LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ claed8()

subroutine claed8 ( integer  K,
integer  N,
integer  QSIZ,
complex, dimension( ldq, * )  Q,
integer  LDQ,
real, dimension( * )  D,
real  RHO,
integer  CUTPNT,
real, dimension( * )  Z,
real, dimension( * )  DLAMDA,
complex, dimension( ldq2, * )  Q2,
integer  LDQ2,
real, dimension( * )  W,
integer, dimension( * )  INDXP,
integer, dimension( * )  INDX,
integer, dimension( * )  INDXQ,
integer, dimension( * )  PERM,
integer  GIVPTR,
integer, dimension( 2, * )  GIVCOL,
real, dimension( 2, * )  GIVNUM,
integer  INFO 
)

CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Download CLAED8 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CLAED8 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 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 REAL 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 REAL
         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 SLAED3.
[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 REAL 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 REAL array, dimension (N)
         Contains a copy of the first K eigenvalues which will be used
         by SLAED3 to form the secular equation.
[out]Q2
          Q2 is COMPLEX 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 SLAED7 in a matrix multiply (SGEMM) 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 REAL array, dimension (N)
         This will hold the first k values of the final
         deflation-altered z-vector and will be passed to SLAED3.
[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 REAL 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 225 of file claed8.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  REAL RHO
236 * ..
237 * .. Array Arguments ..
238  INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
239  $ INDXQ( * ), PERM( * )
240  REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
241  $ Z( * )
242  COMPLEX Q( LDQ, * ), Q2( LDQ2, * )
243 * ..
244 *
245 * =====================================================================
246 *
247 * .. Parameters ..
248  REAL MONE, ZERO, ONE, TWO, EIGHT
249  parameter( mone = -1.0e0, zero = 0.0e0, one = 1.0e0,
250  $ two = 2.0e0, eight = 8.0e0 )
251 * ..
252 * .. Local Scalars ..
253  INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
254  REAL C, EPS, S, T, TAU, TOL
255 * ..
256 * .. External Functions ..
257  INTEGER ISAMAX
258  REAL SLAMCH, SLAPY2
259  EXTERNAL isamax, slamch, slapy2
260 * ..
261 * .. External Subroutines ..
262  EXTERNAL ccopy, clacpy, csrot, scopy, slamrg, sscal,
263  $ xerbla
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( 'CLAED8', -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 sscal( 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 sscal( 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 slamrg( 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 = isamax( n, z, 1 )
339  jmax = isamax( n, d, 1 )
340  eps = slamch( '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 ccopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
352  50 CONTINUE
353  CALL clacpy( '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 = slapy2( 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 csrot( 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 ccopy( 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 scopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
475  CALL clacpy( '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 CLAED8
482 *
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:63
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
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 ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine csrot(N, CX, INCX, CY, INCY, C, S)
CSROT
Definition: csrot.f:98
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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