summaryrefslogtreecommitdiff
path: root/SRC/slarrb.f
blob: 988e25ff07663c93a1c5d854d8bd093e08cefe63 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
*> \brief \b SLARRB provides limited bisection to locate eigenvalues for more accuracy.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLARRB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slarrb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slarrb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarrb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
*                          RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
*                          PIVMIN, SPDIAM, TWIST, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
*       REAL               PIVMIN, RTOL1, RTOL2, SPDIAM
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       REAL               D( * ), LLD( * ), W( * ),
*      $                   WERR( * ), WGAP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>          The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*>          LLD is REAL array, dimension (N-1)
*>          The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] IFIRST
*> \verbatim
*>          IFIRST is INTEGER
*>          The index of the first eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] ILAST
*> \verbatim
*>          ILAST is INTEGER
*>          The index of the last eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*>          RTOL1 is REAL
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (2*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (2*N)
*>          Workspace.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is REAL
*>          The minimum pivot in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*>          SPDIAM is REAL
*>          The spectral diameter of the matrix.
*> \endverbatim
*>
*> \param[in] TWIST
*> \verbatim
*>          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)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          Error flag.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
*  ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
*  =====================================================================
      SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
     $                   RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
     $                   PIVMIN, SPDIAM, TWIST, INFO )
*
*  -- LAPACK auxiliary routine (version 3.7.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2017
*
*     .. Scalar Arguments ..
      INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
      REAL               PIVMIN, RTOL1, RTOL2, SPDIAM
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      REAL               D( * ), LLD( * ), W( * ),
     $                   WERR( * ), WGAP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, TWO, HALF
      PARAMETER        ( ZERO = 0.0E0, TWO = 2.0E0,
     $                   HALF = 0.5E0 )
      INTEGER   MAXITR
*     ..
*     .. Local Scalars ..
      INTEGER            I, I1, II, IP, ITER, K, NEGCNT, NEXT, NINT,
     $                   OLNINT, PREV, R
      REAL               BACK, CVRGD, GAP, LEFT, LGAP, MID, MNWDTH,
     $                   RGAP, RIGHT, TMP, WIDTH
*     ..
*     .. External Functions ..
      INTEGER            SLANEG
      EXTERNAL           SLANEG
*
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         RETURN
      END IF
*
      MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2
      MNWDTH = TWO * PIVMIN
*
      R = TWIST
      IF((R.LT.1).OR.(R.GT.N)) R = N
*
*     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
*     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
*     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
*     for an unconverged interval is set to the index of the next unconverged
*     interval, and is -1 or 0 for a converged interval. Thus a linked
*     list of unconverged intervals is set up.
*
      I1 = IFIRST
*     The number of unconverged intervals
      NINT = 0
*     The last unconverged interval found
      PREV = 0

      RGAP = WGAP( I1-OFFSET )
      DO 75 I = I1, ILAST
         K = 2*I
         II = I - OFFSET
         LEFT = W( II ) - WERR( II )
         RIGHT = W( II ) + WERR( II )
         LGAP = RGAP
         RGAP = WGAP( II )
         GAP = MIN( LGAP, RGAP )

*        Make sure that [LEFT,RIGHT] contains the desired eigenvalue
*        Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT
*
*        Do while( NEGCNT(LEFT).GT.I-1 )
*
         BACK = WERR( II )
 20      CONTINUE
         NEGCNT = SLANEG( N, D, LLD, LEFT, PIVMIN, R )
         IF( NEGCNT.GT.I-1 ) THEN
            LEFT = LEFT - BACK
            BACK = TWO*BACK
            GO TO 20
         END IF
*
*        Do while( NEGCNT(RIGHT).LT.I )
*        Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT
*
         BACK = WERR( II )
 50      CONTINUE

         NEGCNT = SLANEG( N, D, LLD, RIGHT, PIVMIN, R )
          IF( NEGCNT.LT.I ) THEN
             RIGHT = RIGHT + BACK
             BACK = TWO*BACK
             GO TO 50
          END IF
         WIDTH = HALF*ABS( LEFT - RIGHT )
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
         CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
         IF( WIDTH.LE.CVRGD .OR. WIDTH.LE.MNWDTH ) THEN
*           This interval has already converged and does not need refinement.
*           (Note that the gaps might change through refining the
*            eigenvalues, however, they can only get bigger.)
*           Remove it from the list.
            IWORK( K-1 ) = -1
*           Make sure that I1 always points to the first unconverged interval
            IF((I.EQ.I1).AND.(I.LT.ILAST)) I1 = I + 1
            IF((PREV.GE.I1).AND.(I.LE.ILAST)) IWORK( 2*PREV-1 ) = I + 1
         ELSE
*           unconverged interval found
            PREV = I
            NINT = NINT + 1
            IWORK( K-1 ) = I + 1
            IWORK( K ) = NEGCNT
         END IF
         WORK( K-1 ) = LEFT
         WORK( K ) = RIGHT
 75   CONTINUE

*
*     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
*     and while (ITER.LT.MAXITR)
*
      ITER = 0
 80   CONTINUE
      PREV = I1 - 1
      I = I1
      OLNINT = NINT

      DO 100 IP = 1, OLNINT
         K = 2*I
         II = I - OFFSET
         RGAP = WGAP( II )
         LGAP = RGAP
         IF(II.GT.1) LGAP = WGAP( II-1 )
         GAP = MIN( LGAP, RGAP )
         NEXT = IWORK( K-1 )
         LEFT = WORK( K-1 )
         RIGHT = WORK( K )
         MID = HALF*( LEFT + RIGHT )

*        semiwidth of interval
         WIDTH = RIGHT - MID
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
         CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
         IF( ( WIDTH.LE.CVRGD ) .OR. ( WIDTH.LE.MNWDTH ).OR.
     $       ( ITER.EQ.MAXITR ) )THEN
*           reduce number of unconverged intervals
            NINT = NINT - 1
*           Mark interval as converged.
            IWORK( K-1 ) = 0
            IF( I1.EQ.I ) THEN
               I1 = NEXT
            ELSE
*              Prev holds the last unconverged interval previously examined
               IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
            END IF
            I = NEXT
            GO TO 100
         END IF
         PREV = I
*
*        Perform one bisection step
*
         NEGCNT = SLANEG( N, D, LLD, MID, PIVMIN, R )
         IF( NEGCNT.LE.I-1 ) THEN
            WORK( K-1 ) = MID
         ELSE
            WORK( K ) = MID
         END IF
         I = NEXT
 100  CONTINUE
      ITER = ITER + 1
*     do another loop if there are still unconverged intervals
*     However, in the last iteration, all intervals are accepted
*     since this is the best we can do.
      IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
*
*
*     At this point, all the intervals have converged
      DO 110 I = IFIRST, ILAST
         K = 2*I
         II = I - OFFSET
*        All intervals marked by '0' have been refined.
         IF( IWORK( K-1 ).EQ.0 ) THEN
            W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
            WERR( II ) = WORK( K ) - W( II )
         END IF
 110  CONTINUE
*
      DO 111 I = IFIRST+1, ILAST
         K = 2*I
         II = I - OFFSET
         WGAP( II-1 ) = MAX( ZERO,
     $                     W(II) - WERR (II) - W( II-1 ) - WERR( II-1 ))
 111  CONTINUE

      RETURN
*
*     End of SLARRB
*
      END