dlarrj.f

          SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
         $                   RTOL, OFFSET, W, WERR, WORK, IWORK,
         $                   PIVMIN, SPDIAM, INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            IFIRST, ILAST, INFO, N, OFFSET
          DOUBLE PRECISION   PIVMIN, RTOL, SPDIAM
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   D( * ), E2( * ), W( * ),
         $                   WERR( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  Given the initial eigenvalue approximations of T, DLARRJ
    *  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) DOUBLE PRECISION array, dimension (N)
    *          The N diagonal elements of T.
    *
    *  E2      (input) DOUBLE PRECISION 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) DOUBLE PRECISION
    *          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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (2*N)
    *          Workspace.
    *
    *  IWORK   (workspace) INTEGER array, dimension (2*N)
    *          Workspace.
    *
    *  PIVMIN  (input) DOUBLE PRECISION
    *          The minimum pivot in the Sturm sequence for T.
    *
    *  SPDIAM  (input) DOUBLE PRECISION
    *          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 ..
          DOUBLE PRECISION   ZERO, ONE, TWO, HALF
          PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
         $                   HALF = 0.5D0 )
          INTEGER   MAXITR
    *     ..
    *     .. Local Scalars ..
          INTEGER            CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
         $                   OLNINT, P, PREV, SAVI1
          DOUBLE PRECISION   DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
    *
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX
    *     ..
    *     .. 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 = MAX( ABS( 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 = MAX( ABS( 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 DLARRJ
    *
          END