 LAPACK 3.11.0 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( * ) 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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