dlarrb()

subroutine dlarrb	(	integer	n,
		double precision, dimension( * )	d,
		double precision, dimension( * )	lld,
		integer	ifirst,
		integer	ilast,
		double precision	rtol1,
		double precision	rtol2,
		integer	offset,
		double precision, dimension( * )	w,
		double precision, dimension( * )	wgap,
		double precision, dimension( * )	werr,
		double precision, dimension( * )	work,
		integer, dimension( * )	iwork,
		double precision	pivmin,
		double precision	spdiam,
		integer	twist,
		integer	info
	)

DLARRB provides limited bisection to locate eigenvalues for more accuracy.

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Purpose:

 Given the relatively robust representation(RRR) L D L^T, DLARRB
 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.

Parameters

[in]	N	N is INTEGER The order of the matrix.
[in]	D	D is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D.
[in]	LLD	LLD is DOUBLE PRECISION array, dimension (N-1) The (N-1) elements L(i)*L(i)*D(i).
[in]	IFIRST	IFIRST is INTEGER The index of the first eigenvalue to be computed.
[in]	ILAST	ILAST is INTEGER The index of the last eigenvalue to be computed.
[in]	RTOL1	RTOL1 is DOUBLE PRECISION
[in]	RTOL2	RTOL2 is DOUBLE PRECISION Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) where GAP is the (estimated) distance to the nearest eigenvalue.
[in]	OFFSET	OFFSET is 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.
[in,out]	W	W is 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.
[in,out]	WGAP	WGAP is DOUBLE PRECISION 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 = ILAST then WGAP(IFIRST-OFFSET) must be set to ZERO. On output, these gaps are refined.
[in,out]	WERR	WERR is 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.
[out]	WORK	WORK is DOUBLE PRECISION array, dimension (2*N) Workspace.
[out]	IWORK	IWORK is INTEGER array, dimension (2*N) Workspace.
[in]	PIVMIN	PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence.
[in]	SPDIAM	SPDIAM is DOUBLE PRECISION The spectral diameter of the matrix.
[in]	TWIST	TWIST is 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)
[out]	INFO	INFO is INTEGER Error flag.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Definition at line 193 of file dlarrb.f.

*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
      DOUBLE PRECISION   PIVMIN, RTOL1, RTOL2, SPDIAM
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), LLD( * ), W( * ),
     $                   WERR( * ), WGAP( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, TWO, HALF
      parameter( zero = 0.0d0, two = 2.0d0,
     $                   half = 0.5d0 )
      INTEGER   MAXITR
*     ..
*     .. Local Scalars ..
      INTEGER            I, I1, II, IP, ITER, K, NEGCNT, NEXT, NINT,
     $                   OLNINT, PREV, R
      DOUBLE PRECISION   BACK, CVRGD, GAP, LEFT, LGAP, MID, MNWDTH,
     $                   RGAP, RIGHT, TMP, WIDTH
*     ..
*     .. External Functions ..
      INTEGER            DLANEG
      EXTERNAL           dlaneg
*
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min
*     ..
*     .. Executable Statements ..
*
      info = 0
*
*     Quick return if possible
*
      IF( n.LE.0 ) THEN
         RETURN
      END IF
*
      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 = dlaneg( 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 = dlaneg( 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 = dlaneg( 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 DLARRB
*

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