slarrb.f

          SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
         $                   RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
         $                   PIVMIN, SPDIAM, TWIST, 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, TWIST
          REAL               PIVMIN, RTOL1, RTOL2, SPDIAM
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          REAL               D( * ), LLD( * ), W( * ),
         $                   WERR( * ), WGAP( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  Given the relatively robust representation(RRR) L D L^T, SLARRB
    *  does "limited" bisection to refine the eigenvalues of L D L^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 and their gaps are input in WERR
    *  and WGAP, respectively. 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 the diagonal matrix D.
    *
    *  LLD     (input) REAL             array, dimension (N-1)
    *          The (N-1) elements L(i)*L(i)*D(i).
    *
    *  IFIRST  (input) INTEGER
    *          The index of the first eigenvalue to be computed.
    *
    *  ILAST   (input) INTEGER
    *          The index of the last eigenvalue to be computed.
    *
    *  RTOL1   (input) REAL
    *  RTOL2   (input) REAL
    *          Tolerance for the convergence of the bisection intervals.
    *          An interval [LEFT,RIGHT] has converged if
    *          RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
    *          where GAP is the (estimated) distance to the nearest
    *          eigenvalue.
    *
    *  OFFSET  (input) INTEGER
    *          Offset for the arrays W, WGAP 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 throug
    *          ILAST.
    *          On output, these estimates are refined.
    *
    *  WGAP    (input/output) REAL             array, dimension (N-1)
    *          On input, the (estimated) gaps between consecutive
    *          eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between
    *          eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST
    *          then WGAP(IFIRST-OFFSET) must be set to ZERO.
    *          On output, these gaps 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) DOUBLE PRECISION
    *          The minimum pivot in the Sturm sequence.
    *
    *  SPDIAM  (input) DOUBLE PRECISION
    *          The spectral diameter of the matrix.
    *
    *  TWIST   (input) INTEGER
    *          The twist index for the twisted factorization that is used
    *          for the negcount.
    *          TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T
    *          TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T
    *          TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)
    *
    *  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, TWO, HALF
          PARAMETER        ( ZERO = 0.0E0, TWO = 2.0E0,
         $                   HALF = 0.5E0 )
          INTEGER   MAXITR
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, I1, II, IP, ITER, K, NEGCNT, NEXT, NINT,
         $                   OLNINT, PREV, R
          REAL               BACK, CVRGD, GAP, LEFT, LGAP, MID, MNWDTH,
         $                   RGAP, RIGHT, TMP, WIDTH
    *     ..
    *     .. External Functions ..
          INTEGER            SLANEG
          EXTERNAL           SLANEG
    *
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
          INFO = 0
    *
          MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
         $           LOG( TWO ) ) + 2
          MNWDTH = TWO * PIVMIN
    *
          R = TWIST
          IF((R.LT.1).OR.(R.GT.N)) R = N
    *
    *     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
    *     The number of unconverged intervals
          NINT = 0
    *     The last unconverged interval found
          PREV = 0

          RGAP = WGAP( I1-OFFSET )
          DO 75 I = I1, ILAST
             K = 2*I
             II = I - OFFSET
             LEFT = W( II ) - WERR( II )
             RIGHT = W( II ) + WERR( II )
             LGAP = RGAP
             RGAP = WGAP( II )
             GAP = MIN( LGAP, RGAP )

    *        Make sure that [LEFT,RIGHT] contains the desired eigenvalue
    *        Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT
    *
    *        Do while( NEGCNT(LEFT).GT.I-1 )
    *
             BACK = WERR( II )
     20      CONTINUE
             NEGCNT = SLANEG( N, D, LLD, LEFT, PIVMIN, R )
             IF( NEGCNT.GT.I-1 ) THEN
                LEFT = LEFT - BACK
                BACK = TWO*BACK
                GO TO 20
             END IF
    *
    *        Do while( NEGCNT(RIGHT).LT.I )
    *        Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT
    *
             BACK = WERR( II )
     50      CONTINUE

             NEGCNT = SLANEG( N, D, LLD, RIGHT, PIVMIN, R )
              IF( NEGCNT.LT.I ) THEN
                 RIGHT = RIGHT + BACK
                 BACK = TWO*BACK
                 GO TO 50
              END IF
             WIDTH = HALF*ABS( LEFT - RIGHT )
             TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
             CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
             IF( WIDTH.LE.CVRGD .OR. WIDTH.LE.MNWDTH ) 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.ILAST)) I1 = I + 1
                IF((PREV.GE.I1).AND.(I.LE.ILAST)) IWORK( 2*PREV-1 ) = I + 1
             ELSE
    *           unconverged interval found
                PREV = I
                NINT = NINT + 1
                IWORK( K-1 ) = I + 1
                IWORK( K ) = NEGCNT
             END IF
             WORK( K-1 ) = LEFT
             WORK( K ) = RIGHT
     75   CONTINUE

    *
    *     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 IP = 1, OLNINT
             K = 2*I
             II = I - OFFSET
             RGAP = WGAP( II )
             LGAP = RGAP
             IF(II.GT.1) LGAP = WGAP( II-1 )
             GAP = MIN( LGAP, RGAP )
             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 ) )
             CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
             IF( ( WIDTH.LE.CVRGD ) .OR. ( WIDTH.LE.MNWDTH ).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
    *
             NEGCNT = SLANEG( N, D, LLD, MID, PIVMIN, R )
             IF( NEGCNT.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 = IFIRST, 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
    *
          DO 111 I = IFIRST+1, ILAST
             K = 2*I
             II = I - OFFSET
             WGAP( II-1 ) = MAX( ZERO,
         $                     W(II) - WERR (II) - W( II-1 ) - WERR( II-1 ))
     111  CONTINUE

          RETURN
    *
    *     End of SLARRB
    *
          END