 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine dlaebz ( integer IJOB, integer NITMAX, integer N, integer MMAX, integer MINP, integer NBMIN, double precision ABSTOL, double precision RELTOL, double precision PIVMIN, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) E2, integer, dimension( * ) NVAL, double precision, dimension( mmax, * ) AB, double precision, dimension( * ) C, integer MOUT, integer, dimension( mmax, * ) NAB, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.

Purpose:
DLAEBZ contains the iteration loops which compute and use the
function N(w), which is the count of eigenvalues of a symmetric
tridiagonal matrix T less than or equal to its argument  w.  It
performs a choice of two types of loops:

IJOB=1, followed by
IJOB=2: It takes as input a list of intervals and returns a list of
sufficiently small intervals whose union contains the same
eigenvalues as the union of the original intervals.
The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
The output interval (AB(j,1),AB(j,2)] will contain
eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.

IJOB=3: It performs a binary search in each input interval
(AB(j,1),AB(j,2)] for a point  w(j)  such that
N(w(j))=NVAL(j), and uses  C(j)  as the starting point of
the search.  If such a w(j) is found, then on output
AB(j,1)=AB(j,2)=w.  If no such w(j) is found, then on output
(AB(j,1),AB(j,2)] will be a small interval containing the
point where N(w) jumps through NVAL(j), unless that point
lies outside the initial interval.

Note that the intervals are in all cases half-open intervals,
i.e., of the form  (a,b] , which includes  b  but not  a .

To avoid underflow, the matrix should be scaled so that its largest
element is no greater than  overflow**(1/2) * underflow**(1/4)
in absolute value.  To assure the most accurate computation
of small eigenvalues, the matrix should be scaled to be
not much smaller than that, either.

See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford
University, July 21, 1966

Note: the arguments are, in general, *not* checked for unreasonable
values.
Parameters
 [in] IJOB IJOB is INTEGER Specifies what is to be done: = 1: Compute NAB for the initial intervals. = 2: Perform bisection iteration to find eigenvalues of T. = 3: Perform bisection iteration to invert N(w), i.e., to find a point which has a specified number of eigenvalues of T to its left. Other values will cause DLAEBZ to return with INFO=-1. [in] NITMAX NITMAX is INTEGER The maximum number of "levels" of bisection to be performed, i.e., an interval of width W will not be made smaller than 2^(-NITMAX) * W. If not all intervals have converged after NITMAX iterations, then INFO is set to the number of non-converged intervals. [in] N N is INTEGER The dimension n of the tridiagonal matrix T. It must be at least 1. [in] MMAX MMAX is INTEGER The maximum number of intervals. If more than MMAX intervals are generated, then DLAEBZ will quit with INFO=MMAX+1. [in] MINP MINP is INTEGER The initial number of intervals. It may not be greater than MMAX. [in] NBMIN NBMIN is INTEGER The smallest number of intervals that should be processed using a vector loop. If zero, then only the scalar loop will be used. [in] ABSTOL ABSTOL is DOUBLE PRECISION The minimum (absolute) width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. This must be at least zero. [in] RELTOL RELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon. [in] PIVMIN PIVMIN is DOUBLE PRECISION The minimum absolute value of a "pivot" in the Sturm sequence loop. This must be at least max |e(j)**2|*safe_min and at least safe_min, where safe_min is at least the smallest number that can divide one without overflow. [in] D D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T. [in] E E is DOUBLE PRECISION array, dimension (N) The offdiagonal elements of the tridiagonal matrix T in positions 1 through N-1. E(N) is arbitrary. [in] E2 E2 is DOUBLE PRECISION array, dimension (N) The squares of the offdiagonal elements of the tridiagonal matrix T. E2(N) is ignored. [in,out] NVAL NVAL is INTEGER array, dimension (MINP) If IJOB=1 or 2, not referenced. If IJOB=3, the desired values of N(w). The elements of NVAL will be reordered to correspond with the intervals in AB. Thus, NVAL(j) on output will not, in general be the same as NVAL(j) on input, but it will correspond with the interval (AB(j,1),AB(j,2)] on output. [in,out] AB AB is DOUBLE PRECISION array, dimension (MMAX,2) The endpoints of the intervals. AB(j,1) is a(j), the left endpoint of the j-th interval, and AB(j,2) is b(j), the right endpoint of the j-th interval. The input intervals will, in general, be modified, split, and reordered by the calculation. [in,out] C C is DOUBLE PRECISION array, dimension (MMAX) If IJOB=1, ignored. If IJOB=2, workspace. If IJOB=3, then on input C(j) should be initialized to the first search point in the binary search. [out] MOUT MOUT is INTEGER If IJOB=1, the number of eigenvalues in the intervals. If IJOB=2 or 3, the number of intervals output. If IJOB=3, MOUT will equal MINP. [in,out] NAB NAB is INTEGER array, dimension (MMAX,2) If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). If IJOB=2, then on input, NAB(i,j) should be set. It must satisfy the condition: N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), which means that in interval i only eigenvalues NAB(i,1)+1,...,NAB(i,2) will be considered. Usually, NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with IJOB=1. On output, NAB(i,j) will contain max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of the input interval that the output interval (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the the input values of NAB(k,1) and NAB(k,2). If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), unless N(w) > NVAL(i) for all search points w , in which case NAB(i,1) will not be modified, i.e., the output value will be the same as the input value (modulo reorderings -- see NVAL and AB), or unless N(w) < NVAL(i) for all search points w , in which case NAB(i,2) will not be modified. Normally, NAB should be set to some distinctive value(s) before DLAEBZ is called. [out] WORK WORK is DOUBLE PRECISION array, dimension (MMAX) Workspace. [out] IWORK IWORK is INTEGER array, dimension (MMAX) Workspace. [out] INFO INFO is INTEGER = 0: All intervals converged. = 1--MMAX: The last INFO intervals did not converge. = MMAX+1: More than MMAX intervals were generated.
Date
September 2012
Further Details:
This routine is intended to be called only by other LAPACK
routines, thus the interface is less user-friendly.  It is intended
for two purposes:

(a) finding eigenvalues.  In this case, DLAEBZ should have one or
more initial intervals set up in AB, and DLAEBZ should be called
with IJOB=1.  This sets up NAB, and also counts the eigenvalues.
Intervals with no eigenvalues would usually be thrown out at
this point.  Also, if not all the eigenvalues in an interval i
are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
eigenvalue.  DLAEBZ is then called with IJOB=2 and MMAX
no smaller than the value of MOUT returned by the call with
IJOB=1.  After this (IJOB=2) call, eigenvalues NAB(i,1)+1
through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
tolerance specified by ABSTOL and RELTOL.

(b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
AB to contain 2 search intervals, both initially (a,b).  One
NVAL element should contain  f-1  and the other should contain  l
, while C should contain a and b, resp.  NAB(i,1) should be -1
and NAB(i,2) should be N+1, to flag an error if the desired
interval does not lie in (a,b).  DLAEBZ is then called with
IJOB=3.  On exit, if w(f-1) < w(f), then one of the intervals --
j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
>= 0, then the interval will have  N(AB(j,1))=NAB(j,1)=f-k and
N(AB(j,2))=NAB(j,2)=f+r.  The cases w(l) < w(l+1) and
w(l-r)=...=w(l+k) are handled similarly.

Definition at line 321 of file dlaebz.f.

321 *
322 * -- LAPACK auxiliary routine (version 3.4.2) --
323 * -- LAPACK is a software package provided by Univ. of Tennessee, --
324 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
325 * September 2012
326 *
327 * .. Scalar Arguments ..
328  INTEGER ijob, info, minp, mmax, mout, n, nbmin, nitmax
329  DOUBLE PRECISION abstol, pivmin, reltol
330 * ..
331 * .. Array Arguments ..
332  INTEGER iwork( * ), nab( mmax, * ), nval( * )
333  DOUBLE PRECISION ab( mmax, * ), c( * ), d( * ), e( * ), e2( * ),
334  \$ work( * )
335 * ..
336 *
337 * =====================================================================
338 *
339 * .. Parameters ..
340  DOUBLE PRECISION zero, two, half
341  parameter ( zero = 0.0d0, two = 2.0d0,
342  \$ half = 1.0d0 / two )
343 * ..
344 * .. Local Scalars ..
345  INTEGER itmp1, itmp2, j, ji, jit, jp, kf, kfnew, kl,
346  \$ klnew
347  DOUBLE PRECISION tmp1, tmp2
348 * ..
349 * .. Intrinsic Functions ..
350  INTRINSIC abs, max, min
351 * ..
352 * .. Executable Statements ..
353 *
354 * Check for Errors
355 *
356  info = 0
357  IF( ijob.LT.1 .OR. ijob.GT.3 ) THEN
358  info = -1
359  RETURN
360  END IF
361 *
362 * Initialize NAB
363 *
364  IF( ijob.EQ.1 ) THEN
365 *
366 * Compute the number of eigenvalues in the initial intervals.
367 *
368  mout = 0
369  DO 30 ji = 1, minp
370  DO 20 jp = 1, 2
371  tmp1 = d( 1 ) - ab( ji, jp )
372  IF( abs( tmp1 ).LT.pivmin )
373  \$ tmp1 = -pivmin
374  nab( ji, jp ) = 0
375  IF( tmp1.LE.zero )
376  \$ nab( ji, jp ) = 1
377 *
378  DO 10 j = 2, n
379  tmp1 = d( j ) - e2( j-1 ) / tmp1 - ab( ji, jp )
380  IF( abs( tmp1 ).LT.pivmin )
381  \$ tmp1 = -pivmin
382  IF( tmp1.LE.zero )
383  \$ nab( ji, jp ) = nab( ji, jp ) + 1
384  10 CONTINUE
385  20 CONTINUE
386  mout = mout + nab( ji, 2 ) - nab( ji, 1 )
387  30 CONTINUE
388  RETURN
389  END IF
390 *
391 * Initialize for loop
392 *
393 * KF and KL have the following meaning:
394 * Intervals 1,...,KF-1 have converged.
395 * Intervals KF,...,KL still need to be refined.
396 *
397  kf = 1
398  kl = minp
399 *
400 * If IJOB=2, initialize C.
401 * If IJOB=3, use the user-supplied starting point.
402 *
403  IF( ijob.EQ.2 ) THEN
404  DO 40 ji = 1, minp
405  c( ji ) = half*( ab( ji, 1 )+ab( ji, 2 ) )
406  40 CONTINUE
407  END IF
408 *
409 * Iteration loop
410 *
411  DO 130 jit = 1, nitmax
412 *
413 * Loop over intervals
414 *
415  IF( kl-kf+1.GE.nbmin .AND. nbmin.GT.0 ) THEN
416 *
417 * Begin of Parallel Version of the loop
418 *
419  DO 60 ji = kf, kl
420 *
421 * Compute N(c), the number of eigenvalues less than c
422 *
423  work( ji ) = d( 1 ) - c( ji )
424  iwork( ji ) = 0
425  IF( work( ji ).LE.pivmin ) THEN
426  iwork( ji ) = 1
427  work( ji ) = min( work( ji ), -pivmin )
428  END IF
429 *
430  DO 50 j = 2, n
431  work( ji ) = d( j ) - e2( j-1 ) / work( ji ) - c( ji )
432  IF( work( ji ).LE.pivmin ) THEN
433  iwork( ji ) = iwork( ji ) + 1
434  work( ji ) = min( work( ji ), -pivmin )
435  END IF
436  50 CONTINUE
437  60 CONTINUE
438 *
439  IF( ijob.LE.2 ) THEN
440 *
441 * IJOB=2: Choose all intervals containing eigenvalues.
442 *
443  klnew = kl
444  DO 70 ji = kf, kl
445 *
446 * Insure that N(w) is monotone
447 *
448  iwork( ji ) = min( nab( ji, 2 ),
449  \$ max( nab( ji, 1 ), iwork( ji ) ) )
450 *
451 * Update the Queue -- add intervals if both halves
452 * contain eigenvalues.
453 *
454  IF( iwork( ji ).EQ.nab( ji, 2 ) ) THEN
455 *
456 * No eigenvalue in the upper interval:
457 * just use the lower interval.
458 *
459  ab( ji, 2 ) = c( ji )
460 *
461  ELSE IF( iwork( ji ).EQ.nab( ji, 1 ) ) THEN
462 *
463 * No eigenvalue in the lower interval:
464 * just use the upper interval.
465 *
466  ab( ji, 1 ) = c( ji )
467  ELSE
468  klnew = klnew + 1
469  IF( klnew.LE.mmax ) THEN
470 *
471 * Eigenvalue in both intervals -- add upper to
472 * queue.
473 *
474  ab( klnew, 2 ) = ab( ji, 2 )
475  nab( klnew, 2 ) = nab( ji, 2 )
476  ab( klnew, 1 ) = c( ji )
477  nab( klnew, 1 ) = iwork( ji )
478  ab( ji, 2 ) = c( ji )
479  nab( ji, 2 ) = iwork( ji )
480  ELSE
481  info = mmax + 1
482  END IF
483  END IF
484  70 CONTINUE
485  IF( info.NE.0 )
486  \$ RETURN
487  kl = klnew
488  ELSE
489 *
490 * IJOB=3: Binary search. Keep only the interval containing
491 * w s.t. N(w) = NVAL
492 *
493  DO 80 ji = kf, kl
494  IF( iwork( ji ).LE.nval( ji ) ) THEN
495  ab( ji, 1 ) = c( ji )
496  nab( ji, 1 ) = iwork( ji )
497  END IF
498  IF( iwork( ji ).GE.nval( ji ) ) THEN
499  ab( ji, 2 ) = c( ji )
500  nab( ji, 2 ) = iwork( ji )
501  END IF
502  80 CONTINUE
503  END IF
504 *
505  ELSE
506 *
507 * End of Parallel Version of the loop
508 *
509 * Begin of Serial Version of the loop
510 *
511  klnew = kl
512  DO 100 ji = kf, kl
513 *
514 * Compute N(w), the number of eigenvalues less than w
515 *
516  tmp1 = c( ji )
517  tmp2 = d( 1 ) - tmp1
518  itmp1 = 0
519  IF( tmp2.LE.pivmin ) THEN
520  itmp1 = 1
521  tmp2 = min( tmp2, -pivmin )
522  END IF
523 *
524  DO 90 j = 2, n
525  tmp2 = d( j ) - e2( j-1 ) / tmp2 - tmp1
526  IF( tmp2.LE.pivmin ) THEN
527  itmp1 = itmp1 + 1
528  tmp2 = min( tmp2, -pivmin )
529  END IF
530  90 CONTINUE
531 *
532  IF( ijob.LE.2 ) THEN
533 *
534 * IJOB=2: Choose all intervals containing eigenvalues.
535 *
536 * Insure that N(w) is monotone
537 *
538  itmp1 = min( nab( ji, 2 ),
539  \$ max( nab( ji, 1 ), itmp1 ) )
540 *
541 * Update the Queue -- add intervals if both halves
542 * contain eigenvalues.
543 *
544  IF( itmp1.EQ.nab( ji, 2 ) ) THEN
545 *
546 * No eigenvalue in the upper interval:
547 * just use the lower interval.
548 *
549  ab( ji, 2 ) = tmp1
550 *
551  ELSE IF( itmp1.EQ.nab( ji, 1 ) ) THEN
552 *
553 * No eigenvalue in the lower interval:
554 * just use the upper interval.
555 *
556  ab( ji, 1 ) = tmp1
557  ELSE IF( klnew.LT.mmax ) THEN
558 *
559 * Eigenvalue in both intervals -- add upper to queue.
560 *
561  klnew = klnew + 1
562  ab( klnew, 2 ) = ab( ji, 2 )
563  nab( klnew, 2 ) = nab( ji, 2 )
564  ab( klnew, 1 ) = tmp1
565  nab( klnew, 1 ) = itmp1
566  ab( ji, 2 ) = tmp1
567  nab( ji, 2 ) = itmp1
568  ELSE
569  info = mmax + 1
570  RETURN
571  END IF
572  ELSE
573 *
574 * IJOB=3: Binary search. Keep only the interval
575 * containing w s.t. N(w) = NVAL
576 *
577  IF( itmp1.LE.nval( ji ) ) THEN
578  ab( ji, 1 ) = tmp1
579  nab( ji, 1 ) = itmp1
580  END IF
581  IF( itmp1.GE.nval( ji ) ) THEN
582  ab( ji, 2 ) = tmp1
583  nab( ji, 2 ) = itmp1
584  END IF
585  END IF
586  100 CONTINUE
587  kl = klnew
588 *
589  END IF
590 *
591 * Check for convergence
592 *
593  kfnew = kf
594  DO 110 ji = kf, kl
595  tmp1 = abs( ab( ji, 2 )-ab( ji, 1 ) )
596  tmp2 = max( abs( ab( ji, 2 ) ), abs( ab( ji, 1 ) ) )
597  IF( tmp1.LT.max( abstol, pivmin, reltol*tmp2 ) .OR.
598  \$ nab( ji, 1 ).GE.nab( ji, 2 ) ) THEN
599 *
600 * Converged -- Swap with position KFNEW,
601 * then increment KFNEW
602 *
603  IF( ji.GT.kfnew ) THEN
604  tmp1 = ab( ji, 1 )
605  tmp2 = ab( ji, 2 )
606  itmp1 = nab( ji, 1 )
607  itmp2 = nab( ji, 2 )
608  ab( ji, 1 ) = ab( kfnew, 1 )
609  ab( ji, 2 ) = ab( kfnew, 2 )
610  nab( ji, 1 ) = nab( kfnew, 1 )
611  nab( ji, 2 ) = nab( kfnew, 2 )
612  ab( kfnew, 1 ) = tmp1
613  ab( kfnew, 2 ) = tmp2
614  nab( kfnew, 1 ) = itmp1
615  nab( kfnew, 2 ) = itmp2
616  IF( ijob.EQ.3 ) THEN
617  itmp1 = nval( ji )
618  nval( ji ) = nval( kfnew )
619  nval( kfnew ) = itmp1
620  END IF
621  END IF
622  kfnew = kfnew + 1
623  END IF
624  110 CONTINUE
625  kf = kfnew
626 *
627 * Choose Midpoints
628 *
629  DO 120 ji = kf, kl
630  c( ji ) = half*( ab( ji, 1 )+ab( ji, 2 ) )
631  120 CONTINUE
632 *
633 * If no more intervals to refine, quit.
634 *
635  IF( kf.GT.kl )
636  \$ GO TO 140
637  130 CONTINUE
638 *
639 * Converged
640 *
641  140 CONTINUE
642  info = max( kl+1-kf, 0 )
643  mout = kl
644 *
645  RETURN
646 *
647 * End of DLAEBZ
648 *

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