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SUBROUTINE DLARRK( N, IW, GL, GU,
$ D, E2, PIVMIN, RELTOL, W, WERR, INFO)
IMPLICIT NONE
*
* -- LAPACK auxiliary routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, IW, N
DOUBLE PRECISION PIVMIN, RELTOL, GL, GU, W, WERR
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E2( * )
* ..
*
* Purpose
* =======
*
* DLARRK computes one eigenvalue of a symmetric tridiagonal
* matrix T to suitable accuracy. This is an auxiliary code to be
* called from DSTEMR.
*
* To avoid overflow, the matrix must be scaled so that its
* largest element is no greater than overflow**(1/2) *
* underflow**(1/4) in absolute value, and for greatest
* accuracy, it should not be much smaller than that.
*
* See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
* Matrix", Report CS41, Computer Science Dept., Stanford
* University, July 21, 1966.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the tridiagonal matrix T. N >= 0.
*
* IW (input) INTEGER
* The index of the eigenvalues to be returned.
*
* GL (input) DOUBLE PRECISION
* GU (input) DOUBLE PRECISION
* An upper and a lower bound on the eigenvalue.
*
* D (input) DOUBLE PRECISION array, dimension (N)
* The n diagonal elements of the tridiagonal matrix T.
*
* E2 (input) DOUBLE PRECISION array, dimension (N-1)
* The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
*
* PIVMIN (input) DOUBLE PRECISION
* The minimum pivot allowed in the Sturm sequence for T.
*
* RELTOL (input) DOUBLE PRECISION
* The minimum relative width of an interval. When an interval
* is narrower than RELTOL times the larger (in
* magnitude) endpoint, then it is considered to be
* sufficiently small, i.e., converged. Note: this should
* always be at least radix*machine epsilon.
*
* W (output) DOUBLE PRECISION
*
* WERR (output) DOUBLE PRECISION
* The error bound on the corresponding eigenvalue approximation
* in W.
*
* INFO (output) INTEGER
* = 0: Eigenvalue converged
* = -1: Eigenvalue did NOT converge
*
* Internal Parameters
* ===================
*
* FUDGE DOUBLE PRECISION, default = 2
* A "fudge factor" to widen the Gershgorin intervals.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION FUDGE, HALF, TWO, ZERO
PARAMETER ( HALF = 0.5D0, TWO = 2.0D0,
$ FUDGE = TWO, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, IT, ITMAX, NEGCNT
DOUBLE PRECISION ATOLI, EPS, LEFT, MID, RIGHT, RTOLI, TMP1,
$ TMP2, TNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX
* ..
* .. Executable Statements ..
*
* Get machine constants
EPS = DLAMCH( 'P' )
TNORM = MAX( ABS( GL ), ABS( GU ) )
RTOLI = RELTOL
ATOLI = FUDGE*TWO*PIVMIN
ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
$ LOG( TWO ) ) + 2
INFO = -1
LEFT = GL - FUDGE*TNORM*EPS*N - FUDGE*TWO*PIVMIN
RIGHT = GU + FUDGE*TNORM*EPS*N + FUDGE*TWO*PIVMIN
IT = 0
10 CONTINUE
*
* Check if interval converged or maximum number of iterations reached
*
TMP1 = ABS( RIGHT - LEFT )
TMP2 = MAX( ABS(RIGHT), ABS(LEFT) )
IF( TMP1.LT.MAX( ATOLI, PIVMIN, RTOLI*TMP2 ) ) THEN
INFO = 0
GOTO 30
ENDIF
IF(IT.GT.ITMAX)
$ GOTO 30
*
* Count number of negative pivots for mid-point
*
IT = IT + 1
MID = HALF * (LEFT + RIGHT)
NEGCNT = 0
TMP1 = D( 1 ) - MID
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
IF( TMP1.LE.ZERO )
$ NEGCNT = NEGCNT + 1
*
DO 20 I = 2, N
TMP1 = D( I ) - E2( I-1 ) / TMP1 - MID
IF( ABS( TMP1 ).LT.PIVMIN )
$ TMP1 = -PIVMIN
IF( TMP1.LE.ZERO )
$ NEGCNT = NEGCNT + 1
20 CONTINUE
IF(NEGCNT.GE.IW) THEN
RIGHT = MID
ELSE
LEFT = MID
ENDIF
GOTO 10
30 CONTINUE
*
* Converged or maximum number of iterations reached
*
W = HALF * (LEFT + RIGHT)
WERR = HALF * ABS( RIGHT - LEFT )
RETURN
*
* End of DLARRK
*
END
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