 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zhbevx_2stage()

 subroutine zhbevx_2stage ( character JOBZ, character RANGE, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:
``` ZHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian band matrix A using the 2stage technique for
the reduction to tridiagonal.  Eigenvalues and eigenvectors
can be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.```
Parameters
 [in] JOBZ ``` JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.``` [in] RANGE ``` RANGE is CHARACTER*1 = 'A': all eigenvalues will be found; = 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.``` [in,out] AB ``` AB is COMPLEX*16 array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.``` [out] Q ``` Q is COMPLEX*16 array, dimension (LDQ, N) If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.``` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).``` [in] VL ``` VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] VU ``` VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] IL ``` IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] IU ``` IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] ABSTOL ``` ABSTOL is DOUBLE PRECISION The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AB to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.``` [out] M ``` M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.``` [out] W ``` W is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX*16 array, dimension (LDZ, max(1,M)) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).``` [out] WORK ` WORK is COMPLEX*16 array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.``` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (7*N)` [out] IWORK ` IWORK is INTEGER array, dimension (5*N)` [out] IFAIL ``` IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.```
Further Details:
```  All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196 ```

Definition at line 323 of file zhbevx_2stage.f.

327 *
328  IMPLICIT NONE
329 *
330 * -- LAPACK driver routine --
331 * -- LAPACK is a software package provided by Univ. of Tennessee, --
332 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
333 *
334 * .. Scalar Arguments ..
335  CHARACTER JOBZ, RANGE, UPLO
336  INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N, LWORK
337  DOUBLE PRECISION ABSTOL, VL, VU
338 * ..
339 * .. Array Arguments ..
340  INTEGER IFAIL( * ), IWORK( * )
341  DOUBLE PRECISION RWORK( * ), W( * )
342  COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
343  \$ Z( LDZ, * )
344 * ..
345 *
346 * =====================================================================
347 *
348 * .. Parameters ..
349  DOUBLE PRECISION ZERO, ONE
350  parameter( zero = 0.0d0, one = 1.0d0 )
351  COMPLEX*16 CZERO, CONE
352  parameter( czero = ( 0.0d0, 0.0d0 ),
353  \$ cone = ( 1.0d0, 0.0d0 ) )
354 * ..
355 * .. Local Scalars ..
356  LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ,
357  \$ LQUERY
358  CHARACTER ORDER
359  INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
360  \$ INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1,
361  \$ LLWORK, LWMIN, LHTRD, LWTRD, IB, INDHOUS,
362  \$ J, JJ, NSPLIT
363  DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
364  \$ SIGMA, SMLNUM, TMP1, VLL, VUU
365  COMPLEX*16 CTMP1
366 * ..
367 * .. External Functions ..
368  LOGICAL LSAME
369  INTEGER ILAENV2STAGE
370  DOUBLE PRECISION DLAMCH, ZLANHB
371  EXTERNAL lsame, dlamch, zlanhb, ilaenv2stage
372 * ..
373 * .. External Subroutines ..
374  EXTERNAL dcopy, dscal, dstebz, dsterf, xerbla, zcopy,
377 * ..
378 * .. Intrinsic Functions ..
379  INTRINSIC dble, max, min, sqrt
380 * ..
381 * .. Executable Statements ..
382 *
383 * Test the input parameters.
384 *
385  wantz = lsame( jobz, 'V' )
386  alleig = lsame( range, 'A' )
387  valeig = lsame( range, 'V' )
388  indeig = lsame( range, 'I' )
389  lower = lsame( uplo, 'L' )
390  lquery = ( lwork.EQ.-1 )
391 *
392  info = 0
393  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
394  info = -1
395  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
396  info = -2
397  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
398  info = -3
399  ELSE IF( n.LT.0 ) THEN
400  info = -4
401  ELSE IF( kd.LT.0 ) THEN
402  info = -5
403  ELSE IF( ldab.LT.kd+1 ) THEN
404  info = -7
405  ELSE IF( wantz .AND. ldq.LT.max( 1, n ) ) THEN
406  info = -9
407  ELSE
408  IF( valeig ) THEN
409  IF( n.GT.0 .AND. vu.LE.vl )
410  \$ info = -11
411  ELSE IF( indeig ) THEN
412  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
413  info = -12
414  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
415  info = -13
416  END IF
417  END IF
418  END IF
419  IF( info.EQ.0 ) THEN
420  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) )
421  \$ info = -18
422  END IF
423 *
424  IF( info.EQ.0 ) THEN
425  IF( n.LE.1 ) THEN
426  lwmin = 1
427  work( 1 ) = lwmin
428  ELSE
429  ib = ilaenv2stage( 2, 'ZHETRD_HB2ST', jobz,
430  \$ n, kd, -1, -1 )
431  lhtrd = ilaenv2stage( 3, 'ZHETRD_HB2ST', jobz,
432  \$ n, kd, ib, -1 )
433  lwtrd = ilaenv2stage( 4, 'ZHETRD_HB2ST', jobz,
434  \$ n, kd, ib, -1 )
435  lwmin = lhtrd + lwtrd
436  work( 1 ) = lwmin
437  ENDIF
438 *
439  IF( lwork.LT.lwmin .AND. .NOT.lquery )
440  \$ info = -20
441  END IF
442 *
443  IF( info.NE.0 ) THEN
444  CALL xerbla( 'ZHBEVX_2STAGE', -info )
445  RETURN
446  ELSE IF( lquery ) THEN
447  RETURN
448  END IF
449 *
450 * Quick return if possible
451 *
452  m = 0
453  IF( n.EQ.0 )
454  \$ RETURN
455 *
456  IF( n.EQ.1 ) THEN
457  m = 1
458  IF( lower ) THEN
459  ctmp1 = ab( 1, 1 )
460  ELSE
461  ctmp1 = ab( kd+1, 1 )
462  END IF
463  tmp1 = dble( ctmp1 )
464  IF( valeig ) THEN
465  IF( .NOT.( vl.LT.tmp1 .AND. vu.GE.tmp1 ) )
466  \$ m = 0
467  END IF
468  IF( m.EQ.1 ) THEN
469  w( 1 ) = dble( ctmp1 )
470  IF( wantz )
471  \$ z( 1, 1 ) = cone
472  END IF
473  RETURN
474  END IF
475 *
476 * Get machine constants.
477 *
478  safmin = dlamch( 'Safe minimum' )
479  eps = dlamch( 'Precision' )
480  smlnum = safmin / eps
481  bignum = one / smlnum
482  rmin = sqrt( smlnum )
483  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
484 *
485 * Scale matrix to allowable range, if necessary.
486 *
487  iscale = 0
488  abstll = abstol
489  IF( valeig ) THEN
490  vll = vl
491  vuu = vu
492  ELSE
493  vll = zero
494  vuu = zero
495  END IF
496  anrm = zlanhb( 'M', uplo, n, kd, ab, ldab, rwork )
497  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
498  iscale = 1
499  sigma = rmin / anrm
500  ELSE IF( anrm.GT.rmax ) THEN
501  iscale = 1
502  sigma = rmax / anrm
503  END IF
504  IF( iscale.EQ.1 ) THEN
505  IF( lower ) THEN
506  CALL zlascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
507  ELSE
508  CALL zlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
509  END IF
510  IF( abstol.GT.0 )
511  \$ abstll = abstol*sigma
512  IF( valeig ) THEN
513  vll = vl*sigma
514  vuu = vu*sigma
515  END IF
516  END IF
517 *
518 * Call ZHBTRD_HB2ST to reduce Hermitian band matrix to tridiagonal form.
519 *
520  indd = 1
521  inde = indd + n
522  indrwk = inde + n
523 *
524  indhous = 1
525  indwrk = indhous + lhtrd
526  llwork = lwork - indwrk + 1
527 *
528  CALL zhetrd_hb2st( 'N', jobz, uplo, n, kd, ab, ldab,
529  \$ rwork( indd ), rwork( inde ), work( indhous ),
530  \$ lhtrd, work( indwrk ), llwork, iinfo )
531 *
532 * If all eigenvalues are desired and ABSTOL is less than or equal
533 * to zero, then call DSTERF or ZSTEQR. If this fails for some
534 * eigenvalue, then try DSTEBZ.
535 *
536  test = .false.
537  IF (indeig) THEN
538  IF (il.EQ.1 .AND. iu.EQ.n) THEN
539  test = .true.
540  END IF
541  END IF
542  IF ((alleig .OR. test) .AND. (abstol.LE.zero)) THEN
543  CALL dcopy( n, rwork( indd ), 1, w, 1 )
544  indee = indrwk + 2*n
545  IF( .NOT.wantz ) THEN
546  CALL dcopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
547  CALL dsterf( n, w, rwork( indee ), info )
548  ELSE
549  CALL zlacpy( 'A', n, n, q, ldq, z, ldz )
550  CALL dcopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
551  CALL zsteqr( jobz, n, w, rwork( indee ), z, ldz,
552  \$ rwork( indrwk ), info )
553  IF( info.EQ.0 ) THEN
554  DO 10 i = 1, n
555  ifail( i ) = 0
556  10 CONTINUE
557  END IF
558  END IF
559  IF( info.EQ.0 ) THEN
560  m = n
561  GO TO 30
562  END IF
563  info = 0
564  END IF
565 *
566 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
567 *
568  IF( wantz ) THEN
569  order = 'B'
570  ELSE
571  order = 'E'
572  END IF
573  indibl = 1
574  indisp = indibl + n
575  indiwk = indisp + n
576  CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
577  \$ rwork( indd ), rwork( inde ), m, nsplit, w,
578  \$ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
579  \$ iwork( indiwk ), info )
580 *
581  IF( wantz ) THEN
582  CALL zstein( n, rwork( indd ), rwork( inde ), m, w,
583  \$ iwork( indibl ), iwork( indisp ), z, ldz,
584  \$ rwork( indrwk ), iwork( indiwk ), ifail, info )
585 *
586 * Apply unitary matrix used in reduction to tridiagonal
587 * form to eigenvectors returned by ZSTEIN.
588 *
589  DO 20 j = 1, m
590  CALL zcopy( n, z( 1, j ), 1, work( 1 ), 1 )
591  CALL zgemv( 'N', n, n, cone, q, ldq, work, 1, czero,
592  \$ z( 1, j ), 1 )
593  20 CONTINUE
594  END IF
595 *
596 * If matrix was scaled, then rescale eigenvalues appropriately.
597 *
598  30 CONTINUE
599  IF( iscale.EQ.1 ) THEN
600  IF( info.EQ.0 ) THEN
601  imax = m
602  ELSE
603  imax = info - 1
604  END IF
605  CALL dscal( imax, one / sigma, w, 1 )
606  END IF
607 *
608 * If eigenvalues are not in order, then sort them, along with
609 * eigenvectors.
610 *
611  IF( wantz ) THEN
612  DO 50 j = 1, m - 1
613  i = 0
614  tmp1 = w( j )
615  DO 40 jj = j + 1, m
616  IF( w( jj ).LT.tmp1 ) THEN
617  i = jj
618  tmp1 = w( jj )
619  END IF
620  40 CONTINUE
621 *
622  IF( i.NE.0 ) THEN
623  itmp1 = iwork( indibl+i-1 )
624  w( i ) = w( j )
625  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
626  w( j ) = tmp1
627  iwork( indibl+j-1 ) = itmp1
628  CALL zswap( n, z( 1, i ), 1, z( 1, j ), 1 )
629  IF( info.NE.0 ) THEN
630  itmp1 = ifail( i )
631  ifail( i ) = ifail( j )
632  ifail( j ) = itmp1
633  END IF
634  END IF
635  50 CONTINUE
636  END IF
637 *
638 * Set WORK(1) to optimal workspace size.
639 *
640  work( 1 ) = lwmin
641 *
642  RETURN
643 *
644 * End of ZHBEVX_2STAGE
645 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:149
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
DSTEBZ
Definition: dstebz.f:273
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: zlascl.f:143
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
double precision function zlanhb(NORM, UPLO, N, K, AB, LDAB, WORK)
ZLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlanhb.f:132
subroutine zsteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
ZSTEQR
Definition: zsteqr.f:132
subroutine zstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
ZSTEIN
Definition: zstein.f:182
subroutine zhetrd_hb2st(STAGE1, VECT, UPLO, N, KD, AB, LDAB, D, E, HOUS, LHOUS, WORK, LWORK, INFO)
ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T
Definition: zhetrd_hb2st.F:230
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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