LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ slarrb()

 subroutine slarrb ( integer N, real, dimension( * ) D, real, dimension( * ) LLD, integer IFIRST, integer ILAST, real RTOL1, real RTOL2, integer OFFSET, real, dimension( * ) W, real, dimension( * ) WGAP, real, dimension( * ) WERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, real PIVMIN, real SPDIAM, integer TWIST, integer INFO )

SLARRB provides limited bisection to locate eigenvalues for more accuracy.

Purpose:
``` Given the relatively robust representation(RRR) L D L^T, SLARRB
does "limited" bisection to refine the eigenvalues of L D L^T,
W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
guesses for these eigenvalues are input in W, the corresponding estimate
of the error in these guesses and their gaps are input in WERR
and WGAP, respectively. During bisection, intervals
[left, right] are maintained by storing their mid-points and
semi-widths in the arrays W and WERR respectively.```
Parameters
 [in] N ``` N is INTEGER The order of the matrix.``` [in] D ``` D is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D.``` [in] LLD ``` LLD is REAL array, dimension (N-1) The (N-1) elements L(i)*L(i)*D(i).``` [in] IFIRST ``` IFIRST is INTEGER The index of the first eigenvalue to be computed.``` [in] ILAST ``` ILAST is INTEGER The index of the last eigenvalue to be computed.``` [in] RTOL1 ` RTOL1 is REAL` [in] RTOL2 ``` RTOL2 is REAL Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) where GAP is the (estimated) distance to the nearest eigenvalue.``` [in] OFFSET ``` OFFSET is INTEGER Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET through ILAST-OFFSET elements of these arrays are to be used.``` [in,out] W ``` W is REAL array, dimension (N) On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined.``` [in,out] WGAP ``` WGAP is REAL array, dimension (N-1) On input, the (estimated) gaps between consecutive eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST then WGAP(IFIRST-OFFSET) must be set to ZERO. On output, these gaps are refined.``` [in,out] WERR ``` WERR is REAL array, dimension (N) On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined.``` [out] WORK ``` WORK is REAL array, dimension (2*N) Workspace.``` [out] IWORK ``` IWORK is INTEGER array, dimension (2*N) Workspace.``` [in] PIVMIN ``` PIVMIN is REAL The minimum pivot in the Sturm sequence.``` [in] SPDIAM ``` SPDIAM is REAL The spectral diameter of the matrix.``` [in] TWIST ``` TWIST is INTEGER The twist index for the twisted factorization that is used for the negcount. TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)``` [out] INFO ``` INFO is INTEGER Error flag.```
Date
June 2017
Contributors:
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

Definition at line 198 of file slarrb.f.

198 *
199 * -- LAPACK auxiliary routine (version 3.7.1) --
200 * -- LAPACK is a software package provided by Univ. of Tennessee, --
201 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
202 * June 2017
203 *
204 * .. Scalar Arguments ..
205  INTEGER ifirst, ilast, info, n, offset, twist
206  REAL pivmin, rtol1, rtol2, spdiam
207 * ..
208 * .. Array Arguments ..
209  INTEGER iwork( * )
210  REAL d( * ), lld( * ), w( * ),
211  \$ werr( * ), wgap( * ), work( * )
212 * ..
213 *
214 * =====================================================================
215 *
216 * .. Parameters ..
217  REAL zero, two, half
218  parameter( zero = 0.0e0, two = 2.0e0,
219  \$ half = 0.5e0 )
220  INTEGER maxitr
221 * ..
222 * .. Local Scalars ..
223  INTEGER i, i1, ii, ip, iter, k, negcnt, next, nint,
224  \$ olnint, prev, r
225  REAL back, cvrgd, gap, left, lgap, mid, mnwdth,
226  \$ rgap, right, tmp, width
227 * ..
228 * .. External Functions ..
229  INTEGER slaneg
230  EXTERNAL slaneg
231 *
232 * ..
233 * .. Intrinsic Functions ..
234  INTRINSIC abs, max, min
235 * ..
236 * .. Executable Statements ..
237 *
238  info = 0
239 *
240 * Quick return if possible
241 *
242  IF( n.LE.0 ) THEN
243  RETURN
244  END IF
245 *
246  maxitr = int( ( log( spdiam+pivmin )-log( pivmin ) ) /
247  \$ log( two ) ) + 2
248  mnwdth = two * pivmin
249 *
250  r = twist
251  IF((r.LT.1).OR.(r.GT.n)) r = n
252 *
253 * Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
254 * The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
255 * Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
256 * for an unconverged interval is set to the index of the next unconverged
257 * interval, and is -1 or 0 for a converged interval. Thus a linked
258 * list of unconverged intervals is set up.
259 *
260  i1 = ifirst
261 * The number of unconverged intervals
262  nint = 0
263 * The last unconverged interval found
264  prev = 0
265
266  rgap = wgap( i1-offset )
267  DO 75 i = i1, ilast
268  k = 2*i
269  ii = i - offset
270  left = w( ii ) - werr( ii )
271  right = w( ii ) + werr( ii )
272  lgap = rgap
273  rgap = wgap( ii )
274  gap = min( lgap, rgap )
275
276 * Make sure that [LEFT,RIGHT] contains the desired eigenvalue
277 * Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT
278 *
279 * Do while( NEGCNT(LEFT).GT.I-1 )
280 *
281  back = werr( ii )
282  20 CONTINUE
283  negcnt = slaneg( n, d, lld, left, pivmin, r )
284  IF( negcnt.GT.i-1 ) THEN
285  left = left - back
286  back = two*back
287  GO TO 20
288  END IF
289 *
290 * Do while( NEGCNT(RIGHT).LT.I )
291 * Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT
292 *
293  back = werr( ii )
294  50 CONTINUE
295
296  negcnt = slaneg( n, d, lld, right, pivmin, r )
297  IF( negcnt.LT.i ) THEN
298  right = right + back
299  back = two*back
300  GO TO 50
301  END IF
302  width = half*abs( left - right )
303  tmp = max( abs( left ), abs( right ) )
304  cvrgd = max(rtol1*gap,rtol2*tmp)
305  IF( width.LE.cvrgd .OR. width.LE.mnwdth ) THEN
306 * This interval has already converged and does not need refinement.
307 * (Note that the gaps might change through refining the
308 * eigenvalues, however, they can only get bigger.)
309 * Remove it from the list.
310  iwork( k-1 ) = -1
311 * Make sure that I1 always points to the first unconverged interval
312  IF((i.EQ.i1).AND.(i.LT.ilast)) i1 = i + 1
313  IF((prev.GE.i1).AND.(i.LE.ilast)) iwork( 2*prev-1 ) = i + 1
314  ELSE
315 * unconverged interval found
316  prev = i
317  nint = nint + 1
318  iwork( k-1 ) = i + 1
319  iwork( k ) = negcnt
320  END IF
321  work( k-1 ) = left
322  work( k ) = right
323  75 CONTINUE
324
325 *
326 * Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
327 * and while (ITER.LT.MAXITR)
328 *
329  iter = 0
330  80 CONTINUE
331  prev = i1 - 1
332  i = i1
333  olnint = nint
334
335  DO 100 ip = 1, olnint
336  k = 2*i
337  ii = i - offset
338  rgap = wgap( ii )
339  lgap = rgap
340  IF(ii.GT.1) lgap = wgap( ii-1 )
341  gap = min( lgap, rgap )
342  next = iwork( k-1 )
343  left = work( k-1 )
344  right = work( k )
345  mid = half*( left + right )
346
347 * semiwidth of interval
348  width = right - mid
349  tmp = max( abs( left ), abs( right ) )
350  cvrgd = max(rtol1*gap,rtol2*tmp)
351  IF( ( width.LE.cvrgd ) .OR. ( width.LE.mnwdth ).OR.
352  \$ ( iter.EQ.maxitr ) )THEN
353 * reduce number of unconverged intervals
354  nint = nint - 1
355 * Mark interval as converged.
356  iwork( k-1 ) = 0
357  IF( i1.EQ.i ) THEN
358  i1 = next
359  ELSE
360 * Prev holds the last unconverged interval previously examined
361  IF(prev.GE.i1) iwork( 2*prev-1 ) = next
362  END IF
363  i = next
364  GO TO 100
365  END IF
366  prev = i
367 *
368 * Perform one bisection step
369 *
370  negcnt = slaneg( n, d, lld, mid, pivmin, r )
371  IF( negcnt.LE.i-1 ) THEN
372  work( k-1 ) = mid
373  ELSE
374  work( k ) = mid
375  END IF
376  i = next
377  100 CONTINUE
378  iter = iter + 1
379 * do another loop if there are still unconverged intervals
380 * However, in the last iteration, all intervals are accepted
381 * since this is the best we can do.
382  IF( ( nint.GT.0 ).AND.(iter.LE.maxitr) ) GO TO 80
383 *
384 *
385 * At this point, all the intervals have converged
386  DO 110 i = ifirst, ilast
387  k = 2*i
388  ii = i - offset
389 * All intervals marked by '0' have been refined.
390  IF( iwork( k-1 ).EQ.0 ) THEN
391  w( ii ) = half*( work( k-1 )+work( k ) )
392  werr( ii ) = work( k ) - w( ii )
393  END IF
394  110 CONTINUE
395 *
396  DO 111 i = ifirst+1, ilast
397  k = 2*i
398  ii = i - offset
399  wgap( ii-1 ) = max( zero,
400  \$ w(ii) - werr(ii) - w( ii-1 ) - werr( ii-1 ))
401  111 CONTINUE
402
403  RETURN
404 *
405 * End of SLARRB
406 *
integer function slaneg(N, D, LLD, SIGMA, PIVMIN, R)
SLANEG computes the Sturm count.
Definition: slaneg.f:120
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