LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ ssyevr()

subroutine ssyevr ( 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,
integer, dimension( * )  ISUPPZ,
real, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer  LIWORK,
integer  INFO 
)

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

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

Purpose:
 SSYEVR computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
 selected by specifying either a range of values or a range of
 indices for the desired eigenvalues.

 SSYEVR first reduces the matrix A to tridiagonal form T with a call
 to SSYTRD.  Then, whenever possible, SSYEVR calls SSTEMR to compute
 the eigenspectrum using Relatively Robust Representations.  SSTEMR
 computes eigenvalues by the dqds algorithm, while orthogonal
 eigenvectors are computed from various "good" L D L^T representations
 (also known as Relatively Robust Representations). Gram-Schmidt
 orthogonalization is avoided as far as possible. More specifically,
 the various steps of the algorithm are as follows.

 For each unreduced block (submatrix) of T,
    (a) Compute T - sigma I  = L D L^T, so that L and D
        define all the wanted eigenvalues to high relative accuracy.
        This means that small relative changes in the entries of D and L
        cause only small relative changes in the eigenvalues and
        eigenvectors. The standard (unfactored) representation of the
        tridiagonal matrix T does not have this property in general.
    (b) Compute the eigenvalues to suitable accuracy.
        If the eigenvectors are desired, the algorithm attains full
        accuracy of the computed eigenvalues only right before
        the corresponding vectors have to be computed, see steps c) and d).
    (c) For each cluster of close eigenvalues, select a new
        shift close to the cluster, find a new factorization, and refine
        the shifted eigenvalues to suitable accuracy.
    (d) For each eigenvalue with a large enough relative separation compute
        the corresponding eigenvector by forming a rank revealing twisted
        factorization. Go back to (c) for any clusters that remain.

 The desired accuracy of the output can be specified by the input
 parameter ABSTOL.

 For more details, see SSTEMR's documentation and:
 - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
 - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
   2004.  Also LAPACK Working Note 154.
 - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
   tridiagonal eigenvalue/eigenvector problem",
   Computer Science Division Technical Report No. UCB/CSD-97-971,
   UC Berkeley, May 1997.


 Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested
 on machines which conform to the ieee-754 floating point standard.
 SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and
 when partial spectrum requests are made.

 Normal execution of SSTEMR may create NaNs and infinities and
 hence may abort due to a floating point exception in environments
 which do not handle NaNs and infinities in the ieee standard default
 manner.
Parameters
[in]JOBZ
          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.
[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.
          For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
          SSTEIN are called
[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.

          See "Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy," by Demmel and
          Kahan, LAPACK Working Note #3.

          If high relative accuracy is important, set ABSTOL to
          SLAMCH( 'Safe minimum' ).  Doing so will guarantee that
          eigenvalues are computed to high relative accuracy when
          possible in future releases.  The current code does not
          make any guarantees about high relative accuracy, but
          future releases will. See J. Barlow and J. Demmel,
          "Computing Accurate Eigensystems of Scaled Diagonally
          Dominant Matrices", LAPACK Working Note #7, for a discussion
          of which matrices define their eigenvalues to high relative
          accuracy.
[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)
          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 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.
          Supplying N columns is always safe.
[in]LDZ
          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).
[out]ISUPPZ
          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal
          matrix). The support of the eigenvectors of A is typically
          1:N because of the orthogonal transformations applied by SORMTR.
          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
[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 dimension of the array WORK.  LWORK >= max(1,26*N).
          For optimal efficiency, LWORK >= (NB+6)*N,
          where NB is the max of the blocksize for SSYTRD and SORMTR
          returned by ILAENV.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.
[out]IWORK
          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
[in]LIWORK
          LIWORK is INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  Internal error
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 333 of file ssyevr.f.

336 *
337 * -- LAPACK driver routine --
338 * -- LAPACK is a software package provided by Univ. of Tennessee, --
339 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
340 *
341 * .. Scalar Arguments ..
342  CHARACTER JOBZ, RANGE, UPLO
343  INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
344  REAL ABSTOL, VL, VU
345 * ..
346 * .. Array Arguments ..
347  INTEGER ISUPPZ( * ), IWORK( * )
348  REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
349 * ..
350 *
351 * =====================================================================
352 *
353 * .. Parameters ..
354  REAL ZERO, ONE, TWO
355  parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0 )
356 * ..
357 * .. Local Scalars ..
358  LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
359  $ WANTZ, TRYRAC
360  CHARACTER ORDER
361  INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
362  $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
363  $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
364  $ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT
365  REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
366  $ SIGMA, SMLNUM, TMP1, VLL, VUU
367 * ..
368 * .. External Functions ..
369  LOGICAL LSAME
370  INTEGER ILAENV
371  REAL SLAMCH, SLANSY
372  EXTERNAL lsame, ilaenv, slamch, slansy
373 * ..
374 * .. External Subroutines ..
375  EXTERNAL scopy, sormtr, sscal, sstebz, sstemr, sstein,
377 * ..
378 * .. Intrinsic Functions ..
379  INTRINSIC max, min, sqrt
380 * ..
381 * .. Executable Statements ..
382 *
383 * Test the input parameters.
384 *
385  ieeeok = ilaenv( 10, 'SSYEVR', 'N', 1, 2, 3, 4 )
386 *
387  lower = lsame( uplo, 'L' )
388  wantz = lsame( jobz, 'V' )
389  alleig = lsame( range, 'A' )
390  valeig = lsame( range, 'V' )
391  indeig = lsame( range, 'I' )
392 *
393  lquery = ( ( lwork.EQ.-1 ) .OR. ( liwork.EQ.-1 ) )
394 *
395  lwmin = max( 1, 26*n )
396  liwmin = max( 1, 10*n )
397 *
398  info = 0
399  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
400  info = -1
401  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
402  info = -2
403  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
404  info = -3
405  ELSE IF( n.LT.0 ) THEN
406  info = -4
407  ELSE IF( lda.LT.max( 1, n ) ) THEN
408  info = -6
409  ELSE
410  IF( valeig ) THEN
411  IF( n.GT.0 .AND. vu.LE.vl )
412  $ info = -8
413  ELSE IF( indeig ) THEN
414  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
415  info = -9
416  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
417  info = -10
418  END IF
419  END IF
420  END IF
421  IF( info.EQ.0 ) THEN
422  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
423  info = -15
424  END IF
425  END IF
426 *
427  IF( info.EQ.0 ) THEN
428  nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
429  nb = max( nb, ilaenv( 1, 'SORMTR', uplo, n, -1, -1, -1 ) )
430  lwkopt = max( ( nb+1 )*n, lwmin )
431  work( 1 ) = lwkopt
432  iwork( 1 ) = liwmin
433 *
434  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
435  info = -18
436  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
437  info = -20
438  END IF
439  END IF
440 *
441  IF( info.NE.0 ) THEN
442  CALL xerbla( 'SSYEVR', -info )
443  RETURN
444  ELSE IF( lquery ) THEN
445  RETURN
446  END IF
447 *
448 * Quick return if possible
449 *
450  m = 0
451  IF( n.EQ.0 ) THEN
452  work( 1 ) = 1
453  RETURN
454  END IF
455 *
456  IF( n.EQ.1 ) THEN
457  work( 1 ) = 26
458  IF( alleig .OR. indeig ) THEN
459  m = 1
460  w( 1 ) = a( 1, 1 )
461  ELSE
462  IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
463  m = 1
464  w( 1 ) = a( 1, 1 )
465  END IF
466  END IF
467  IF( wantz ) THEN
468  z( 1, 1 ) = one
469  isuppz( 1 ) = 1
470  isuppz( 2 ) = 1
471  END IF
472  RETURN
473  END IF
474 *
475 * Get machine constants.
476 *
477  safmin = slamch( 'Safe minimum' )
478  eps = slamch( 'Precision' )
479  smlnum = safmin / eps
480  bignum = one / smlnum
481  rmin = sqrt( smlnum )
482  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
483 *
484 * Scale matrix to allowable range, if necessary.
485 *
486  iscale = 0
487  abstll = abstol
488  IF (valeig) THEN
489  vll = vl
490  vuu = vu
491  END IF
492  anrm = slansy( 'M', uplo, n, a, lda, work )
493  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
494  iscale = 1
495  sigma = rmin / anrm
496  ELSE IF( anrm.GT.rmax ) THEN
497  iscale = 1
498  sigma = rmax / anrm
499  END IF
500  IF( iscale.EQ.1 ) THEN
501  IF( lower ) THEN
502  DO 10 j = 1, n
503  CALL sscal( n-j+1, sigma, a( j, j ), 1 )
504  10 CONTINUE
505  ELSE
506  DO 20 j = 1, n
507  CALL sscal( j, sigma, a( 1, j ), 1 )
508  20 CONTINUE
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 * Initialize indices into workspaces. Note: The IWORK indices are
519 * used only if SSTERF or SSTEMR fail.
520 
521 * WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
522 * elementary reflectors used in SSYTRD.
523  indtau = 1
524 * WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
525  indd = indtau + n
526 * WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
527 * tridiagonal matrix from SSYTRD.
528  inde = indd + n
529 * WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
530 * -written by SSTEMR (the SSTERF path copies the diagonal to W).
531  inddd = inde + n
532 * WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
533 * -written while computing the eigenvalues in SSTERF and SSTEMR.
534  indee = inddd + n
535 * INDWK is the starting offset of the left-over workspace, and
536 * LLWORK is the remaining workspace size.
537  indwk = indee + n
538  llwork = lwork - indwk + 1
539 
540 * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
541 * stores the block indices of each of the M<=N eigenvalues.
542  indibl = 1
543 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
544 * stores the starting and finishing indices of each block.
545  indisp = indibl + n
546 * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
547 * that corresponding to eigenvectors that fail to converge in
548 * SSTEIN. This information is discarded; if any fail, the driver
549 * returns INFO > 0.
550  indifl = indisp + n
551 * INDIWO is the offset of the remaining integer workspace.
552  indiwo = indifl + n
553 
554 *
555 * Call SSYTRD to reduce symmetric matrix to tridiagonal form.
556 *
557  CALL ssytrd( uplo, n, a, lda, work( indd ), work( inde ),
558  $ work( indtau ), work( indwk ), llwork, iinfo )
559 *
560 * If all eigenvalues are desired
561 * then call SSTERF or SSTEMR and SORMTR.
562 *
563  test = .false.
564  IF( indeig ) THEN
565  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
566  test = .true.
567  END IF
568  END IF
569  IF( ( alleig.OR.test ) .AND. ( ieeeok.EQ.1 ) ) THEN
570  IF( .NOT.wantz ) THEN
571  CALL scopy( n, work( indd ), 1, w, 1 )
572  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
573  CALL ssterf( n, w, work( indee ), info )
574  ELSE
575  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
576  CALL scopy( n, work( indd ), 1, work( inddd ), 1 )
577 *
578  IF (abstol .LE. two*n*eps) THEN
579  tryrac = .true.
580  ELSE
581  tryrac = .false.
582  END IF
583  CALL sstemr( jobz, 'A', n, work( inddd ), work( indee ),
584  $ vl, vu, il, iu, m, w, z, ldz, n, isuppz,
585  $ tryrac, work( indwk ), lwork, iwork, liwork,
586  $ info )
587 *
588 *
589 *
590 * Apply orthogonal matrix used in reduction to tridiagonal
591 * form to eigenvectors returned by SSTEMR.
592 *
593  IF( wantz .AND. info.EQ.0 ) THEN
594  indwkn = inde
595  llwrkn = lwork - indwkn + 1
596  CALL sormtr( 'L', uplo, 'N', n, m, a, lda,
597  $ work( indtau ), z, ldz, work( indwkn ),
598  $ llwrkn, iinfo )
599  END IF
600  END IF
601 *
602 *
603  IF( info.EQ.0 ) THEN
604 * Everything worked. Skip SSTEBZ/SSTEIN. IWORK(:) are
605 * undefined.
606  m = n
607  GO TO 30
608  END IF
609  info = 0
610  END IF
611 *
612 * Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
613 * Also call SSTEBZ and SSTEIN if SSTEMR fails.
614 *
615  IF( wantz ) THEN
616  order = 'B'
617  ELSE
618  order = 'E'
619  END IF
620 
621  CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
622  $ work( indd ), work( inde ), m, nsplit, w,
623  $ iwork( indibl ), iwork( indisp ), work( indwk ),
624  $ iwork( indiwo ), info )
625 *
626  IF( wantz ) THEN
627  CALL sstein( n, work( indd ), work( inde ), m, w,
628  $ iwork( indibl ), iwork( indisp ), z, ldz,
629  $ work( indwk ), iwork( indiwo ), iwork( indifl ),
630  $ info )
631 *
632 * Apply orthogonal matrix used in reduction to tridiagonal
633 * form to eigenvectors returned by SSTEIN.
634 *
635  indwkn = inde
636  llwrkn = lwork - indwkn + 1
637  CALL sormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
638  $ ldz, work( indwkn ), llwrkn, iinfo )
639  END IF
640 *
641 * If matrix was scaled, then rescale eigenvalues appropriately.
642 *
643 * Jump here if SSTEMR/SSTEIN succeeded.
644  30 CONTINUE
645  IF( iscale.EQ.1 ) THEN
646  IF( info.EQ.0 ) THEN
647  imax = m
648  ELSE
649  imax = info - 1
650  END IF
651  CALL sscal( imax, one / sigma, w, 1 )
652  END IF
653 *
654 * If eigenvalues are not in order, then sort them, along with
655 * eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
656 * It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do
657 * not return this detailed information to the user.
658 *
659  IF( wantz ) THEN
660  DO 50 j = 1, m - 1
661  i = 0
662  tmp1 = w( j )
663  DO 40 jj = j + 1, m
664  IF( w( jj ).LT.tmp1 ) THEN
665  i = jj
666  tmp1 = w( jj )
667  END IF
668  40 CONTINUE
669 *
670  IF( i.NE.0 ) THEN
671  w( i ) = w( j )
672  w( j ) = tmp1
673  CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
674  END IF
675  50 CONTINUE
676  END IF
677 *
678 * Set WORK(1) to optimal workspace size.
679 *
680  work( 1 ) = lwkopt
681  iwork( 1 ) = liwmin
682 *
683  RETURN
684 *
685 * End of SSYEVR
686 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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 sstemr(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
SSTEMR
Definition: sstemr.f:321
subroutine sstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSTEIN
Definition: sstein.f:174
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(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
SSYTRD
Definition: ssytrd.f:192
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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