Annotation of rpl/lapack/lapack/dlarrj.f, revision 1.2

1.1       bertrand    1:       SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
                      2:      $                   RTOL, OFFSET, W, WERR, WORK, IWORK,
                      3:      $                   PIVMIN, SPDIAM, INFO )
                      4: *
                      5: *  -- LAPACK auxiliary routine (version 3.2) --
                      6: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                      7: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                      8: *     November 2006
                      9: *
                     10: *     .. Scalar Arguments ..
                     11:       INTEGER            IFIRST, ILAST, INFO, N, OFFSET
                     12:       DOUBLE PRECISION   PIVMIN, RTOL, SPDIAM
                     13: *     ..
                     14: *     .. Array Arguments ..
                     15:       INTEGER            IWORK( * )
                     16:       DOUBLE PRECISION   D( * ), E2( * ), W( * ),
                     17:      $                   WERR( * ), WORK( * )
                     18: *     ..
                     19: *
                     20: *  Purpose
                     21: *  =======
                     22: *
                     23: *  Given the initial eigenvalue approximations of T, DLARRJ
                     24: *  does  bisection to refine the eigenvalues of T,
                     25: *  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
                     26: *  guesses for these eigenvalues are input in W, the corresponding estimate
                     27: *  of the error in these guesses in WERR. During bisection, intervals
                     28: *  [left, right] are maintained by storing their mid-points and
                     29: *  semi-widths in the arrays W and WERR respectively.
                     30: *
                     31: *  Arguments
                     32: *  =========
                     33: *
                     34: *  N       (input) INTEGER
                     35: *          The order of the matrix.
                     36: *
                     37: *  D       (input) DOUBLE PRECISION array, dimension (N)
                     38: *          The N diagonal elements of T.
                     39: *
                     40: *  E2      (input) DOUBLE PRECISION array, dimension (N-1)
                     41: *          The Squares of the (N-1) subdiagonal elements of T.
                     42: *
                     43: *  IFIRST  (input) INTEGER
                     44: *          The index of the first eigenvalue to be computed.
                     45: *
                     46: *  ILAST   (input) INTEGER
                     47: *          The index of the last eigenvalue to be computed.
                     48: *
                     49: *  RTOL   (input) DOUBLE PRECISION
                     50: *          Tolerance for the convergence of the bisection intervals.
                     51: *          An interval [LEFT,RIGHT] has converged if
                     52: *          RIGHT-LEFT.LT.RTOL*MAX(|LEFT|,|RIGHT|).
                     53: *
                     54: *  OFFSET  (input) INTEGER
                     55: *          Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
                     56: *          through ILAST-OFFSET elements of these arrays are to be used.
                     57: *
                     58: *  W       (input/output) DOUBLE PRECISION array, dimension (N)
                     59: *          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
                     60: *          estimates of the eigenvalues of L D L^T indexed IFIRST through
                     61: *          ILAST.
                     62: *          On output, these estimates are refined.
                     63: *
                     64: *  WERR    (input/output) DOUBLE PRECISION array, dimension (N)
                     65: *          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
                     66: *          the errors in the estimates of the corresponding elements in W.
                     67: *          On output, these errors are refined.
                     68: *
                     69: *  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
                     70: *          Workspace.
                     71: *
                     72: *  IWORK   (workspace) INTEGER array, dimension (2*N)
                     73: *          Workspace.
                     74: *
                     75: *  PIVMIN  (input) DOUBLE PRECISION
                     76: *          The minimum pivot in the Sturm sequence for T.
                     77: *
                     78: *  SPDIAM  (input) DOUBLE PRECISION
                     79: *          The spectral diameter of T.
                     80: *
                     81: *  INFO    (output) INTEGER
                     82: *          Error flag.
                     83: *
                     84: *  Further Details
                     85: *  ===============
                     86: *
                     87: *  Based on contributions by
                     88: *     Beresford Parlett, University of California, Berkeley, USA
                     89: *     Jim Demmel, University of California, Berkeley, USA
                     90: *     Inderjit Dhillon, University of Texas, Austin, USA
                     91: *     Osni Marques, LBNL/NERSC, USA
                     92: *     Christof Voemel, University of California, Berkeley, USA
                     93: *
                     94: *  =====================================================================
                     95: *
                     96: *     .. Parameters ..
                     97:       DOUBLE PRECISION   ZERO, ONE, TWO, HALF
                     98:       PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
                     99:      $                   HALF = 0.5D0 )
                    100:       INTEGER   MAXITR
                    101: *     ..
                    102: *     .. Local Scalars ..
                    103:       INTEGER            CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
                    104:      $                   OLNINT, P, PREV, SAVI1
                    105:       DOUBLE PRECISION   DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
                    106: *
                    107: *     ..
                    108: *     .. Intrinsic Functions ..
                    109:       INTRINSIC          ABS, MAX
                    110: *     ..
                    111: *     .. Executable Statements ..
                    112: *
                    113:       INFO = 0
                    114: *
                    115:       MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
                    116:      $           LOG( TWO ) ) + 2
                    117: *
                    118: *     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
                    119: *     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
                    120: *     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
                    121: *     for an unconverged interval is set to the index of the next unconverged
                    122: *     interval, and is -1 or 0 for a converged interval. Thus a linked
                    123: *     list of unconverged intervals is set up.
                    124: *
                    125: 
                    126:       I1 = IFIRST
                    127:       I2 = ILAST
                    128: *     The number of unconverged intervals
                    129:       NINT = 0
                    130: *     The last unconverged interval found
                    131:       PREV = 0
                    132:       DO 75 I = I1, I2
                    133:          K = 2*I
                    134:          II = I - OFFSET
                    135:          LEFT = W( II ) - WERR( II )
                    136:          MID = W(II)
                    137:          RIGHT = W( II ) + WERR( II )
                    138:          WIDTH = RIGHT - MID
                    139:          TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
                    140: 
                    141: *        The following test prevents the test of converged intervals
                    142:          IF( WIDTH.LT.RTOL*TMP ) THEN
                    143: *           This interval has already converged and does not need refinement.
                    144: *           (Note that the gaps might change through refining the
                    145: *            eigenvalues, however, they can only get bigger.)
                    146: *           Remove it from the list.
                    147:             IWORK( K-1 ) = -1
                    148: *           Make sure that I1 always points to the first unconverged interval
                    149:             IF((I.EQ.I1).AND.(I.LT.I2)) I1 = I + 1
                    150:             IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
                    151:          ELSE
                    152: *           unconverged interval found
                    153:             PREV = I
                    154: *           Make sure that [LEFT,RIGHT] contains the desired eigenvalue
                    155: *
                    156: *           Do while( CNT(LEFT).GT.I-1 )
                    157: *
                    158:             FAC = ONE
                    159:  20         CONTINUE
                    160:             CNT = 0
                    161:             S = LEFT
                    162:             DPLUS = D( 1 ) - S
                    163:             IF( DPLUS.LT.ZERO ) CNT = CNT + 1
                    164:             DO 30 J = 2, N
                    165:                DPLUS = D( J ) - S - E2( J-1 )/DPLUS
                    166:                IF( DPLUS.LT.ZERO ) CNT = CNT + 1
                    167:  30         CONTINUE
                    168:             IF( CNT.GT.I-1 ) THEN
                    169:                LEFT = LEFT - WERR( II )*FAC
                    170:                FAC = TWO*FAC
                    171:                GO TO 20
                    172:             END IF
                    173: *
                    174: *           Do while( CNT(RIGHT).LT.I )
                    175: *
                    176:             FAC = ONE
                    177:  50         CONTINUE
                    178:             CNT = 0
                    179:             S = RIGHT
                    180:             DPLUS = D( 1 ) - S
                    181:             IF( DPLUS.LT.ZERO ) CNT = CNT + 1
                    182:             DO 60 J = 2, N
                    183:                DPLUS = D( J ) - S - E2( J-1 )/DPLUS
                    184:                IF( DPLUS.LT.ZERO ) CNT = CNT + 1
                    185:  60         CONTINUE
                    186:             IF( CNT.LT.I ) THEN
                    187:                RIGHT = RIGHT + WERR( II )*FAC
                    188:                FAC = TWO*FAC
                    189:                GO TO 50
                    190:             END IF
                    191:             NINT = NINT + 1
                    192:             IWORK( K-1 ) = I + 1
                    193:             IWORK( K ) = CNT
                    194:          END IF
                    195:          WORK( K-1 ) = LEFT
                    196:          WORK( K ) = RIGHT
                    197:  75   CONTINUE
                    198: 
                    199: 
                    200:       SAVI1 = I1
                    201: *
                    202: *     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
                    203: *     and while (ITER.LT.MAXITR)
                    204: *
                    205:       ITER = 0
                    206:  80   CONTINUE
                    207:       PREV = I1 - 1
                    208:       I = I1
                    209:       OLNINT = NINT
                    210: 
                    211:       DO 100 P = 1, OLNINT
                    212:          K = 2*I
                    213:          II = I - OFFSET
                    214:          NEXT = IWORK( K-1 )
                    215:          LEFT = WORK( K-1 )
                    216:          RIGHT = WORK( K )
                    217:          MID = HALF*( LEFT + RIGHT )
                    218: 
                    219: *        semiwidth of interval
                    220:          WIDTH = RIGHT - MID
                    221:          TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
                    222: 
                    223:          IF( ( WIDTH.LT.RTOL*TMP ) .OR.
                    224:      $      (ITER.EQ.MAXITR) )THEN
                    225: *           reduce number of unconverged intervals
                    226:             NINT = NINT - 1
                    227: *           Mark interval as converged.
                    228:             IWORK( K-1 ) = 0
                    229:             IF( I1.EQ.I ) THEN
                    230:                I1 = NEXT
                    231:             ELSE
                    232: *              Prev holds the last unconverged interval previously examined
                    233:                IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
                    234:             END IF
                    235:             I = NEXT
                    236:             GO TO 100
                    237:          END IF
                    238:          PREV = I
                    239: *
                    240: *        Perform one bisection step
                    241: *
                    242:          CNT = 0
                    243:          S = MID
                    244:          DPLUS = D( 1 ) - S
                    245:          IF( DPLUS.LT.ZERO ) CNT = CNT + 1
                    246:          DO 90 J = 2, N
                    247:             DPLUS = D( J ) - S - E2( J-1 )/DPLUS
                    248:             IF( DPLUS.LT.ZERO ) CNT = CNT + 1
                    249:  90      CONTINUE
                    250:          IF( CNT.LE.I-1 ) THEN
                    251:             WORK( K-1 ) = MID
                    252:          ELSE
                    253:             WORK( K ) = MID
                    254:          END IF
                    255:          I = NEXT
                    256: 
                    257:  100  CONTINUE
                    258:       ITER = ITER + 1
                    259: *     do another loop if there are still unconverged intervals
                    260: *     However, in the last iteration, all intervals are accepted
                    261: *     since this is the best we can do.
                    262:       IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
                    263: *
                    264: *
                    265: *     At this point, all the intervals have converged
                    266:       DO 110 I = SAVI1, ILAST
                    267:          K = 2*I
                    268:          II = I - OFFSET
                    269: *        All intervals marked by '0' have been refined.
                    270:          IF( IWORK( K-1 ).EQ.0 ) THEN
                    271:             W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
                    272:             WERR( II ) = WORK( K ) - W( II )
                    273:          END IF
                    274:  110  CONTINUE
                    275: *
                    276: 
                    277:       RETURN
                    278: *
                    279: *     End of DLARRJ
                    280: *
                    281:       END

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