givnum( 1, givptr ) = c
416  givnum( 2, givptr ) = s
417  CALL zdrot( qsiz, q( 1, indxq( indx( jlam ) ) ), 1,
418  \$ q( 1, indxq( indx( j ) ) ), 1, c, s )
419  t = d( jlam )*c*c + d( j )*s*s
420  d( j ) = d( jlam )*s*s + d( j )*c*c
421  d( jlam ) = t
422  k2 = k2 - 1
423  i = 1
424  80 CONTINUE
425  IF( k2+i.LE.n ) THEN
426  IF( d( jlam ).LT.d( indxp( k2+i ) ) ) THEN
427  indxp( k2+i-1 ) = indxp( k2+i )
428  indxp( k2+i ) = jlam
429  i = i + 1
430  GO TO 80
431  ELSE
432  indxp( k2+i-1 ) = jlam
433  END IF
434  ELSE
435  indxp( k2+i-1 ) = jlam
436  END IF
437  jlam = j
438  ELSE
439  k = k + 1
440  w( k ) = z( jlam )
441  dlamda( k ) = d( jlam )
442  indxp( k ) = jlam
443  jlam = j
444  END IF
445  END IF
446  GO TO 70
447  90 CONTINUE
448 *
449 * Record the last eigenvalue.
450 *
451  k = k + 1
452  w( k ) = z( jlam )
453  dlamda( k ) = d( jlam )
454  indxp( k ) = jlam
455 *
456  100 CONTINUE
457 *
458 * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
459 * and Q2 respectively. The eigenvalues/vectors which were not
460 * deflated go into the first K slots of DLAMDA and Q2 respectively,
461 * while those which were deflated go into the last N - K slots.
462 *
463  DO 110 j = 1, n
464  jp = indxp( j )
465  dlamda( j ) = d( jp )
466  perm( j ) = indxq( indx( jp ) )
467  CALL zcopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
468  110 CONTINUE
469 *
470 * The deflated eigenvalues and their corresponding vectors go back
471 * into the last N - K slots of D and Q respectively.
472 *
473  IF( k.LT.n ) THEN
474  CALL dcopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
475  CALL zlacpy( 'A', qsiz, n-k, q2( 1, k+1 ), ldq2, q( 1, k+1 ),
476  \$ ldq )
477  END IF
478 *
479  RETURN
480 *
481 * End of ZLAED8
482 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
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 zdrot(N, ZX, INCX, ZY, INCY, C, S)
ZDROT
Definition: zdrot.f:98
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
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