LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ dsyevx_2stage()

subroutine dsyevx_2stage ( character  JOBZ,
character  RANGE,
character  UPLO,
integer  N,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision  VL,
double precision  VU,
integer  IL,
integer  IU,
double precision  ABSTOL,
integer  M,
double precision, dimension( * )  W,
double precision, dimension( ldz, * )  Z,
integer  LDZ,
double precision, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer, dimension( * )  IFAIL,
integer  INFO 
)

DSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Download DSYEVX_2STAGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DSYEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric 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 DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric 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 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 A 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)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.
[out]Z
          Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 + 3*N
                                             = N*KD + N*max(KD+1,FACTOPTNB) 
                                               + max(2*KD*KD, KD*NTHREADS) 
                                               + (KD+1)*N + 3*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]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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 297 of file dsyevx_2stage.f.

300 *
301  IMPLICIT NONE
302 *
303 * -- LAPACK driver routine --
304 * -- LAPACK is a software package provided by Univ. of Tennessee, --
305 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
306 *
307 * .. Scalar Arguments ..
308  CHARACTER JOBZ, RANGE, UPLO
309  INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
310  DOUBLE PRECISION ABSTOL, VL, VU
311 * ..
312 * .. Array Arguments ..
313  INTEGER IFAIL( * ), IWORK( * )
314  DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
315 * ..
316 *
317 * =====================================================================
318 *
319 * .. Parameters ..
320  DOUBLE PRECISION ZERO, ONE
321  parameter( zero = 0.0d+0, one = 1.0d+0 )
322 * ..
323 * .. Local Scalars ..
324  LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
325  $ WANTZ
326  CHARACTER ORDER
327  INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
328  $ INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
329  $ ITMP1, J, JJ, LLWORK, LLWRKN,
330  $ NSPLIT, LWMIN, LHTRD, LWTRD, KD, IB, INDHOUS
331  DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
332  $ SIGMA, SMLNUM, TMP1, VLL, VUU
333 * ..
334 * .. External Functions ..
335  LOGICAL LSAME
336  INTEGER ILAENV2STAGE
337  DOUBLE PRECISION DLAMCH, DLANSY
338  EXTERNAL lsame, dlamch, dlansy, ilaenv2stage
339 * ..
340 * .. External Subroutines ..
341  EXTERNAL dcopy, dlacpy, dorgtr, dormtr, dscal, dstebz,
343  $ dsytrd_2stage
344 * ..
345 * .. Intrinsic Functions ..
346  INTRINSIC max, min, sqrt
347 * ..
348 * .. Executable Statements ..
349 *
350 * Test the input parameters.
351 *
352  lower = lsame( uplo, 'L' )
353  wantz = lsame( jobz, 'V' )
354  alleig = lsame( range, 'A' )
355  valeig = lsame( range, 'V' )
356  indeig = lsame( range, 'I' )
357  lquery = ( lwork.EQ.-1 )
358 *
359  info = 0
360  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
361  info = -1
362  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
363  info = -2
364  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
365  info = -3
366  ELSE IF( n.LT.0 ) THEN
367  info = -4
368  ELSE IF( lda.LT.max( 1, n ) ) THEN
369  info = -6
370  ELSE
371  IF( valeig ) THEN
372  IF( n.GT.0 .AND. vu.LE.vl )
373  $ info = -8
374  ELSE IF( indeig ) THEN
375  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
376  info = -9
377  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
378  info = -10
379  END IF
380  END IF
381  END IF
382  IF( info.EQ.0 ) THEN
383  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
384  info = -15
385  END IF
386  END IF
387 *
388  IF( info.EQ.0 ) THEN
389  IF( n.LE.1 ) THEN
390  lwmin = 1
391  work( 1 ) = lwmin
392  ELSE
393  kd = ilaenv2stage( 1, 'DSYTRD_2STAGE', jobz,
394  $ n, -1, -1, -1 )
395  ib = ilaenv2stage( 2, 'DSYTRD_2STAGE', jobz,
396  $ n, kd, -1, -1 )
397  lhtrd = ilaenv2stage( 3, 'DSYTRD_2STAGE', jobz,
398  $ n, kd, ib, -1 )
399  lwtrd = ilaenv2stage( 4, 'DSYTRD_2STAGE', jobz,
400  $ n, kd, ib, -1 )
401  lwmin = max( 8*n, 3*n + lhtrd + lwtrd )
402  work( 1 ) = lwmin
403  END IF
404 *
405  IF( lwork.LT.lwmin .AND. .NOT.lquery )
406  $ info = -17
407  END IF
408 *
409  IF( info.NE.0 ) THEN
410  CALL xerbla( 'DSYEVX_2STAGE', -info )
411  RETURN
412  ELSE IF( lquery ) THEN
413  RETURN
414  END IF
415 *
416 * Quick return if possible
417 *
418  m = 0
419  IF( n.EQ.0 ) THEN
420  RETURN
421  END IF
422 *
423  IF( n.EQ.1 ) THEN
424  IF( alleig .OR. indeig ) THEN
425  m = 1
426  w( 1 ) = a( 1, 1 )
427  ELSE
428  IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
429  m = 1
430  w( 1 ) = a( 1, 1 )
431  END IF
432  END IF
433  IF( wantz )
434  $ z( 1, 1 ) = one
435  RETURN
436  END IF
437 *
438 * Get machine constants.
439 *
440  safmin = dlamch( 'Safe minimum' )
441  eps = dlamch( 'Precision' )
442  smlnum = safmin / eps
443  bignum = one / smlnum
444  rmin = sqrt( smlnum )
445  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
446 *
447 * Scale matrix to allowable range, if necessary.
448 *
449  iscale = 0
450  abstll = abstol
451  IF( valeig ) THEN
452  vll = vl
453  vuu = vu
454  END IF
455  anrm = dlansy( 'M', uplo, n, a, lda, work )
456  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
457  iscale = 1
458  sigma = rmin / anrm
459  ELSE IF( anrm.GT.rmax ) THEN
460  iscale = 1
461  sigma = rmax / anrm
462  END IF
463  IF( iscale.EQ.1 ) THEN
464  IF( lower ) THEN
465  DO 10 j = 1, n
466  CALL dscal( n-j+1, sigma, a( j, j ), 1 )
467  10 CONTINUE
468  ELSE
469  DO 20 j = 1, n
470  CALL dscal( j, sigma, a( 1, j ), 1 )
471  20 CONTINUE
472  END IF
473  IF( abstol.GT.0 )
474  $ abstll = abstol*sigma
475  IF( valeig ) THEN
476  vll = vl*sigma
477  vuu = vu*sigma
478  END IF
479  END IF
480 *
481 * Call DSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
482 *
483  indtau = 1
484  inde = indtau + n
485  indd = inde + n
486  indhous = indd + n
487  indwrk = indhous + lhtrd
488  llwork = lwork - indwrk + 1
489 *
490  CALL dsytrd_2stage( jobz, uplo, n, a, lda, work( indd ),
491  $ work( inde ), work( indtau ), work( indhous ),
492  $ lhtrd, work( indwrk ), llwork, iinfo )
493 *
494 * If all eigenvalues are desired and ABSTOL is less than or equal to
495 * zero, then call DSTERF or DORGTR and SSTEQR. If this fails for
496 * some eigenvalue, then try DSTEBZ.
497 *
498  test = .false.
499  IF( indeig ) THEN
500  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
501  test = .true.
502  END IF
503  END IF
504  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
505  CALL dcopy( n, work( indd ), 1, w, 1 )
506  indee = indwrk + 2*n
507  IF( .NOT.wantz ) THEN
508  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
509  CALL dsterf( n, w, work( indee ), info )
510  ELSE
511  CALL dlacpy( 'A', n, n, a, lda, z, ldz )
512  CALL dorgtr( uplo, n, z, ldz, work( indtau ),
513  $ work( indwrk ), llwork, iinfo )
514  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
515  CALL dsteqr( jobz, n, w, work( indee ), z, ldz,
516  $ work( indwrk ), info )
517  IF( info.EQ.0 ) THEN
518  DO 30 i = 1, n
519  ifail( i ) = 0
520  30 CONTINUE
521  END IF
522  END IF
523  IF( info.EQ.0 ) THEN
524  m = n
525  GO TO 40
526  END IF
527  info = 0
528  END IF
529 *
530 * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
531 *
532  IF( wantz ) THEN
533  order = 'B'
534  ELSE
535  order = 'E'
536  END IF
537  indibl = 1
538  indisp = indibl + n
539  indiwo = indisp + n
540  CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
541  $ work( indd ), work( inde ), m, nsplit, w,
542  $ iwork( indibl ), iwork( indisp ), work( indwrk ),
543  $ iwork( indiwo ), info )
544 *
545  IF( wantz ) THEN
546  CALL dstein( n, work( indd ), work( inde ), m, w,
547  $ iwork( indibl ), iwork( indisp ), z, ldz,
548  $ work( indwrk ), iwork( indiwo ), ifail, info )
549 *
550 * Apply orthogonal matrix used in reduction to tridiagonal
551 * form to eigenvectors returned by DSTEIN.
552 *
553  indwkn = inde
554  llwrkn = lwork - indwkn + 1
555  CALL dormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
556  $ ldz, work( indwkn ), llwrkn, iinfo )
557  END IF
558 *
559 * If matrix was scaled, then rescale eigenvalues appropriately.
560 *
561  40 CONTINUE
562  IF( iscale.EQ.1 ) THEN
563  IF( info.EQ.0 ) THEN
564  imax = m
565  ELSE
566  imax = info - 1
567  END IF
568  CALL dscal( imax, one / sigma, w, 1 )
569  END IF
570 *
571 * If eigenvalues are not in order, then sort them, along with
572 * eigenvectors.
573 *
574  IF( wantz ) THEN
575  DO 60 j = 1, m - 1
576  i = 0
577  tmp1 = w( j )
578  DO 50 jj = j + 1, m
579  IF( w( jj ).LT.tmp1 ) THEN
580  i = jj
581  tmp1 = w( jj )
582  END IF
583  50 CONTINUE
584 *
585  IF( i.NE.0 ) THEN
586  itmp1 = iwork( indibl+i-1 )
587  w( i ) = w( j )
588  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
589  w( j ) = tmp1
590  iwork( indibl+j-1 ) = itmp1
591  CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
592  IF( info.NE.0 ) THEN
593  itmp1 = ifail( i )
594  ifail( i ) = ifail( j )
595  ifail( j ) = itmp1
596  END IF
597  END IF
598  60 CONTINUE
599  END IF
600 *
601 * Set WORK(1) to optimal workspace size.
602 *
603  work( 1 ) = lwmin
604 *
605  RETURN
606 *
607 * End of DSYEVX_2STAGE
608 *
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 dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
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 dsteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
DSTEQR
Definition: dsteqr.f:131
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
subroutine dstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSTEIN
Definition: dstein.f:174
subroutine dormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMTR
Definition: dormtr.f:171
subroutine dorgtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
DORGTR
Definition: dorgtr.f:123
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
subroutine dsytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
DSYTRD_2STAGE
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