1*> \brief \b DLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
22*                          RTOL, OFFSET, W, WERR, WORK, IWORK,
23*                          PIVMIN, SPDIAM, INFO )
24*
25*       .. Scalar Arguments ..
26*       INTEGER            IFIRST, ILAST, INFO, N, OFFSET
27*       DOUBLE PRECISION   PIVMIN, RTOL, SPDIAM
28*       ..
29*       .. Array Arguments ..
30*       INTEGER            IWORK( * )
31*       DOUBLE PRECISION   D( * ), E2( * ), W( * ),
32*      $                   WERR( * ), WORK( * )
33*       ..
34*
35*
36*> \par Purpose:
37*  =============
38*>
39*> \verbatim
40*>
41*> Given the initial eigenvalue approximations of T, DLARRJ
42*> does  bisection to refine the eigenvalues of T,
43*> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
44*> guesses for these eigenvalues are input in W, the corresponding estimate
45*> of the error in these guesses in WERR. During bisection, intervals
46*> [left, right] are maintained by storing their mid-points and
47*> semi-widths in the arrays W and WERR respectively.
48*> \endverbatim
49*
50*  Arguments:
51*  ==========
52*
53*> \param[in] N
54*> \verbatim
55*>          N is INTEGER
56*>          The order of the matrix.
57*> \endverbatim
58*>
59*> \param[in] D
60*> \verbatim
61*>          D is DOUBLE PRECISION array, dimension (N)
62*>          The N diagonal elements of T.
63*> \endverbatim
64*>
65*> \param[in] E2
66*> \verbatim
67*>          E2 is DOUBLE PRECISION array, dimension (N-1)
68*>          The Squares of the (N-1) subdiagonal elements of T.
69*> \endverbatim
70*>
71*> \param[in] IFIRST
72*> \verbatim
73*>          IFIRST is INTEGER
74*>          The index of the first eigenvalue to be computed.
75*> \endverbatim
76*>
77*> \param[in] ILAST
78*> \verbatim
79*>          ILAST is INTEGER
80*>          The index of the last eigenvalue to be computed.
81*> \endverbatim
82*>
83*> \param[in] RTOL
84*> \verbatim
85*>          RTOL is DOUBLE PRECISION
86*>          Tolerance for the convergence of the bisection intervals.
87*>          An interval [LEFT,RIGHT] has converged if
88*>          RIGHT-LEFT < RTOL*MAX(|LEFT|,|RIGHT|).
89*> \endverbatim
90*>
91*> \param[in] OFFSET
92*> \verbatim
93*>          OFFSET is INTEGER
94*>          Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
95*>          through ILAST-OFFSET elements of these arrays are to be used.
96*> \endverbatim
97*>
98*> \param[in,out] W
99*> \verbatim
100*>          W is DOUBLE PRECISION array, dimension (N)
101*>          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
102*>          estimates of the eigenvalues of L D L^T indexed IFIRST through
103*>          ILAST.
104*>          On output, these estimates are refined.
105*> \endverbatim
106*>
107*> \param[in,out] WERR
108*> \verbatim
109*>          WERR is DOUBLE PRECISION array, dimension (N)
110*>          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
111*>          the errors in the estimates of the corresponding elements in W.
112*>          On output, these errors are refined.
113*> \endverbatim
114*>
115*> \param[out] WORK
116*> \verbatim
117*>          WORK is DOUBLE PRECISION array, dimension (2*N)
118*>          Workspace.
119*> \endverbatim
120*>
121*> \param[out] IWORK
122*> \verbatim
123*>          IWORK is INTEGER array, dimension (2*N)
124*>          Workspace.
125*> \endverbatim
126*>
127*> \param[in] PIVMIN
128*> \verbatim
129*>          PIVMIN is DOUBLE PRECISION
130*>          The minimum pivot in the Sturm sequence for T.
131*> \endverbatim
132*>
133*> \param[in] SPDIAM
134*> \verbatim
135*>          SPDIAM is DOUBLE PRECISION
136*>          The spectral diameter of T.
137*> \endverbatim
138*>
139*> \param[out] INFO
140*> \verbatim
141*>          INFO is INTEGER
142*>          Error flag.
143*> \endverbatim
144*
145*  Authors:
146*  ========
147*
148*> \author Univ. of Tennessee
149*> \author Univ. of California Berkeley
150*> \author Univ. of Colorado Denver
151*> \author NAG Ltd.
152*
153*> \date June 2017
154*
155*> \ingroup OTHERauxiliary
156*
157*> \par Contributors:
158*  ==================
159*>
160*> Beresford Parlett, University of California, Berkeley, USA \n
161*> Jim Demmel, University of California, Berkeley, USA \n
162*> Inderjit Dhillon, University of Texas, Austin, USA \n
163*> Osni Marques, LBNL/NERSC, USA \n
164*> Christof Voemel, University of California, Berkeley, USA
165*
166*  =====================================================================
167      SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
168     $                   RTOL, OFFSET, W, WERR, WORK, IWORK,
169     $                   PIVMIN, SPDIAM, INFO )
170*
171*  -- LAPACK auxiliary routine (version 3.7.1) --
172*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
173*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
174*     June 2017
175*
176*     .. Scalar Arguments ..
177      INTEGER            IFIRST, ILAST, INFO, N, OFFSET
178      DOUBLE PRECISION   PIVMIN, RTOL, SPDIAM
179*     ..
180*     .. Array Arguments ..
181      INTEGER            IWORK( * )
182      DOUBLE PRECISION   D( * ), E2( * ), W( * ),
183     $                   WERR( * ), WORK( * )
184*     ..
185*
186*  =====================================================================
187*
188*     .. Parameters ..
189      DOUBLE PRECISION   ZERO, ONE, TWO, HALF
190      PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
191     $                   HALF = 0.5D0 )
192      INTEGER   MAXITR
193*     ..
194*     .. Local Scalars ..
195      INTEGER            CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
196     $                   OLNINT, P, PREV, SAVI1
197      DOUBLE PRECISION   DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
198*
199*     ..
200*     .. Intrinsic Functions ..
201      INTRINSIC          ABS, MAX
202*     ..
203*     .. Executable Statements ..
204*
205      INFO = 0
206*
207*     Quick return if possible
208*
209      IF( N.LE.0 ) THEN
210         RETURN
211      END IF
212*
213      MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
214     $           LOG( TWO ) ) + 2
215*
216*     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
217*     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
218*     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
219*     for an unconverged interval is set to the index of the next unconverged
220*     interval, and is -1 or 0 for a converged interval. Thus a linked
221*     list of unconverged intervals is set up.
222*
223
224      I1 = IFIRST
225      I2 = ILAST
226*     The number of unconverged intervals
227      NINT = 0
228*     The last unconverged interval found
229      PREV = 0
230      DO 75 I = I1, I2
231         K = 2*I
232         II = I - OFFSET
233         LEFT = W( II ) - WERR( II )
234         MID = W(II)
235         RIGHT = W( II ) + WERR( II )
236         WIDTH = RIGHT - MID
237         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
238
239*        The following test prevents the test of converged intervals
240         IF( WIDTH.LT.RTOL*TMP ) THEN
241*           This interval has already converged and does not need refinement.
242*           (Note that the gaps might change through refining the
243*            eigenvalues, however, they can only get bigger.)
244*           Remove it from the list.
245            IWORK( K-1 ) = -1
246*           Make sure that I1 always points to the first unconverged interval
247            IF((I.EQ.I1).AND.(I.LT.I2)) I1 = I + 1
248            IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
249         ELSE
250*           unconverged interval found
251            PREV = I
252*           Make sure that [LEFT,RIGHT] contains the desired eigenvalue
253*
254*           Do while( CNT(LEFT).GT.I-1 )
255*
256            FAC = ONE
257 20         CONTINUE
258            CNT = 0
259            S = LEFT
260            DPLUS = D( 1 ) - S
261            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
262            DO 30 J = 2, N
263               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
264               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
265 30         CONTINUE
266            IF( CNT.GT.I-1 ) THEN
267               LEFT = LEFT - WERR( II )*FAC
268               FAC = TWO*FAC
269               GO TO 20
270            END IF
271*
272*           Do while( CNT(RIGHT).LT.I )
273*
274            FAC = ONE
275 50         CONTINUE
276            CNT = 0
277            S = RIGHT
278            DPLUS = D( 1 ) - S
279            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
280            DO 60 J = 2, N
281               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
282               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
283 60         CONTINUE
284            IF( CNT.LT.I ) THEN
285               RIGHT = RIGHT + WERR( II )*FAC
286               FAC = TWO*FAC
287               GO TO 50
288            END IF
289            NINT = NINT + 1
290            IWORK( K-1 ) = I + 1
291            IWORK( K ) = CNT
292         END IF
293         WORK( K-1 ) = LEFT
294         WORK( K ) = RIGHT
295 75   CONTINUE
296
297
298      SAVI1 = I1
299*
300*     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
301*     and while (ITER.LT.MAXITR)
302*
303      ITER = 0
304 80   CONTINUE
305      PREV = I1 - 1
306      I = I1
307      OLNINT = NINT
308
309      DO 100 P = 1, OLNINT
310         K = 2*I
311         II = I - OFFSET
312         NEXT = IWORK( K-1 )
313         LEFT = WORK( K-1 )
314         RIGHT = WORK( K )
315         MID = HALF*( LEFT + RIGHT )
316
317*        semiwidth of interval
318         WIDTH = RIGHT - MID
319         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
320
321         IF( ( WIDTH.LT.RTOL*TMP ) .OR.
322     $      (ITER.EQ.MAXITR) )THEN
323*           reduce number of unconverged intervals
324            NINT = NINT - 1
325*           Mark interval as converged.
326            IWORK( K-1 ) = 0
327            IF( I1.EQ.I ) THEN
328               I1 = NEXT
329            ELSE
330*              Prev holds the last unconverged interval previously examined
331               IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
332            END IF
333            I = NEXT
334            GO TO 100
335         END IF
336         PREV = I
337*
338*        Perform one bisection step
339*
340         CNT = 0
341         S = MID
342         DPLUS = D( 1 ) - S
343         IF( DPLUS.LT.ZERO ) CNT = CNT + 1
344         DO 90 J = 2, N
345            DPLUS = D( J ) - S - E2( J-1 )/DPLUS
346            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
347 90      CONTINUE
348         IF( CNT.LE.I-1 ) THEN
349            WORK( K-1 ) = MID
350         ELSE
351            WORK( K ) = MID
352         END IF
353         I = NEXT
354
355 100  CONTINUE
356      ITER = ITER + 1
357*     do another loop if there are still unconverged intervals
358*     However, in the last iteration, all intervals are accepted
359*     since this is the best we can do.
360      IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
361*
362*
363*     At this point, all the intervals have converged
364      DO 110 I = SAVI1, ILAST
365         K = 2*I
366         II = I - OFFSET
367*        All intervals marked by '0' have been refined.
368         IF( IWORK( K-1 ).EQ.0 ) THEN
369            W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
370            WERR( II ) = WORK( K ) - W( II )
371         END IF
372 110  CONTINUE
373*
374
375      RETURN
376*
377*     End of DLARRJ
378*
379      END
380