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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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*> \ingroup OTHERauxiliary
154*
155*> \par Contributors:
156*  ==================
157*>
158*> Beresford Parlett, University of California, Berkeley, USA \n
159*> Jim Demmel, University of California, Berkeley, USA \n
160*> Inderjit Dhillon, University of Texas, Austin, USA \n
161*> Osni Marques, LBNL/NERSC, USA \n
162*> Christof Voemel, University of California, Berkeley, USA
163*
164*  =====================================================================
165      SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
166     $                   RTOL, OFFSET, W, WERR, WORK, IWORK,
167     $                   PIVMIN, SPDIAM, INFO )
168*
169*  -- LAPACK auxiliary routine --
170*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
171*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
172*
173*     .. Scalar Arguments ..
174      INTEGER            IFIRST, ILAST, INFO, N, OFFSET
175      DOUBLE PRECISION   PIVMIN, RTOL, SPDIAM
176*     ..
177*     .. Array Arguments ..
178      INTEGER            IWORK( * )
179      DOUBLE PRECISION   D( * ), E2( * ), W( * ),
180     $                   WERR( * ), WORK( * )
181*     ..
182*
183*  =====================================================================
184*
185*     .. Parameters ..
186      DOUBLE PRECISION   ZERO, ONE, TWO, HALF
187      PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
188     $                   HALF = 0.5D0 )
189      INTEGER   MAXITR
190*     ..
191*     .. Local Scalars ..
192      INTEGER            CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
193     $                   OLNINT, P, PREV, SAVI1
194      DOUBLE PRECISION   DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
195*
196*     ..
197*     .. Intrinsic Functions ..
198      INTRINSIC          ABS, MAX
199*     ..
200*     .. Executable Statements ..
201*
202      INFO = 0
203*
204*     Quick return if possible
205*
206      IF( N.LE.0 ) THEN
207         RETURN
208      END IF
209*
210      MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
211     $           LOG( TWO ) ) + 2
212*
213*     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
214*     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
215*     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
216*     for an unconverged interval is set to the index of the next unconverged
217*     interval, and is -1 or 0 for a converged interval. Thus a linked
218*     list of unconverged intervals is set up.
219*
220
221      I1 = IFIRST
222      I2 = ILAST
223*     The number of unconverged intervals
224      NINT = 0
225*     The last unconverged interval found
226      PREV = 0
227      DO 75 I = I1, I2
228         K = 2*I
229         II = I - OFFSET
230         LEFT = W( II ) - WERR( II )
231         MID = W(II)
232         RIGHT = W( II ) + WERR( II )
233         WIDTH = RIGHT - MID
234         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
235
236*        The following test prevents the test of converged intervals
237         IF( WIDTH.LT.RTOL*TMP ) THEN
238*           This interval has already converged and does not need refinement.
239*           (Note that the gaps might change through refining the
240*            eigenvalues, however, they can only get bigger.)
241*           Remove it from the list.
242            IWORK( K-1 ) = -1
243*           Make sure that I1 always points to the first unconverged interval
244            IF((I.EQ.I1).AND.(I.LT.I2)) I1 = I + 1
245            IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
246         ELSE
247*           unconverged interval found
248            PREV = I
249*           Make sure that [LEFT,RIGHT] contains the desired eigenvalue
250*
251*           Do while( CNT(LEFT).GT.I-1 )
252*
253            FAC = ONE
254 20         CONTINUE
255            CNT = 0
256            S = LEFT
257            DPLUS = D( 1 ) - S
258            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
259            DO 30 J = 2, N
260               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
261               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
262 30         CONTINUE
263            IF( CNT.GT.I-1 ) THEN
264               LEFT = LEFT - WERR( II )*FAC
265               FAC = TWO*FAC
266               GO TO 20
267            END IF
268*
269*           Do while( CNT(RIGHT).LT.I )
270*
271            FAC = ONE
272 50         CONTINUE
273            CNT = 0
274            S = RIGHT
275            DPLUS = D( 1 ) - S
276            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
277            DO 60 J = 2, N
278               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
279               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
280 60         CONTINUE
281            IF( CNT.LT.I ) THEN
282               RIGHT = RIGHT + WERR( II )*FAC
283               FAC = TWO*FAC
284               GO TO 50
285            END IF
286            NINT = NINT + 1
287            IWORK( K-1 ) = I + 1
288            IWORK( K ) = CNT
289         END IF
290         WORK( K-1 ) = LEFT
291         WORK( K ) = RIGHT
292 75   CONTINUE
293
294
295      SAVI1 = I1
296*
297*     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
298*     and while (ITER.LT.MAXITR)
299*
300      ITER = 0
301 80   CONTINUE
302      PREV = I1 - 1
303      I = I1
304      OLNINT = NINT
305
306      DO 100 P = 1, OLNINT
307         K = 2*I
308         II = I - OFFSET
309         NEXT = IWORK( K-1 )
310         LEFT = WORK( K-1 )
311         RIGHT = WORK( K )
312         MID = HALF*( LEFT + RIGHT )
313
314*        semiwidth of interval
315         WIDTH = RIGHT - MID
316         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
317
318         IF( ( WIDTH.LT.RTOL*TMP ) .OR.
319     $      (ITER.EQ.MAXITR) )THEN
320*           reduce number of unconverged intervals
321            NINT = NINT - 1
322*           Mark interval as converged.
323            IWORK( K-1 ) = 0
324            IF( I1.EQ.I ) THEN
325               I1 = NEXT
326            ELSE
327*              Prev holds the last unconverged interval previously examined
328               IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
329            END IF
330            I = NEXT
331            GO TO 100
332         END IF
333         PREV = I
334*
335*        Perform one bisection step
336*
337         CNT = 0
338         S = MID
339         DPLUS = D( 1 ) - S
340         IF( DPLUS.LT.ZERO ) CNT = CNT + 1
341         DO 90 J = 2, N
342            DPLUS = D( J ) - S - E2( J-1 )/DPLUS
343            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
344 90      CONTINUE
345         IF( CNT.LE.I-1 ) THEN
346            WORK( K-1 ) = MID
347         ELSE
348            WORK( K ) = MID
349         END IF
350         I = NEXT
351
352 100  CONTINUE
353      ITER = ITER + 1
354*     do another loop if there are still unconverged intervals
355*     However, in the last iteration, all intervals are accepted
356*     since this is the best we can do.
357      IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
358*
359*
360*     At this point, all the intervals have converged
361      DO 110 I = SAVI1, ILAST
362         K = 2*I
363         II = I - OFFSET
364*        All intervals marked by '0' have been refined.
365         IF( IWORK( K-1 ).EQ.0 ) THEN
366            W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
367            WERR( II ) = WORK( K ) - W( II )
368         END IF
369 110  CONTINUE
370*
371
372      RETURN
373*
374*     End of DLARRJ
375*
376      END
377