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