*> \brief \b SLARRB provides limited bisection to locate eigenvalues for more accuracy.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarrb.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
* RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
* PIVMIN, SPDIAM, TWIST, INFO )
*
* .. 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( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is REAL array, dimension (N)
*> The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*> LLD is REAL array, dimension (N-1)
*> The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] IFIRST
*> \verbatim
*> IFIRST is INTEGER
*> The index of the first eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] ILAST
*> \verbatim
*> ILAST is INTEGER
*> The index of the last eigenvalue to be computed.
*> \endverbatim
*>
*> \param[in] RTOL1
*> \verbatim
*> RTOL1 is REAL
*> \endverbatim
*>
*> \param[in] RTOL2
*> \verbatim
*> RTOL2 is REAL
*> 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.
*> \endverbatim
*>
*> \param[in] OFFSET
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is 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.
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*> WGAP is 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 = ILAST
*> then WGAP(IFIRST-OFFSET) must be set to ZERO.
*> On output, these gaps are refined.
*> \endverbatim
*>
*> \param[in,out] WERR
*> \verbatim
*> WERR is 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is REAL
*> The minimum pivot in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*> SPDIAM is REAL
*> The spectral diameter of the matrix.
*> \endverbatim
*>
*> \param[in] TWIST
*> \verbatim
*> 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)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> Error flag.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
$ RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
$ PIVMIN, SPDIAM, TWIST, INFO )
*
* -- 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
REAL PIVMIN, RTOL1, RTOL2, SPDIAM
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL D( * ), LLD( * ), W( * ),
$ WERR( * ), WGAP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. 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
*
* 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 = 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