 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dstebz()

 subroutine dstebz ( character RANGE, character ORDER, integer N, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, double precision, dimension( * ) D, double precision, dimension( * ) E, integer M, integer NSPLIT, double precision, dimension( * ) W, integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DSTEBZ

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

Purpose:
``` DSTEBZ computes the eigenvalues of a symmetric tridiagonal
matrix T.  The user may ask for all eigenvalues, all eigenvalues
in the half-open interval (VL, VU], or the IL-th through IU-th
eigenvalues.

To avoid overflow, the matrix must be scaled so that its
largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
accuracy, it should not be much smaller than that.

See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.```
Parameters
 [in] RANGE ``` RANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.``` [in] ORDER ``` ORDER is CHARACTER*1 = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest.``` [in] N ``` N is INTEGER The order of the tridiagonal matrix T. N >= 0.``` [in] VL ``` VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. 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. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. 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 tolerance for the eigenvalues. An eigenvalue (or cluster) is considered to be located if it has been determined to lie in an interval whose width is ABSTOL or less. If ABSTOL is less than or equal to zero, then ULP*|T| will be used, where |T| means the 1-norm of T. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero.``` [in] D ``` D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T.``` [in] E ``` E is DOUBLE PRECISION array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T.``` [out] M ``` M is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)``` [out] NSPLIT ``` NSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.``` [out] W ``` W is DOUBLE PRECISION array, dimension (N) On exit, the first M elements of W will contain the eigenvalues. (DSTEBZ may use the remaining N-M elements as workspace.)``` [out] IBLOCK ``` IBLOCK is INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (DSTEBZ may use the remaining N-M elements as workspace.)``` [out] ISPLIT ``` ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.)``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (4*N)` [out] IWORK ` IWORK is INTEGER array, dimension (3*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: some or all of the eigenvalues failed to converge or were not computed: =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found. Effect: M < IU+1-IL Cause: non-monotonic arithmetic, causing the Sturm sequence to be non-monotonic. Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER "FUDGE" may make things work. = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed. Probable cause: your machine has sloppy floating-point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again.```
Internal Parameters:
```  RELFAC  DOUBLE PRECISION, default = 2.0e0
The relative tolerance.  An interval (a,b] lies within
"relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),
where "ulp" is the machine precision (distance from 1 to
the next larger floating point number.)

FUDGE   DOUBLE PRECISION, default = 2
A "fudge factor" to widen the Gershgorin intervals.  Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger.  The default for
publicly released versions should be large enough to handle
the worst machine around.  Note that this has no effect
on accuracy of the solution.```

Definition at line 270 of file dstebz.f.

273 *
274 * -- LAPACK computational routine --
275 * -- LAPACK is a software package provided by Univ. of Tennessee, --
276 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
277 *
278 * .. Scalar Arguments ..
279  CHARACTER ORDER, RANGE
280  INTEGER IL, INFO, IU, M, N, NSPLIT
281  DOUBLE PRECISION ABSTOL, VL, VU
282 * ..
283 * .. Array Arguments ..
284  INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * )
285  DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
286 * ..
287 *
288 * =====================================================================
289 *
290 * .. Parameters ..
291  DOUBLE PRECISION ZERO, ONE, TWO, HALF
292  parameter( zero = 0.0d0, one = 1.0d0, two = 2.0d0,
293  \$ half = 1.0d0 / two )
294  DOUBLE PRECISION FUDGE, RELFAC
295  parameter( fudge = 2.1d0, relfac = 2.0d0 )
296 * ..
297 * .. Local Scalars ..
298  LOGICAL NCNVRG, TOOFEW
299  INTEGER IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO,
300  \$ IM, IN, IOFF, IORDER, IOUT, IRANGE, ITMAX,
301  \$ ITMP1, IW, IWOFF, J, JB, JDISC, JE, NB, NWL,
302  \$ NWU
303  DOUBLE PRECISION ATOLI, BNORM, GL, GU, PIVMIN, RTOLI, SAFEMN,
304  \$ TMP1, TMP2, TNORM, ULP, WKILL, WL, WLU, WU, WUL
305 * ..
306 * .. Local Arrays ..
307  INTEGER IDUMMA( 1 )
308 * ..
309 * .. External Functions ..
310  LOGICAL LSAME
311  INTEGER ILAENV
312  DOUBLE PRECISION DLAMCH
313  EXTERNAL lsame, ilaenv, dlamch
314 * ..
315 * .. External Subroutines ..
316  EXTERNAL dlaebz, xerbla
317 * ..
318 * .. Intrinsic Functions ..
319  INTRINSIC abs, int, log, max, min, sqrt
320 * ..
321 * .. Executable Statements ..
322 *
323  info = 0
324 *
325 * Decode RANGE
326 *
327  IF( lsame( range, 'A' ) ) THEN
328  irange = 1
329  ELSE IF( lsame( range, 'V' ) ) THEN
330  irange = 2
331  ELSE IF( lsame( range, 'I' ) ) THEN
332  irange = 3
333  ELSE
334  irange = 0
335  END IF
336 *
337 * Decode ORDER
338 *
339  IF( lsame( order, 'B' ) ) THEN
340  iorder = 2
341  ELSE IF( lsame( order, 'E' ) ) THEN
342  iorder = 1
343  ELSE
344  iorder = 0
345  END IF
346 *
347 * Check for Errors
348 *
349  IF( irange.LE.0 ) THEN
350  info = -1
351  ELSE IF( iorder.LE.0 ) THEN
352  info = -2
353  ELSE IF( n.LT.0 ) THEN
354  info = -3
355  ELSE IF( irange.EQ.2 ) THEN
356  IF( vl.GE.vu )
357  \$ info = -5
358  ELSE IF( irange.EQ.3 .AND. ( il.LT.1 .OR. il.GT.max( 1, n ) ) )
359  \$ THEN
360  info = -6
361  ELSE IF( irange.EQ.3 .AND. ( iu.LT.min( n, il ) .OR. iu.GT.n ) )
362  \$ THEN
363  info = -7
364  END IF
365 *
366  IF( info.NE.0 ) THEN
367  CALL xerbla( 'DSTEBZ', -info )
368  RETURN
369  END IF
370 *
371 * Initialize error flags
372 *
373  info = 0
374  ncnvrg = .false.
375  toofew = .false.
376 *
377 * Quick return if possible
378 *
379  m = 0
380  IF( n.EQ.0 )
381  \$ RETURN
382 *
383 * Simplifications:
384 *
385  IF( irange.EQ.3 .AND. il.EQ.1 .AND. iu.EQ.n )
386  \$ irange = 1
387 *
388 * Get machine constants
389 * NB is the minimum vector length for vector bisection, or 0
390 * if only scalar is to be done.
391 *
392  safemn = dlamch( 'S' )
393  ulp = dlamch( 'P' )
394  rtoli = ulp*relfac
395  nb = ilaenv( 1, 'DSTEBZ', ' ', n, -1, -1, -1 )
396  IF( nb.LE.1 )
397  \$ nb = 0
398 *
399 * Special Case when N=1
400 *
401  IF( n.EQ.1 ) THEN
402  nsplit = 1
403  isplit( 1 ) = 1
404  IF( irange.EQ.2 .AND. ( vl.GE.d( 1 ) .OR. vu.LT.d( 1 ) ) ) THEN
405  m = 0
406  ELSE
407  w( 1 ) = d( 1 )
408  iblock( 1 ) = 1
409  m = 1
410  END IF
411  RETURN
412  END IF
413 *
414 * Compute Splitting Points
415 *
416  nsplit = 1
417  work( n ) = zero
418  pivmin = one
419 *
420  DO 10 j = 2, n
421  tmp1 = e( j-1 )**2
422  IF( abs( d( j )*d( j-1 ) )*ulp**2+safemn.GT.tmp1 ) THEN
423  isplit( nsplit ) = j - 1
424  nsplit = nsplit + 1
425  work( j-1 ) = zero
426  ELSE
427  work( j-1 ) = tmp1
428  pivmin = max( pivmin, tmp1 )
429  END IF
430  10 CONTINUE
431  isplit( nsplit ) = n
432  pivmin = pivmin*safemn
433 *
434 * Compute Interval and ATOLI
435 *
436  IF( irange.EQ.3 ) THEN
437 *
438 * RANGE='I': Compute the interval containing eigenvalues
439 * IL through IU.
440 *
441 * Compute Gershgorin interval for entire (split) matrix
442 * and use it as the initial interval
443 *
444  gu = d( 1 )
445  gl = d( 1 )
446  tmp1 = zero
447 *
448  DO 20 j = 1, n - 1
449  tmp2 = sqrt( work( j ) )
450  gu = max( gu, d( j )+tmp1+tmp2 )
451  gl = min( gl, d( j )-tmp1-tmp2 )
452  tmp1 = tmp2
453  20 CONTINUE
454 *
455  gu = max( gu, d( n )+tmp1 )
456  gl = min( gl, d( n )-tmp1 )
457  tnorm = max( abs( gl ), abs( gu ) )
458  gl = gl - fudge*tnorm*ulp*n - fudge*two*pivmin
459  gu = gu + fudge*tnorm*ulp*n + fudge*pivmin
460 *
461 * Compute Iteration parameters
462 *
463  itmax = int( ( log( tnorm+pivmin )-log( pivmin ) ) /
464  \$ log( two ) ) + 2
465  IF( abstol.LE.zero ) THEN
466  atoli = ulp*tnorm
467  ELSE
468  atoli = abstol
469  END IF
470 *
471  work( n+1 ) = gl
472  work( n+2 ) = gl
473  work( n+3 ) = gu
474  work( n+4 ) = gu
475  work( n+5 ) = gl
476  work( n+6 ) = gu
477  iwork( 1 ) = -1
478  iwork( 2 ) = -1
479  iwork( 3 ) = n + 1
480  iwork( 4 ) = n + 1
481  iwork( 5 ) = il - 1
482  iwork( 6 ) = iu
483 *
484  CALL dlaebz( 3, itmax, n, 2, 2, nb, atoli, rtoli, pivmin, d, e,
485  \$ work, iwork( 5 ), work( n+1 ), work( n+5 ), iout,
486  \$ iwork, w, iblock, iinfo )
487 *
488  IF( iwork( 6 ).EQ.iu ) THEN
489  wl = work( n+1 )
490  wlu = work( n+3 )
491  nwl = iwork( 1 )
492  wu = work( n+4 )
493  wul = work( n+2 )
494  nwu = iwork( 4 )
495  ELSE
496  wl = work( n+2 )
497  wlu = work( n+4 )
498  nwl = iwork( 2 )
499  wu = work( n+3 )
500  wul = work( n+1 )
501  nwu = iwork( 3 )
502  END IF
503 *
504  IF( nwl.LT.0 .OR. nwl.GE.n .OR. nwu.LT.1 .OR. nwu.GT.n ) THEN
505  info = 4
506  RETURN
507  END IF
508  ELSE
509 *
510 * RANGE='A' or 'V' -- Set ATOLI
511 *
512  tnorm = max( abs( d( 1 ) )+abs( e( 1 ) ),
513  \$ abs( d( n ) )+abs( e( n-1 ) ) )
514 *
515  DO 30 j = 2, n - 1
516  tnorm = max( tnorm, abs( d( j ) )+abs( e( j-1 ) )+
517  \$ abs( e( j ) ) )
518  30 CONTINUE
519 *
520  IF( abstol.LE.zero ) THEN
521  atoli = ulp*tnorm
522  ELSE
523  atoli = abstol
524  END IF
525 *
526  IF( irange.EQ.2 ) THEN
527  wl = vl
528  wu = vu
529  ELSE
530  wl = zero
531  wu = zero
532  END IF
533  END IF
534 *
535 * Find Eigenvalues -- Loop Over Blocks and recompute NWL and NWU.
536 * NWL accumulates the number of eigenvalues .le. WL,
537 * NWU accumulates the number of eigenvalues .le. WU
538 *
539  m = 0
540  iend = 0
541  info = 0
542  nwl = 0
543  nwu = 0
544 *
545  DO 70 jb = 1, nsplit
546  ioff = iend
547  ibegin = ioff + 1
548  iend = isplit( jb )
549  in = iend - ioff
550 *
551  IF( in.EQ.1 ) THEN
552 *
553 * Special Case -- IN=1
554 *
555  IF( irange.EQ.1 .OR. wl.GE.d( ibegin )-pivmin )
556  \$ nwl = nwl + 1
557  IF( irange.EQ.1 .OR. wu.GE.d( ibegin )-pivmin )
558  \$ nwu = nwu + 1
559  IF( irange.EQ.1 .OR. ( wl.LT.d( ibegin )-pivmin .AND. wu.GE.
560  \$ d( ibegin )-pivmin ) ) THEN
561  m = m + 1
562  w( m ) = d( ibegin )
563  iblock( m ) = jb
564  END IF
565  ELSE
566 *
567 * General Case -- IN > 1
568 *
569 * Compute Gershgorin Interval
570 * and use it as the initial interval
571 *
572  gu = d( ibegin )
573  gl = d( ibegin )
574  tmp1 = zero
575 *
576  DO 40 j = ibegin, iend - 1
577  tmp2 = abs( e( j ) )
578  gu = max( gu, d( j )+tmp1+tmp2 )
579  gl = min( gl, d( j )-tmp1-tmp2 )
580  tmp1 = tmp2
581  40 CONTINUE
582 *
583  gu = max( gu, d( iend )+tmp1 )
584  gl = min( gl, d( iend )-tmp1 )
585  bnorm = max( abs( gl ), abs( gu ) )
586  gl = gl - fudge*bnorm*ulp*in - fudge*pivmin
587  gu = gu + fudge*bnorm*ulp*in + fudge*pivmin
588 *
589 * Compute ATOLI for the current submatrix
590 *
591  IF( abstol.LE.zero ) THEN
592  atoli = ulp*max( abs( gl ), abs( gu ) )
593  ELSE
594  atoli = abstol
595  END IF
596 *
597  IF( irange.GT.1 ) THEN
598  IF( gu.LT.wl ) THEN
599  nwl = nwl + in
600  nwu = nwu + in
601  GO TO 70
602  END IF
603  gl = max( gl, wl )
604  gu = min( gu, wu )
605  IF( gl.GE.gu )
606  \$ GO TO 70
607  END IF
608 *
609 * Set Up Initial Interval
610 *
611  work( n+1 ) = gl
612  work( n+in+1 ) = gu
613  CALL dlaebz( 1, 0, in, in, 1, nb, atoli, rtoli, pivmin,
614  \$ d( ibegin ), e( ibegin ), work( ibegin ),
615  \$ idumma, work( n+1 ), work( n+2*in+1 ), im,
616  \$ iwork, w( m+1 ), iblock( m+1 ), iinfo )
617 *
618  nwl = nwl + iwork( 1 )
619  nwu = nwu + iwork( in+1 )
620  iwoff = m - iwork( 1 )
621 *
622 * Compute Eigenvalues
623 *
624  itmax = int( ( log( gu-gl+pivmin )-log( pivmin ) ) /
625  \$ log( two ) ) + 2
626  CALL dlaebz( 2, itmax, in, in, 1, nb, atoli, rtoli, pivmin,
627  \$ d( ibegin ), e( ibegin ), work( ibegin ),
628  \$ idumma, work( n+1 ), work( n+2*in+1 ), iout,
629  \$ iwork, w( m+1 ), iblock( m+1 ), iinfo )
630 *
631 * Copy Eigenvalues Into W and IBLOCK
632 * Use -JB for block number for unconverged eigenvalues.
633 *
634  DO 60 j = 1, iout
635  tmp1 = half*( work( j+n )+work( j+in+n ) )
636 *
637 * Flag non-convergence.
638 *
639  IF( j.GT.iout-iinfo ) THEN
640  ncnvrg = .true.
641  ib = -jb
642  ELSE
643  ib = jb
644  END IF
645  DO 50 je = iwork( j ) + 1 + iwoff,
646  \$ iwork( j+in ) + iwoff
647  w( je ) = tmp1
648  iblock( je ) = ib
649  50 CONTINUE
650  60 CONTINUE
651 *
652  m = m + im
653  END IF
654  70 CONTINUE
655 *
656 * If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
657 * If NWL+1 < IL or NWU > IU, discard extra eigenvalues.
658 *
659  IF( irange.EQ.3 ) THEN
660  im = 0
661  idiscl = il - 1 - nwl
662  idiscu = nwu - iu
663 *
664  IF( idiscl.GT.0 .OR. idiscu.GT.0 ) THEN
665  DO 80 je = 1, m
666  IF( w( je ).LE.wlu .AND. idiscl.GT.0 ) THEN
667  idiscl = idiscl - 1
668  ELSE IF( w( je ).GE.wul .AND. idiscu.GT.0 ) THEN
669  idiscu = idiscu - 1
670  ELSE
671  im = im + 1
672  w( im ) = w( je )
673  iblock( im ) = iblock( je )
674  END IF
675  80 CONTINUE
676  m = im
677  END IF
678  IF( idiscl.GT.0 .OR. idiscu.GT.0 ) THEN
679 *
680 * Code to deal with effects of bad arithmetic:
681 * Some low eigenvalues to be discarded are not in (WL,WLU],
682 * or high eigenvalues to be discarded are not in (WUL,WU]
683 * so just kill off the smallest IDISCL/largest IDISCU
684 * eigenvalues, by simply finding the smallest/largest
685 * eigenvalue(s).
686 *
687 * (If N(w) is monotone non-decreasing, this should never
688 * happen.)
689 *
690  IF( idiscl.GT.0 ) THEN
691  wkill = wu
692  DO 100 jdisc = 1, idiscl
693  iw = 0
694  DO 90 je = 1, m
695  IF( iblock( je ).NE.0 .AND.
696  \$ ( w( je ).LT.wkill .OR. iw.EQ.0 ) ) THEN
697  iw = je
698  wkill = w( je )
699  END IF
700  90 CONTINUE
701  iblock( iw ) = 0
702  100 CONTINUE
703  END IF
704  IF( idiscu.GT.0 ) THEN
705 *
706  wkill = wl
707  DO 120 jdisc = 1, idiscu
708  iw = 0
709  DO 110 je = 1, m
710  IF( iblock( je ).NE.0 .AND.
711  \$ ( w( je ).GT.wkill .OR. iw.EQ.0 ) ) THEN
712  iw = je
713  wkill = w( je )
714  END IF
715  110 CONTINUE
716  iblock( iw ) = 0
717  120 CONTINUE
718  END IF
719  im = 0
720  DO 130 je = 1, m
721  IF( iblock( je ).NE.0 ) THEN
722  im = im + 1
723  w( im ) = w( je )
724  iblock( im ) = iblock( je )
725  END IF
726  130 CONTINUE
727  m = im
728  END IF
729  IF( idiscl.LT.0 .OR. idiscu.LT.0 ) THEN
730  toofew = .true.
731  END IF
732  END IF
733 *
734 * If ORDER='B', do nothing -- the eigenvalues are already sorted
735 * by block.
736 * If ORDER='E', sort the eigenvalues from smallest to largest
737 *
738  IF( iorder.EQ.1 .AND. nsplit.GT.1 ) THEN
739  DO 150 je = 1, m - 1
740  ie = 0
741  tmp1 = w( je )
742  DO 140 j = je + 1, m
743  IF( w( j ).LT.tmp1 ) THEN
744  ie = j
745  tmp1 = w( j )
746  END IF
747  140 CONTINUE
748 *
749  IF( ie.NE.0 ) THEN
750  itmp1 = iblock( ie )
751  w( ie ) = w( je )
752  iblock( ie ) = iblock( je )
753  w( je ) = tmp1
754  iblock( je ) = itmp1
755  END IF
756  150 CONTINUE
757  END IF
758 *
759  info = 0
760  IF( ncnvrg )
761  \$ info = info + 1
762  IF( toofew )
763  \$ info = info + 2
764  RETURN
765 *
766 * End of DSTEBZ
767 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlaebz(IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO)
DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than ...
Definition: dlaebz.f:319
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
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