LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ ssyevx_2stage()

subroutine ssyevx_2stage ( character  JOBZ,
character  RANGE,
character  UPLO,
integer  N,
real, dimension( lda, * )  A,
integer  LDA,
real  VL,
real  VU,
integer  IL,
integer  IU,
real  ABSTOL,
integer  M,
real, dimension( * )  W,
real, dimension( ldz, * )  Z,
integer  LDZ,
real, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer, dimension( * )  IFAIL,
integer  INFO 
)

SSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Download SSYEVX_2STAGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SSYEVX_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 REAL 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 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 REAL 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 REAL 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 ssyevx_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 REAL ABSTOL, VL, VU
311* ..
312* .. Array Arguments ..
313 INTEGER IFAIL( * ), IWORK( * )
314 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
315* ..
316*
317* =====================================================================
318*
319* .. Parameters ..
320 REAL ZERO, ONE
321 parameter( zero = 0.0e+0, one = 1.0e+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 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
332 $ SIGMA, SMLNUM, TMP1, VLL, VUU
333* ..
334* .. External Functions ..
335 LOGICAL LSAME
336 INTEGER ILAENV2STAGE
337 REAL SLAMCH, SLANSY
338 EXTERNAL lsame, slamch, slansy, ilaenv2stage
339* ..
340* .. External Subroutines ..
341 EXTERNAL scopy, slacpy, sorgtr, sormtr, sscal, sstebz,
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, 'SSYTRD_2STAGE', jobz,
394 $ n, -1, -1, -1 )
395 ib = ilaenv2stage( 2, 'SSYTRD_2STAGE', jobz,
396 $ n, kd, -1, -1 )
397 lhtrd = ilaenv2stage( 3, 'SSYTRD_2STAGE', jobz,
398 $ n, kd, ib, -1 )
399 lwtrd = ilaenv2stage( 4, 'SSYTRD_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( 'SSYEVX_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 = slamch( 'Safe minimum' )
441 eps = slamch( '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 = slansy( '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 sscal( n-j+1, sigma, a( j, j ), 1 )
467 10 CONTINUE
468 ELSE
469 DO 20 j = 1, n
470 CALL sscal( 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 SSYTRD_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 ssytrd_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 SSTERF or SORGTR and SSTEQR. If this fails for
496* some eigenvalue, then try SSTEBZ.
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 scopy( n, work( indd ), 1, w, 1 )
506 indee = indwrk + 2*n
507 IF( .NOT.wantz ) THEN
508 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
509 CALL ssterf( n, w, work( indee ), info )
510 ELSE
511 CALL slacpy( 'A', n, n, a, lda, z, ldz )
512 CALL sorgtr( uplo, n, z, ldz, work( indtau ),
513 $ work( indwrk ), llwork, iinfo )
514 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
515 CALL ssteqr( 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 SSTEBZ 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 sstebz( 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 sstein( 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 SSTEIN.
552*
553 indwkn = inde
554 llwrkn = lwork - indwkn + 1
555 CALL sormtr( '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 sscal( 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 sswap( 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 SSYEVX_2STAGE
608*
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:149
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ssteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR
Definition: ssteqr.f:131
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
subroutine sstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ
Definition: sstebz.f:273
subroutine sormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMTR
Definition: sormtr.f:172
subroutine sstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSTEIN
Definition: sstein.f:174
subroutine sorgtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
SORGTR
Definition: sorgtr.f:123
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansy.f:122
subroutine ssytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
SSYTRD_2STAGE
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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