LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dlaebz()

 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.```
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 316 of file dlaebz.f.

319*
320* -- LAPACK auxiliary routine --
321* -- LAPACK is a software package provided by Univ. of Tennessee, --
322* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
323*
324* .. Scalar Arguments ..
325 INTEGER IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
326 DOUBLE PRECISION ABSTOL, PIVMIN, RELTOL
327* ..
328* .. Array Arguments ..
329 INTEGER IWORK( * ), NAB( MMAX, * ), NVAL( * )
330 DOUBLE PRECISION AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
331 \$ WORK( * )
332* ..
333*
334* =====================================================================
335*
336* .. Parameters ..
337 DOUBLE PRECISION ZERO, TWO, HALF
338 parameter( zero = 0.0d0, two = 2.0d0,
339 \$ half = 1.0d0 / two )
340* ..
341* .. Local Scalars ..
342 INTEGER ITMP1, ITMP2, J, JI, JIT, JP, KF, KFNEW, KL,
343 \$ KLNEW
344 DOUBLE PRECISION TMP1, TMP2
345* ..
346* .. Intrinsic Functions ..
347 INTRINSIC abs, max, min
348* ..
349* .. Executable Statements ..
350*
351* Check for Errors
352*
353 info = 0
354 IF( ijob.LT.1 .OR. ijob.GT.3 ) THEN
355 info = -1
356 RETURN
357 END IF
358*
359* Initialize NAB
360*
361 IF( ijob.EQ.1 ) THEN
362*
363* Compute the number of eigenvalues in the initial intervals.
364*
365 mout = 0
366 DO 30 ji = 1, minp
367 DO 20 jp = 1, 2
368 tmp1 = d( 1 ) - ab( ji, jp )
369 IF( abs( tmp1 ).LT.pivmin )
370 \$ tmp1 = -pivmin
371 nab( ji, jp ) = 0
372 IF( tmp1.LE.zero )
373 \$ nab( ji, jp ) = 1
374*
375 DO 10 j = 2, n
376 tmp1 = d( j ) - e2( j-1 ) / tmp1 - ab( ji, jp )
377 IF( abs( tmp1 ).LT.pivmin )
378 \$ tmp1 = -pivmin
379 IF( tmp1.LE.zero )
380 \$ nab( ji, jp ) = nab( ji, jp ) + 1
381 10 CONTINUE
382 20 CONTINUE
383 mout = mout + nab( ji, 2 ) - nab( ji, 1 )
384 30 CONTINUE
385 RETURN
386 END IF
387*
388* Initialize for loop
389*
390* KF and KL have the following meaning:
391* Intervals 1,...,KF-1 have converged.
392* Intervals KF,...,KL still need to be refined.
393*
394 kf = 1
395 kl = minp
396*
397* If IJOB=2, initialize C.
398* If IJOB=3, use the user-supplied starting point.
399*
400 IF( ijob.EQ.2 ) THEN
401 DO 40 ji = 1, minp
402 c( ji ) = half*( ab( ji, 1 )+ab( ji, 2 ) )
403 40 CONTINUE
404 END IF
405*
406* Iteration loop
407*
408 DO 130 jit = 1, nitmax
409*
410* Loop over intervals
411*
412 IF( kl-kf+1.GE.nbmin .AND. nbmin.GT.0 ) THEN
413*
414* Begin of Parallel Version of the loop
415*
416 DO 60 ji = kf, kl
417*
418* Compute N(c), the number of eigenvalues less than c
419*
420 work( ji ) = d( 1 ) - c( ji )
421 iwork( ji ) = 0
422 IF( work( ji ).LE.pivmin ) THEN
423 iwork( ji ) = 1
424 work( ji ) = min( work( ji ), -pivmin )
425 END IF
426*
427 DO 50 j = 2, n
428 work( ji ) = d( j ) - e2( j-1 ) / work( ji ) - c( ji )
429 IF( work( ji ).LE.pivmin ) THEN
430 iwork( ji ) = iwork( ji ) + 1
431 work( ji ) = min( work( ji ), -pivmin )
432 END IF
433 50 CONTINUE
434 60 CONTINUE
435*
436 IF( ijob.LE.2 ) THEN
437*
438* IJOB=2: Choose all intervals containing eigenvalues.
439*
440 klnew = kl
441 DO 70 ji = kf, kl
442*
443* Insure that N(w) is monotone
444*
445 iwork( ji ) = min( nab( ji, 2 ),
446 \$ max( nab( ji, 1 ), iwork( ji ) ) )
447*
448* Update the Queue -- add intervals if both halves
449* contain eigenvalues.
450*
451 IF( iwork( ji ).EQ.nab( ji, 2 ) ) THEN
452*
453* No eigenvalue in the upper interval:
454* just use the lower interval.
455*
456 ab( ji, 2 ) = c( ji )
457*
458 ELSE IF( iwork( ji ).EQ.nab( ji, 1 ) ) THEN
459*
460* No eigenvalue in the lower interval:
461* just use the upper interval.
462*
463 ab( ji, 1 ) = c( ji )
464 ELSE
465 klnew = klnew + 1
466 IF( klnew.LE.mmax ) THEN
467*
468* Eigenvalue in both intervals -- add upper to
469* queue.
470*
471 ab( klnew, 2 ) = ab( ji, 2 )
472 nab( klnew, 2 ) = nab( ji, 2 )
473 ab( klnew, 1 ) = c( ji )
474 nab( klnew, 1 ) = iwork( ji )
475 ab( ji, 2 ) = c( ji )
476 nab( ji, 2 ) = iwork( ji )
477 ELSE
478 info = mmax + 1
479 END IF
480 END IF
481 70 CONTINUE
482 IF( info.NE.0 )
483 \$ RETURN
484 kl = klnew
485 ELSE
486*
487* IJOB=3: Binary search. Keep only the interval containing
488* w s.t. N(w) = NVAL
489*
490 DO 80 ji = kf, kl
491 IF( iwork( ji ).LE.nval( ji ) ) THEN
492 ab( ji, 1 ) = c( ji )
493 nab( ji, 1 ) = iwork( ji )
494 END IF
495 IF( iwork( ji ).GE.nval( ji ) ) THEN
496 ab( ji, 2 ) = c( ji )
497 nab( ji, 2 ) = iwork( ji )
498 END IF
499 80 CONTINUE
500 END IF
501*
502 ELSE
503*
504* End of Parallel Version of the loop
505*
506* Begin of Serial Version of the loop
507*
508 klnew = kl
509 DO 100 ji = kf, kl
510*
511* Compute N(w), the number of eigenvalues less than w
512*
513 tmp1 = c( ji )
514 tmp2 = d( 1 ) - tmp1
515 itmp1 = 0
516 IF( tmp2.LE.pivmin ) THEN
517 itmp1 = 1
518 tmp2 = min( tmp2, -pivmin )
519 END IF
520*
521 DO 90 j = 2, n
522 tmp2 = d( j ) - e2( j-1 ) / tmp2 - tmp1
523 IF( tmp2.LE.pivmin ) THEN
524 itmp1 = itmp1 + 1
525 tmp2 = min( tmp2, -pivmin )
526 END IF
527 90 CONTINUE
528*
529 IF( ijob.LE.2 ) THEN
530*
531* IJOB=2: Choose all intervals containing eigenvalues.
532*
533* Insure that N(w) is monotone
534*
535 itmp1 = min( nab( ji, 2 ),
536 \$ max( nab( ji, 1 ), itmp1 ) )
537*
538* Update the Queue -- add intervals if both halves
539* contain eigenvalues.
540*
541 IF( itmp1.EQ.nab( ji, 2 ) ) THEN
542*
543* No eigenvalue in the upper interval:
544* just use the lower interval.
545*
546 ab( ji, 2 ) = tmp1
547*
548 ELSE IF( itmp1.EQ.nab( ji, 1 ) ) THEN
549*
550* No eigenvalue in the lower interval:
551* just use the upper interval.
552*
553 ab( ji, 1 ) = tmp1
554 ELSE IF( klnew.LT.mmax ) THEN
555*
556* Eigenvalue in both intervals -- add upper to queue.
557*
558 klnew = klnew + 1
559 ab( klnew, 2 ) = ab( ji, 2 )
560 nab( klnew, 2 ) = nab( ji, 2 )
561 ab( klnew, 1 ) = tmp1
562 nab( klnew, 1 ) = itmp1
563 ab( ji, 2 ) = tmp1
564 nab( ji, 2 ) = itmp1
565 ELSE
566 info = mmax + 1
567 RETURN
568 END IF
569 ELSE
570*
571* IJOB=3: Binary search. Keep only the interval
572* containing w s.t. N(w) = NVAL
573*
574 IF( itmp1.LE.nval( ji ) ) THEN
575 ab( ji, 1 ) = tmp1
576 nab( ji, 1 ) = itmp1
577 END IF
578 IF( itmp1.GE.nval( ji ) ) THEN
579 ab( ji, 2 ) = tmp1
580 nab( ji, 2 ) = itmp1
581 END IF
582 END IF
583 100 CONTINUE
584 kl = klnew
585*
586 END IF
587*
588* Check for convergence
589*
590 kfnew = kf
591 DO 110 ji = kf, kl
592 tmp1 = abs( ab( ji, 2 )-ab( ji, 1 ) )
593 tmp2 = max( abs( ab( ji, 2 ) ), abs( ab( ji, 1 ) ) )
594 IF( tmp1.LT.max( abstol, pivmin, reltol*tmp2 ) .OR.
595 \$ nab( ji, 1 ).GE.nab( ji, 2 ) ) THEN
596*
597* Converged -- Swap with position KFNEW,
598* then increment KFNEW
599*
600 IF( ji.GT.kfnew ) THEN
601 tmp1 = ab( ji, 1 )
602 tmp2 = ab( ji, 2 )
603 itmp1 = nab( ji, 1 )
604 itmp2 = nab( ji, 2 )
605 ab( ji, 1 ) = ab( kfnew, 1 )
606 ab( ji, 2 ) = ab( kfnew, 2 )
607 nab( ji, 1 ) = nab( kfnew, 1 )
608 nab( ji, 2 ) = nab( kfnew, 2 )
609 ab( kfnew, 1 ) = tmp1
610 ab( kfnew, 2 ) = tmp2
611 nab( kfnew, 1 ) = itmp1
612 nab( kfnew, 2 ) = itmp2
613 IF( ijob.EQ.3 ) THEN
614 itmp1 = nval( ji )
615 nval( ji ) = nval( kfnew )
616 nval( kfnew ) = itmp1
617 END IF
618 END IF
619 kfnew = kfnew + 1
620 END IF
621 110 CONTINUE
622 kf = kfnew
623*
624* Choose Midpoints
625*
626 DO 120 ji = kf, kl
627 c( ji ) = half*( ab( ji, 1 )+ab( ji, 2 ) )
628 120 CONTINUE
629*
630* If no more intervals to refine, quit.
631*
632 IF( kf.GT.kl )
633 \$ GO TO 140
634 130 CONTINUE
635*
636* Converged
637*
638 140 CONTINUE
639 info = max( kl+1-kf, 0 )
640 mout = kl
641*
642 RETURN
643*
644* End of DLAEBZ
645*
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