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
      SUBROUTINE SLARRJ( N, D, E2, IFIRST, ILAST,
     $                   RTOL, OFFSET, W, WERR, WORK, IWORK,
     $                   PIVMIN, SPDIAM, INFO )
*
*  -- LAPACK auxiliary routine (version 3.2.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2010
*
*     .. Scalar Arguments ..
      INTEGER            IFIRST, ILAST, INFO, N, OFFSET
      REAL               PIVMIN, RTOL, SPDIAM
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      REAL               D( * ), E2( * ), W( * ),
     $                   WERR( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  Given the initial eigenvalue approximations of T, SLARRJ
*  does  bisection to refine the eigenvalues of 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 in WERR. During bisection, intervals
*  [left, right] are maintained by storing their mid-points and
*  semi-widths in the arrays W and WERR respectively.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the matrix.
*
*  D       (input) REAL             array, dimension (N)
*          The N diagonal elements of T.
*
*  E2      (input) REAL             array, dimension (N-1)
*          The Squares of the (N-1) subdiagonal elements of T.
*
*  IFIRST  (input) INTEGER
*          The index of the first eigenvalue to be computed.
*
*  ILAST   (input) INTEGER
*          The index of the last eigenvalue to be computed.
*
*  RTOL   (input) REAL            
*          Tolerance for the convergence of the bisection intervals.
*          An interval [LEFT,RIGHT] has converged if
*          RIGHT-LEFT.LT.RTOL*MAX(|LEFT|,|RIGHT|).
*
*  OFFSET  (input) INTEGER
*          Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
*          through ILAST-OFFSET elements of these arrays are to be used.
*
*  W       (input/output) 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.
*
*  WERR    (input/output) 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.
*
*  WORK    (workspace) REAL             array, dimension (2*N)
*          Workspace.
*
*  IWORK   (workspace) INTEGER array, dimension (2*N)
*          Workspace.
*
*  PIVMIN  (input) REAL
*          The minimum pivot in the Sturm sequence for T.
*
*  SPDIAM  (input) REAL
*          The spectral diameter of T.
*
*  INFO    (output) INTEGER
*          Error flag.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     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
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TWO, HALF
      PARAMETER        ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0,
     $                   HALF = 0.5E0 )
      INTEGER   MAXITR
*     ..
*     .. Local Scalars ..
      INTEGER            CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
     $                   OLNINT, P, PREV, SAVI1
      REAL               DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
*
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABSMAX
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
      MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2
*
*     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
      I2 = ILAST
*     The number of unconverged intervals
      NINT = 0
*     The last unconverged interval found
      PREV = 0
      DO 75 I = I1, I2
         K = 2*I
         II = I - OFFSET
         LEFT = W( II ) - WERR( II )
         MID = W(II)
         RIGHT = W( II ) + WERR( II )
         WIDTH = RIGHT - MID
         TMP = MAXABS( LEFT ), ABS( RIGHT ) )

*        The following test prevents the test of converged intervals
         IF( WIDTH.LT.RTOL*TMP ) 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.I2)) I1 = I + 1
            IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
         ELSE
*           unconverged interval found
            PREV = I
*           Make sure that [LEFT,RIGHT] contains the desired eigenvalue
*
*           Do while( CNT(LEFT).GT.I-1 )
*
            FAC = ONE
 20         CONTINUE
            CNT = 0
            S = LEFT
            DPLUS = D( 1 ) - S
            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
            DO 30 J = 2, N
               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
 30         CONTINUE
            IF( CNT.GT.I-1 ) THEN
               LEFT = LEFT - WERR( II )*FAC
               FAC = TWO*FAC
               GO TO 20
            END IF
*
*           Do while( CNT(RIGHT).LT.I )
*
            FAC = ONE
 50         CONTINUE
            CNT = 0
            S = RIGHT
            DPLUS = D( 1 ) - S
            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
            DO 60 J = 2, N
               DPLUS = D( J ) - S - E2( J-1 )/DPLUS
               IF( DPLUS.LT.ZERO ) CNT = CNT + 1
 60         CONTINUE
            IF( CNT.LT.I ) THEN
               RIGHT = RIGHT + WERR( II )*FAC
               FAC = TWO*FAC
               GO TO 50
            END IF
            NINT = NINT + 1
            IWORK( K-1 ) = I + 1
            IWORK( K ) = CNT
         END IF
         WORK( K-1 ) = LEFT
         WORK( K ) = RIGHT
 75   CONTINUE


      SAVI1 = I1
*
*     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 P = 1, OLNINT
         K = 2*I
         II = I - OFFSET
         NEXT = IWORK( K-1 )
         LEFT = WORK( K-1 )
         RIGHT = WORK( K )
         MID = HALF*( LEFT + RIGHT )

*        semiwidth of interval
         WIDTH = RIGHT - MID
         TMP = MAXABS( LEFT ), ABS( RIGHT ) )

         IF( ( WIDTH.LT.RTOL*TMP ) .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
*
         CNT = 0
         S = MID
         DPLUS = D( 1 ) - S
         IF( DPLUS.LT.ZERO ) CNT = CNT + 1
         DO 90 J = 2, N
            DPLUS = D( J ) - S - E2( J-1 )/DPLUS
            IF( DPLUS.LT.ZERO ) CNT = CNT + 1
 90      CONTINUE
         IF( CNT.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 = SAVI1, 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
*

      RETURN
*
*     End of SLARRJ
*
      END