LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ 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( * )  dlambda,
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.

Download ZLAED8 + dependencies [TGZ] [ZIP] [TXT]

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]DLAMBDA
          DLAMBDA 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

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( * ), DLAMBDA( * ), 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 dlambda( 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, dlambda, 1, 1, indx )
331 DO 40 i = 1, n
332 d( i ) = dlambda( 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 dlambda( 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 dlambda( k ) = d( jlam )
454 indxp( k ) = jlam
455*
456 100 CONTINUE
457*
458* Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
459* and Q2 respectively. The eigenvalues/vectors which were not
460* deflated go into the first K slots of DLAMBDA 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 dlambda( 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, dlambda( 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*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
zcopy
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
dcopy
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
idamax
integer function idamax(n, dx, incx)
IDAMAX
Definition idamax.f:71
zlacpy
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
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
dlamrg
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
dlapy2
double precision function dlapy2(x, y)
DLAPY2 returns sqrt(x2+y2).
Definition dlapy2.f:63
zdrot
subroutine zdrot(n, zx, incx, zy, incy, c, s)
ZDROT
Definition zdrot.f:98
dscal
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
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