1*> \brief \b DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrf.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
22*                          W, WGAP, WERR,
23*                          SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
24*                          DPLUS, LPLUS, WORK, INFO )
25*
26*       .. Scalar Arguments ..
27*       INTEGER            CLSTRT, CLEND, INFO, N
28*       DOUBLE PRECISION   CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
29*       ..
30*       .. Array Arguments ..
31*       DOUBLE PRECISION   D( * ), DPLUS( * ), L( * ), LD( * ),
32*      $          LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
33*       ..
34*
35*
36*> \par Purpose:
37*  =============
38*>
39*> \verbatim
40*>
41*> Given the initial representation L D L^T and its cluster of close
42*> eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
43*> W( CLEND ), DLARRF finds a new relatively robust representation
44*> L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
45*> eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
46*> \endverbatim
47*
48*  Arguments:
49*  ==========
50*
51*> \param[in] N
52*> \verbatim
53*>          N is INTEGER
54*>          The order of the matrix (subblock, if the matrix split).
55*> \endverbatim
56*>
57*> \param[in] D
58*> \verbatim
59*>          D is DOUBLE PRECISION array, dimension (N)
60*>          The N diagonal elements of the diagonal matrix D.
61*> \endverbatim
62*>
63*> \param[in] L
64*> \verbatim
65*>          L is DOUBLE PRECISION array, dimension (N-1)
66*>          The (N-1) subdiagonal elements of the unit bidiagonal
67*>          matrix L.
68*> \endverbatim
69*>
70*> \param[in] LD
71*> \verbatim
72*>          LD is DOUBLE PRECISION array, dimension (N-1)
73*>          The (N-1) elements L(i)*D(i).
74*> \endverbatim
75*>
76*> \param[in] CLSTRT
77*> \verbatim
78*>          CLSTRT is INTEGER
79*>          The index of the first eigenvalue in the cluster.
80*> \endverbatim
81*>
82*> \param[in] CLEND
83*> \verbatim
84*>          CLEND is INTEGER
85*>          The index of the last eigenvalue in the cluster.
86*> \endverbatim
87*>
88*> \param[in] W
89*> \verbatim
90*>          W is DOUBLE PRECISION array, dimension
91*>          dimension is >=  (CLEND-CLSTRT+1)
92*>          The eigenvalue APPROXIMATIONS of L D L^T in ascending order.
93*>          W( CLSTRT ) through W( CLEND ) form the cluster of relatively
94*>          close eigenalues.
95*> \endverbatim
96*>
97*> \param[in,out] WGAP
98*> \verbatim
99*>          WGAP is DOUBLE PRECISION array, dimension
100*>          dimension is >=  (CLEND-CLSTRT+1)
101*>          The separation from the right neighbor eigenvalue in W.
102*> \endverbatim
103*>
104*> \param[in] WERR
105*> \verbatim
106*>          WERR is DOUBLE PRECISION array, dimension
107*>          dimension is  >=  (CLEND-CLSTRT+1)
108*>          WERR contain the semiwidth of the uncertainty
109*>          interval of the corresponding eigenvalue APPROXIMATION in W
110*> \endverbatim
111*>
112*> \param[in] SPDIAM
113*> \verbatim
114*>          SPDIAM is DOUBLE PRECISION
115*>          estimate of the spectral diameter obtained from the
116*>          Gerschgorin intervals
117*> \endverbatim
118*>
119*> \param[in] CLGAPL
120*> \verbatim
121*>          CLGAPL is DOUBLE PRECISION
122*> \endverbatim
123*>
124*> \param[in] CLGAPR
125*> \verbatim
126*>          CLGAPR is DOUBLE PRECISION
127*>          absolute gap on each end of the cluster.
128*>          Set by the calling routine to protect against shifts too close
129*>          to eigenvalues outside the cluster.
130*> \endverbatim
131*>
132*> \param[in] PIVMIN
133*> \verbatim
134*>          PIVMIN is DOUBLE PRECISION
135*>          The minimum pivot allowed in the Sturm sequence.
136*> \endverbatim
137*>
138*> \param[out] SIGMA
139*> \verbatim
140*>          SIGMA is DOUBLE PRECISION
141*>          The shift used to form L(+) D(+) L(+)^T.
142*> \endverbatim
143*>
144*> \param[out] DPLUS
145*> \verbatim
146*>          DPLUS is DOUBLE PRECISION array, dimension (N)
147*>          The N diagonal elements of the diagonal matrix D(+).
148*> \endverbatim
149*>
150*> \param[out] LPLUS
151*> \verbatim
152*>          LPLUS is DOUBLE PRECISION array, dimension (N-1)
153*>          The first (N-1) elements of LPLUS contain the subdiagonal
154*>          elements of the unit bidiagonal matrix L(+).
155*> \endverbatim
156*>
157*> \param[out] WORK
158*> \verbatim
159*>          WORK is DOUBLE PRECISION array, dimension (2*N)
160*>          Workspace.
161*> \endverbatim
162*>
163*> \param[out] INFO
164*> \verbatim
165*>          INFO is INTEGER
166*>          Signals processing OK (=0) or failure (=1)
167*> \endverbatim
168*
169*  Authors:
170*  ========
171*
172*> \author Univ. of Tennessee
173*> \author Univ. of California Berkeley
174*> \author Univ. of Colorado Denver
175*> \author NAG Ltd.
176*
177*> \ingroup OTHERauxiliary
178*
179*> \par Contributors:
180*  ==================
181*>
182*> Beresford Parlett, University of California, Berkeley, USA \n
183*> Jim Demmel, University of California, Berkeley, USA \n
184*> Inderjit Dhillon, University of Texas, Austin, USA \n
185*> Osni Marques, LBNL/NERSC, USA \n
186*> Christof Voemel, University of California, Berkeley, USA
187*
188*  =====================================================================
189      SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
190     $                   W, WGAP, WERR,
191     $                   SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
192     $                   DPLUS, LPLUS, WORK, INFO )
193*
194*  -- LAPACK auxiliary routine --
195*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
196*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197*
198*     .. Scalar Arguments ..
199      INTEGER            CLSTRT, CLEND, INFO, N
200      DOUBLE PRECISION   CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
201*     ..
202*     .. Array Arguments ..
203      DOUBLE PRECISION   D( * ), DPLUS( * ), L( * ), LD( * ),
204     $          LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
205*     ..
206*
207*  =====================================================================
208*
209*     .. Parameters ..
210      DOUBLE PRECISION   FOUR, MAXGROWTH1, MAXGROWTH2, ONE, QUART, TWO
211      PARAMETER          ( ONE = 1.0D0, TWO = 2.0D0, FOUR = 4.0D0,
212     $                     QUART = 0.25D0,
213     $                     MAXGROWTH1 = 8.D0,
214     $                     MAXGROWTH2 = 8.D0 )
215*     ..
216*     .. Local Scalars ..
217      LOGICAL   DORRR1, FORCER, NOFAIL, SAWNAN1, SAWNAN2, TRYRRR1
218      INTEGER            I, INDX, KTRY, KTRYMAX, SLEFT, SRIGHT, SHIFT
219      PARAMETER          ( KTRYMAX = 1, SLEFT = 1, SRIGHT = 2 )
220      DOUBLE PRECISION   AVGAP, BESTSHIFT, CLWDTH, EPS, FACT, FAIL,
221     $                   FAIL2, GROWTHBOUND, LDELTA, LDMAX, LSIGMA,
222     $                   MAX1, MAX2, MINGAP, OLDP, PROD, RDELTA, RDMAX,
223     $                   RRR1, RRR2, RSIGMA, S, SMLGROWTH, TMP, ZNM2
224*     ..
225*     .. External Functions ..
226      LOGICAL DISNAN
227      DOUBLE PRECISION   DLAMCH
228      EXTERNAL           DISNAN, DLAMCH
229*     ..
230*     .. External Subroutines ..
231      EXTERNAL           DCOPY
232*     ..
233*     .. Intrinsic Functions ..
234      INTRINSIC          ABS
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      FACT = DBLE(2**KTRYMAX)
247      EPS = DLAMCH( 'Precision' )
248      SHIFT = 0
249      FORCER = .FALSE.
250
251
252*     Note that we cannot guarantee that for any of the shifts tried,
253*     the factorization has a small or even moderate element growth.
254*     There could be Ritz values at both ends of the cluster and despite
255*     backing off, there are examples where all factorizations tried
256*     (in IEEE mode, allowing zero pivots & infinities) have INFINITE
257*     element growth.
258*     For this reason, we should use PIVMIN in this subroutine so that at
259*     least the L D L^T factorization exists. It can be checked afterwards
260*     whether the element growth caused bad residuals/orthogonality.
261
262*     Decide whether the code should accept the best among all
263*     representations despite large element growth or signal INFO=1
264*     Setting NOFAIL to .FALSE. for quick fix for bug 113
265      NOFAIL = .FALSE.
266*
267
268*     Compute the average gap length of the cluster
269      CLWDTH = ABS(W(CLEND)-W(CLSTRT)) + WERR(CLEND) + WERR(CLSTRT)
270      AVGAP = CLWDTH / DBLE(CLEND-CLSTRT)
271      MINGAP = MIN(CLGAPL, CLGAPR)
272*     Initial values for shifts to both ends of cluster
273      LSIGMA = MIN(W( CLSTRT ),W( CLEND )) - WERR( CLSTRT )
274      RSIGMA = MAX(W( CLSTRT ),W( CLEND )) + WERR( CLEND )
275
276*     Use a small fudge to make sure that we really shift to the outside
277      LSIGMA = LSIGMA - ABS(LSIGMA)* FOUR * EPS
278      RSIGMA = RSIGMA + ABS(RSIGMA)* FOUR * EPS
279
280*     Compute upper bounds for how much to back off the initial shifts
281      LDMAX = QUART * MINGAP + TWO * PIVMIN
282      RDMAX = QUART * MINGAP + TWO * PIVMIN
283
284      LDELTA = MAX(AVGAP,WGAP( CLSTRT ))/FACT
285      RDELTA = MAX(AVGAP,WGAP( CLEND-1 ))/FACT
286*
287*     Initialize the record of the best representation found
288*
289      S = DLAMCH( 'S' )
290      SMLGROWTH = ONE / S
291      FAIL = DBLE(N-1)*MINGAP/(SPDIAM*EPS)
292      FAIL2 = DBLE(N-1)*MINGAP/(SPDIAM*SQRT(EPS))
293      BESTSHIFT = LSIGMA
294*
295*     while (KTRY <= KTRYMAX)
296      KTRY = 0
297      GROWTHBOUND = MAXGROWTH1*SPDIAM
298
299 5    CONTINUE
300      SAWNAN1 = .FALSE.
301      SAWNAN2 = .FALSE.
302*     Ensure that we do not back off too much of the initial shifts
303      LDELTA = MIN(LDMAX,LDELTA)
304      RDELTA = MIN(RDMAX,RDELTA)
305
306*     Compute the element growth when shifting to both ends of the cluster
307*     accept the shift if there is no element growth at one of the two ends
308
309*     Left end
310      S = -LSIGMA
311      DPLUS( 1 ) = D( 1 ) + S
312      IF(ABS(DPLUS(1)).LT.PIVMIN) THEN
313         DPLUS(1) = -PIVMIN
314*        Need to set SAWNAN1 because refined RRR test should not be used
315*        in this case
316         SAWNAN1 = .TRUE.
317      ENDIF
318      MAX1 = ABS( DPLUS( 1 ) )
319      DO 6 I = 1, N - 1
320         LPLUS( I ) = LD( I ) / DPLUS( I )
321         S = S*LPLUS( I )*L( I ) - LSIGMA
322         DPLUS( I+1 ) = D( I+1 ) + S
323         IF(ABS(DPLUS(I+1)).LT.PIVMIN) THEN
324            DPLUS(I+1) = -PIVMIN
325*           Need to set SAWNAN1 because refined RRR test should not be used
326*           in this case
327            SAWNAN1 = .TRUE.
328         ENDIF
329         MAX1 = MAX( MAX1,ABS(DPLUS(I+1)) )
330 6    CONTINUE
331      SAWNAN1 = SAWNAN1 .OR.  DISNAN( MAX1 )
332
333      IF( FORCER .OR.
334     $   (MAX1.LE.GROWTHBOUND .AND. .NOT.SAWNAN1 ) ) THEN
335         SIGMA = LSIGMA
336         SHIFT = SLEFT
337         GOTO 100
338      ENDIF
339
340*     Right end
341      S = -RSIGMA
342      WORK( 1 ) = D( 1 ) + S
343      IF(ABS(WORK(1)).LT.PIVMIN) THEN
344         WORK(1) = -PIVMIN
345*        Need to set SAWNAN2 because refined RRR test should not be used
346*        in this case
347         SAWNAN2 = .TRUE.
348      ENDIF
349      MAX2 = ABS( WORK( 1 ) )
350      DO 7 I = 1, N - 1
351         WORK( N+I ) = LD( I ) / WORK( I )
352         S = S*WORK( N+I )*L( I ) - RSIGMA
353         WORK( I+1 ) = D( I+1 ) + S
354         IF(ABS(WORK(I+1)).LT.PIVMIN) THEN
355            WORK(I+1) = -PIVMIN
356*           Need to set SAWNAN2 because refined RRR test should not be used
357*           in this case
358            SAWNAN2 = .TRUE.
359         ENDIF
360         MAX2 = MAX( MAX2,ABS(WORK(I+1)) )
361 7    CONTINUE
362      SAWNAN2 = SAWNAN2 .OR.  DISNAN( MAX2 )
363
364      IF( FORCER .OR.
365     $   (MAX2.LE.GROWTHBOUND .AND. .NOT.SAWNAN2 ) ) THEN
366         SIGMA = RSIGMA
367         SHIFT = SRIGHT
368         GOTO 100
369      ENDIF
370*     If we are at this point, both shifts led to too much element growth
371
372*     Record the better of the two shifts (provided it didn't lead to NaN)
373      IF(SAWNAN1.AND.SAWNAN2) THEN
374*        both MAX1 and MAX2 are NaN
375         GOTO 50
376      ELSE
377         IF( .NOT.SAWNAN1 ) THEN
378            INDX = 1
379            IF(MAX1.LE.SMLGROWTH) THEN
380               SMLGROWTH = MAX1
381               BESTSHIFT = LSIGMA
382            ENDIF
383         ENDIF
384         IF( .NOT.SAWNAN2 ) THEN
385            IF(SAWNAN1 .OR. MAX2.LE.MAX1) INDX = 2
386            IF(MAX2.LE.SMLGROWTH) THEN
387               SMLGROWTH = MAX2
388               BESTSHIFT = RSIGMA
389            ENDIF
390         ENDIF
391      ENDIF
392
393*     If we are here, both the left and the right shift led to
394*     element growth. If the element growth is moderate, then
395*     we may still accept the representation, if it passes a
396*     refined test for RRR. This test supposes that no NaN occurred.
397*     Moreover, we use the refined RRR test only for isolated clusters.
398      IF((CLWDTH.LT.MINGAP/DBLE(128)) .AND.
399     $   (MIN(MAX1,MAX2).LT.FAIL2)
400     $  .AND.(.NOT.SAWNAN1).AND.(.NOT.SAWNAN2)) THEN
401         DORRR1 = .TRUE.
402      ELSE
403         DORRR1 = .FALSE.
404      ENDIF
405      TRYRRR1 = .TRUE.
406      IF( TRYRRR1 .AND. DORRR1 ) THEN
407      IF(INDX.EQ.1) THEN
408         TMP = ABS( DPLUS( N ) )
409         ZNM2 = ONE
410         PROD = ONE
411         OLDP = ONE
412         DO 15 I = N-1, 1, -1
413            IF( PROD .LE. EPS ) THEN
414               PROD =
415     $         ((DPLUS(I+1)*WORK(N+I+1))/(DPLUS(I)*WORK(N+I)))*OLDP
416            ELSE
417               PROD = PROD*ABS(WORK(N+I))
418            END IF
419            OLDP = PROD
420            ZNM2 = ZNM2 + PROD**2
421            TMP = MAX( TMP, ABS( DPLUS( I ) * PROD ))
422 15      CONTINUE
423         RRR1 = TMP/( SPDIAM * SQRT( ZNM2 ) )
424         IF (RRR1.LE.MAXGROWTH2) THEN
425            SIGMA = LSIGMA
426            SHIFT = SLEFT
427            GOTO 100
428         ENDIF
429      ELSE IF(INDX.EQ.2) THEN
430         TMP = ABS( WORK( N ) )
431         ZNM2 = ONE
432         PROD = ONE
433         OLDP = ONE
434         DO 16 I = N-1, 1, -1
435            IF( PROD .LE. EPS ) THEN
436               PROD = ((WORK(I+1)*LPLUS(I+1))/(WORK(I)*LPLUS(I)))*OLDP
437            ELSE
438               PROD = PROD*ABS(LPLUS(I))
439            END IF
440            OLDP = PROD
441            ZNM2 = ZNM2 + PROD**2
442            TMP = MAX( TMP, ABS( WORK( I ) * PROD ))
443 16      CONTINUE
444         RRR2 = TMP/( SPDIAM * SQRT( ZNM2 ) )
445         IF (RRR2.LE.MAXGROWTH2) THEN
446            SIGMA = RSIGMA
447            SHIFT = SRIGHT
448            GOTO 100
449         ENDIF
450      END IF
451      ENDIF
452
453 50   CONTINUE
454
455      IF (KTRY.LT.KTRYMAX) THEN
456*        If we are here, both shifts failed also the RRR test.
457*        Back off to the outside
458         LSIGMA = MAX( LSIGMA - LDELTA,
459     $     LSIGMA - LDMAX)
460         RSIGMA = MIN( RSIGMA + RDELTA,
461     $     RSIGMA + RDMAX )
462         LDELTA = TWO * LDELTA
463         RDELTA = TWO * RDELTA
464         KTRY = KTRY + 1
465         GOTO 5
466      ELSE
467*        None of the representations investigated satisfied our
468*        criteria. Take the best one we found.
469         IF((SMLGROWTH.LT.FAIL).OR.NOFAIL) THEN
470            LSIGMA = BESTSHIFT
471            RSIGMA = BESTSHIFT
472            FORCER = .TRUE.
473            GOTO 5
474         ELSE
475            INFO = 1
476            RETURN
477         ENDIF
478      END IF
479
480 100  CONTINUE
481      IF (SHIFT.EQ.SLEFT) THEN
482      ELSEIF (SHIFT.EQ.SRIGHT) THEN
483*        store new L and D back into DPLUS, LPLUS
484         CALL DCOPY( N, WORK, 1, DPLUS, 1 )
485         CALL DCOPY( N-1, WORK(N+1), 1, LPLUS, 1 )
486      ENDIF
487
488      RETURN
489*
490*     End of DLARRF
491*
492      END
493