SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1,
$ RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
$ PIVMIN, SPDIAM, TWIST, INFO )
*
* -- LAPACK auxiliary routine (version 3.2.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2010
*
* .. 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) REAL
* The minimum pivot in the Sturm sequence.
*
* SPDIAM (input) REAL
* 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