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SUBROUTINE DLARRB( N, D, L, LD, LLD, IFIRST, ILAST, SIGMA, RELTOL,
$ W, WGAP, WERR, WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine (instru to count ops, version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER IFIRST, ILAST, INFO, N
DOUBLE PRECISION RELTOL, SIGMA
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ), W( * ),
$ WERR( * ), WGAP( * ), WORK( * )
* ..
* Common block to return operation count
* .. Common blocks ..
COMMON / LATIME / OPS, ITCNT
* ..
* .. Scalars in Common ..
DOUBLE PRECISION ITCNT, OPS
* ..
*
* Purpose
* =======
*
* Given the relatively robust representation(RRR) L D L^T, DLARRB
* does ``limited'' bisection to locate the eigenvalues of L D L^T,
* W( IFIRST ) thru' W( ILAST ), to more accuracy. 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 the diagonal matrix D.
*
* L (input) DOUBLE PRECISION array, dimension (N-1)
* The n-1 subdiagonal elements of the unit bidiagonal matrix L.
*
* LD (input) DOUBLE PRECISION array, dimension (N-1)
* The n-1 elements L(i)*D(i).
*
* LLD (input) DOUBLE PRECISION array, dimension (N-1)
* The n-1 elements L(i)*L(i)*D(i).
*
* IFIRST (input) INTEGER
* The index of the first eigenvalue in the cluster.
*
* ILAST (input) INTEGER
* The index of the last eigenvalue in the cluster.
*
* SIGMA (input) DOUBLE PRECISION
* The shift used to form L D L^T (see DLARRF).
*
* RELTOL (input) DOUBLE PRECISION
* The relative tolerance.
*
* W (input/output) DOUBLE PRECISION array, dimension (N)
* On input, W( IFIRST ) thru' W( ILAST ) are estimates of the
* corresponding eigenvalues of L D L^T.
* On output, these estimates are ``refined''.
*
* WGAP (input/output) DOUBLE PRECISION array, dimension (N)
* The gaps between the eigenvalues of L D L^T. Very small
* gaps are changed on output.
*
* WERR (input/output) DOUBLE PRECISION array, dimension (N)
* On input, WERR( IFIRST ) thru' WERR( ILAST ) are the errors
* in the estimates W( IFIRST ) thru' W( ILAST ).
* On output, these are the ``refined'' errors.
*
*****Reminder to Inder --- WORK is never used in this subroutine *****
* WORK (input) DOUBLE PRECISION array, dimension (???)
* Workspace.
*
* IWORK (input) INTEGER array, dimension (2*N)
* Workspace.
*
*****Reminder to Inder --- INFO is never set in this subroutine ******
* INFO (output) INTEGER
* Error flag.
*
* Further Details
* ===============
*
* Based on contributions by
* Inderjit Dhillon, IBM Almaden, USA
* Osni Marques, LBNL/NERSC, USA
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, TWO, HALF
PARAMETER ( ZERO = 0.0D0, TWO = 2.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
INTEGER CNT, I, I1, I2, INITI1, INITI2, J, K, NCNVRG,
$ NEIG, NINT, NRIGHT, OLNINT
DOUBLE PRECISION DELTA, EPS, GAP, LEFT, MID, PERT, RIGHT, S,
$ THRESH, TMP, WIDTH
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
EPS = DLAMCH( 'Precision' )
I1 = IFIRST
I2 = IFIRST
NEIG = ILAST - IFIRST + 1
NCNVRG = 0
THRESH = RELTOL
DO 10 I = IFIRST, ILAST
OPS = OPS + DBLE( 3 )
IWORK( I ) = 0
PERT = EPS*( ABS( SIGMA )+ABS( W( I ) ) )
WERR( I ) = WERR( I ) + PERT
IF( WGAP( I ).LT.PERT )
$ WGAP( I ) = PERT
10 CONTINUE
DO 20 I = I1, ILAST
IF( I.EQ.1 ) THEN
GAP = WGAP( I )
ELSE IF( I.EQ.N ) THEN
GAP = WGAP( I-1 )
ELSE
GAP = MIN( WGAP( I-1 ), WGAP( I ) )
END IF
OPS = OPS + DBLE( 1 )
IF( WERR( I ).LT.THRESH*GAP ) THEN
NCNVRG = NCNVRG + 1
IWORK( I ) = 1
IF( I1.EQ.I )
$ I1 = I1 + 1
ELSE
I2 = I
END IF
20 CONTINUE
*
* Initialize the unconverged intervals.
*
I = I1
NINT = 0
RIGHT = ZERO
30 CONTINUE
IF( I.LE.I2 ) THEN
IF( IWORK( I ).EQ.0 ) THEN
DELTA = EPS
OPS = OPS + DBLE( 1 )
LEFT = W( I ) - WERR( I )
*
* Do while( CNT(LEFT).GT.I-1 )
*
40 CONTINUE
IF( I.GT.I1 .AND. LEFT.LE.RIGHT ) THEN
LEFT = RIGHT
CNT = I - 1
ELSE
S = -LEFT
CNT = 0
DO 50 J = 1, N - 1
OPS = OPS + DBLE( 5 )
TMP = D( J ) + S
S = S*( LD( J ) / TMP )*L( J ) - LEFT
IF( TMP.LT.ZERO )
$ CNT = CNT + 1
50 CONTINUE
TMP = D( N ) + S
IF( TMP.LT.ZERO )
$ CNT = CNT + 1
IF( CNT.GT.I-1 ) THEN
OPS = OPS + DBLE( 4 )
DELTA = TWO*DELTA
LEFT = LEFT - ( ABS( SIGMA )+ABS( LEFT ) )*DELTA
GO TO 40
END IF
END IF
OPS = OPS + DBLE( 1 )
DELTA = EPS
RIGHT = W( I ) + WERR( I )
*
* Do while( CNT(RIGHT).LT.I )
*
60 CONTINUE
S = -RIGHT
CNT = 0
OPS = OPS + DBLE( 5*( N-1 )+1 )
DO 70 J = 1, N - 1
TMP = D( J ) + S
S = S*( LD( J ) / TMP )*L( J ) - RIGHT
IF( TMP.LT.ZERO )
$ CNT = CNT + 1
70 CONTINUE
TMP = D( N ) + S
IF( TMP.LT.ZERO )
$ CNT = CNT + 1
IF( CNT.LT.I ) THEN
OPS = OPS + DBLE( 4 )
DELTA = TWO*DELTA
RIGHT = RIGHT + ( ABS( SIGMA )+ABS( RIGHT ) )*DELTA
GO TO 60
END IF
WERR( I ) = LEFT
W( I ) = RIGHT
IWORK( N+I ) = CNT
NINT = NINT + 1
I = CNT + 1
ELSE
I = I + 1
END IF
GO TO 30
END IF
*
* While( NCNVRG.LT.NEIG )
*
INITI1 = I1
INITI2 = I2
80 CONTINUE
IF( NCNVRG.LT.NEIG ) THEN
OLNINT = NINT
I = I1
DO 100 K = 1, OLNINT
NRIGHT = IWORK( N+I )
IF( IWORK( I ).EQ.0 ) THEN
OPS = OPS + DBLE( 2 )
MID = HALF*( WERR( I )+W( I ) )
S = -MID
CNT = 0
OPS = OPS + DBLE( 5*( N-1 )+1 )
DO 90 J = 1, N - 1
TMP = D( J ) + S
S = S*( LD( J ) / TMP )*L( J ) - MID
IF( TMP.LT.ZERO )
$ CNT = CNT + 1
90 CONTINUE
TMP = D( N ) + S
IF( TMP.LT.ZERO )
$ CNT = CNT + 1
CNT = MAX( I-1, MIN( NRIGHT, CNT ) )
IF( I.EQ.NRIGHT ) THEN
IF( I.EQ.IFIRST ) THEN
OPS = OPS + DBLE( 1 )
GAP = WERR( I+1 ) - W( I )
ELSE IF( I.EQ.ILAST ) THEN
OPS = OPS + DBLE( 1 )
GAP = WERR( I ) - W( I-1 )
ELSE
OPS = OPS + DBLE( 2 )
GAP = MIN( WERR( I+1 )-W( I ), WERR( I )-W( I-1 ) )
END IF
OPS = OPS + DBLE( 2 )
WIDTH = W( I ) - MID
IF( WIDTH.LT.THRESH*GAP ) THEN
NCNVRG = NCNVRG + 1
IWORK( I ) = 1
IF( I1.EQ.I ) THEN
I1 = I1 + 1
NINT = NINT - 1
END IF
END IF
END IF
IF( IWORK( I ).EQ.0 )
$ I2 = K
IF( CNT.EQ.I-1 ) THEN
WERR( I ) = MID
ELSE IF( CNT.EQ.NRIGHT ) THEN
W( I ) = MID
ELSE
IWORK( N+I ) = CNT
NINT = NINT + 1
WERR( CNT+1 ) = MID
W( CNT+1 ) = W( I )
W( I ) = MID
I = CNT + 1
IWORK( N+I ) = NRIGHT
END IF
END IF
I = NRIGHT + 1
100 CONTINUE
NINT = NINT - OLNINT + I2
GO TO 80
END IF
DO 110 I = INITI1, INITI2
OPS = OPS + DBLE( 3 )
W( I ) = HALF*( WERR( I )+W( I ) )
WERR( I ) = W( I ) - WERR( I )
110 CONTINUE
*
RETURN
*
* End of DLARRB
*
END
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