 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ cheevx_2stage()

 subroutine cheevx_2stage ( character JOBZ, character RANGE, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

CHEEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Download CHEEVX_2STAGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` CHEEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian 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,out] A ``` A is COMPLEX array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] VL ``` VL is REAL 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 REAL 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 REAL 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 A to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('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 REAL array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX 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 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal 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, 8*N, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice 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 size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] RWORK ` RWORK is REAL 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.```
Date
June 2016
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).
Denver, Colorado, USA, 2013.
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 308 of file cheevx_2stage.f.

308 *
309  IMPLICIT NONE
310 *
311 * -- LAPACK driver routine (version 3.8.0) --
312 * -- LAPACK is a software package provided by Univ. of Tennessee, --
313 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
314 * June 2016
315 *
316 * .. Scalar Arguments ..
317  CHARACTER jobz, range, uplo
318  INTEGER il, info, iu, lda, ldz, lwork, m, n
319  REAL abstol, vl, vu
320 * ..
321 * .. Array Arguments ..
322  INTEGER ifail( * ), iwork( * )
323  REAL rwork( * ), w( * )
324  COMPLEX a( lda, * ), work( * ), z( ldz, * )
325 * ..
326 *
327 * =====================================================================
328 *
329 * .. Parameters ..
330  REAL zero, one
331  parameter( zero = 0.0e+0, one = 1.0e+0 )
332  COMPLEX cone
333  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
334 * ..
335 * .. Local Scalars ..
336  LOGICAL alleig, indeig, lower, lquery, test, valeig,
337  \$ wantz
338  CHARACTER order
339  INTEGER i, iinfo, imax, indd, inde, indee, indibl,
340  \$ indisp, indiwk, indrwk, indtau, indwrk, iscale,
341  \$ itmp1, j, jj, llwork,
342  \$ nsplit, lwmin, lhtrd, lwtrd, kd, ib, indhous
343  REAL abstll, anrm, bignum, eps, rmax, rmin, safmin,
344  \$ sigma, smlnum, tmp1, vll, vuu
345 * ..
346 * .. External Functions ..
347  LOGICAL lsame
348  INTEGER ilaenv2stage
349  REAL slamch, clanhe
350  EXTERNAL lsame, slamch, clanhe, ilaenv2stage
351 * ..
352 * .. External Subroutines ..
353  EXTERNAL scopy, sscal, sstebz, ssterf, xerbla, csscal,
355  \$ chetrd_2stage
356 * ..
357 * .. Intrinsic Functions ..
358  INTRINSIC REAL, max, min, sqrt
359 * ..
360 * .. Executable Statements ..
361 *
362 * Test the input parameters.
363 *
364  lower = lsame( uplo, 'L' )
365  wantz = lsame( jobz, 'V' )
366  alleig = lsame( range, 'A' )
367  valeig = lsame( range, 'V' )
368  indeig = lsame( range, 'I' )
369  lquery = ( lwork.EQ.-1 )
370 *
371  info = 0
372  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
373  info = -1
374  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
375  info = -2
376  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
377  info = -3
378  ELSE IF( n.LT.0 ) THEN
379  info = -4
380  ELSE IF( lda.LT.max( 1, n ) ) THEN
381  info = -6
382  ELSE
383  IF( valeig ) THEN
384  IF( n.GT.0 .AND. vu.LE.vl )
385  \$ info = -8
386  ELSE IF( indeig ) THEN
387  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
388  info = -9
389  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
390  info = -10
391  END IF
392  END IF
393  END IF
394  IF( info.EQ.0 ) THEN
395  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
396  info = -15
397  END IF
398  END IF
399 *
400  IF( info.EQ.0 ) THEN
401  IF( n.LE.1 ) THEN
402  lwmin = 1
403  work( 1 ) = lwmin
404  ELSE
405  kd = ilaenv2stage( 1, 'CHETRD_2STAGE', jobz,
406  \$ n, -1, -1, -1 )
407  ib = ilaenv2stage( 2, 'CHETRD_2STAGE', jobz,
408  \$ n, kd, -1, -1 )
409  lhtrd = ilaenv2stage( 3, 'CHETRD_2STAGE', jobz,
410  \$ n, kd, ib, -1 )
411  lwtrd = ilaenv2stage( 4, 'CHETRD_2STAGE', jobz,
412  \$ n, kd, ib, -1 )
413  lwmin = n + lhtrd + lwtrd
414  work( 1 ) = lwmin
415  END IF
416 *
417  IF( lwork.LT.lwmin .AND. .NOT.lquery )
418  \$ info = -17
419  END IF
420 *
421  IF( info.NE.0 ) THEN
422  CALL xerbla( 'CHEEVX_2STAGE', -info )
423  RETURN
424  ELSE IF( lquery ) THEN
425  RETURN
426  END IF
427 *
428 * Quick return if possible
429 *
430  m = 0
431  IF( n.EQ.0 ) THEN
432  RETURN
433  END IF
434 *
435  IF( n.EQ.1 ) THEN
436  IF( alleig .OR. indeig ) THEN
437  m = 1
438  w( 1 ) = REAL( A( 1, 1 ) )
439  ELSE IF( valeig ) THEN
440  IF( vl.LT.REAL( A( 1, 1 ) ) .AND. vu.GE.REAL( A( 1, 1 ) ) )
441  \$ THEN
442  m = 1
443  w( 1 ) = REAL( A( 1, 1 ) )
444  END IF
445  END IF
446  IF( wantz )
447  \$ z( 1, 1 ) = cone
448  RETURN
449  END IF
450 *
451 * Get machine constants.
452 *
453  safmin = slamch( 'Safe minimum' )
454  eps = slamch( 'Precision' )
455  smlnum = safmin / eps
456  bignum = one / smlnum
457  rmin = sqrt( smlnum )
458  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
459 *
460 * Scale matrix to allowable range, if necessary.
461 *
462  iscale = 0
463  abstll = abstol
464  IF( valeig ) THEN
465  vll = vl
466  vuu = vu
467  END IF
468  anrm = clanhe( 'M', uplo, n, a, lda, rwork )
469  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
470  iscale = 1
471  sigma = rmin / anrm
472  ELSE IF( anrm.GT.rmax ) THEN
473  iscale = 1
474  sigma = rmax / anrm
475  END IF
476  IF( iscale.EQ.1 ) THEN
477  IF( lower ) THEN
478  DO 10 j = 1, n
479  CALL csscal( n-j+1, sigma, a( j, j ), 1 )
480  10 CONTINUE
481  ELSE
482  DO 20 j = 1, n
483  CALL csscal( j, sigma, a( 1, j ), 1 )
484  20 CONTINUE
485  END IF
486  IF( abstol.GT.0 )
487  \$ abstll = abstol*sigma
488  IF( valeig ) THEN
489  vll = vl*sigma
490  vuu = vu*sigma
491  END IF
492  END IF
493 *
494 * Call CHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
495 *
496  indd = 1
497  inde = indd + n
498  indrwk = inde + n
499  indtau = 1
500  indhous = indtau + n
501  indwrk = indhous + lhtrd
502  llwork = lwork - indwrk + 1
503 *
504  CALL chetrd_2stage( jobz, uplo, n, a, lda, rwork( indd ),
505  \$ rwork( inde ), work( indtau ),
506  \$ work( indhous ), lhtrd, work( indwrk ),
507  \$ llwork, iinfo )
508 *
509 * If all eigenvalues are desired and ABSTOL is less than or equal to
510 * zero, then call SSTERF or CUNGTR and CSTEQR. If this fails for
511 * some eigenvalue, then try SSTEBZ.
512 *
513  test = .false.
514  IF( indeig ) THEN
515  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
516  test = .true.
517  END IF
518  END IF
519  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
520  CALL scopy( n, rwork( indd ), 1, w, 1 )
521  indee = indrwk + 2*n
522  IF( .NOT.wantz ) THEN
523  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
524  CALL ssterf( n, w, rwork( indee ), info )
525  ELSE
526  CALL clacpy( 'A', n, n, a, lda, z, ldz )
527  CALL cungtr( uplo, n, z, ldz, work( indtau ),
528  \$ work( indwrk ), llwork, iinfo )
529  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
530  CALL csteqr( jobz, n, w, rwork( indee ), z, ldz,
531  \$ rwork( indrwk ), info )
532  IF( info.EQ.0 ) THEN
533  DO 30 i = 1, n
534  ifail( i ) = 0
535  30 CONTINUE
536  END IF
537  END IF
538  IF( info.EQ.0 ) THEN
539  m = n
540  GO TO 40
541  END IF
542  info = 0
543  END IF
544 *
545 * Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
546 *
547  IF( wantz ) THEN
548  order = 'B'
549  ELSE
550  order = 'E'
551  END IF
552  indibl = 1
553  indisp = indibl + n
554  indiwk = indisp + n
555  CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
556  \$ rwork( indd ), rwork( inde ), m, nsplit, w,
557  \$ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
558  \$ iwork( indiwk ), info )
559 *
560  IF( wantz ) THEN
561  CALL cstein( n, rwork( indd ), rwork( inde ), m, w,
562  \$ iwork( indibl ), iwork( indisp ), z, ldz,
563  \$ rwork( indrwk ), iwork( indiwk ), ifail, info )
564 *
565 * Apply unitary matrix used in reduction to tridiagonal
566 * form to eigenvectors returned by CSTEIN.
567 *
568  CALL cunmtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
569  \$ ldz, work( indwrk ), llwork, iinfo )
570  END IF
571 *
572 * If matrix was scaled, then rescale eigenvalues appropriately.
573 *
574  40 CONTINUE
575  IF( iscale.EQ.1 ) THEN
576  IF( info.EQ.0 ) THEN
577  imax = m
578  ELSE
579  imax = info - 1
580  END IF
581  CALL sscal( imax, one / sigma, w, 1 )
582  END IF
583 *
584 * If eigenvalues are not in order, then sort them, along with
585 * eigenvectors.
586 *
587  IF( wantz ) THEN
588  DO 60 j = 1, m - 1
589  i = 0
590  tmp1 = w( j )
591  DO 50 jj = j + 1, m
592  IF( w( jj ).LT.tmp1 ) THEN
593  i = jj
594  tmp1 = w( jj )
595  END IF
596  50 CONTINUE
597 *
598  IF( i.NE.0 ) THEN
599  itmp1 = iwork( indibl+i-1 )
600  w( i ) = w( j )
601  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
602  w( j ) = tmp1
603  iwork( indibl+j-1 ) = itmp1
604  CALL cswap( n, z( 1, i ), 1, z( 1, j ), 1 )
605  IF( info.NE.0 ) THEN
606  itmp1 = ifail( i )
607  ifail( i ) = ifail( j )
608  ifail( j ) = itmp1
609  END IF
610  END IF
611  60 CONTINUE
612  END IF
613 *
614 * Set WORK(1) to optimal complex workspace size.
615 *
616  work( 1 ) = lwmin
617 *
618  RETURN
619 *
620 * End of CHEEVX_2STAGE
621 *
subroutine cunmtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMTR
Definition: cunmtr.f:174
subroutine sstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ
Definition: sstebz.f:275
subroutine cungtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
CUNGTR
Definition: cungtr.f:125
subroutine csteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
CSTEQR
Definition: csteqr.f:134
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81
subroutine chetrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
CHETRD_2STAGE
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:83
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:151
subroutine cstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
CSTEIN
Definition: cstein.f:184
real function clanhe(NORM, UPLO, N, A, LDA, WORK)
CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix.
Definition: clanhe.f:126
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:88
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:80
Here is the call graph for this function:
Here is the caller graph for this function: