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*> \brief \b DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrk.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrk.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrk.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRK( N, IW, GL, GU,
* D, E2, PIVMIN, RELTOL, W, WERR, INFO)
*
* .. Scalar Arguments ..
* INTEGER INFO, IW, N
* DOUBLE PRECISION PIVMIN, RELTOL, GL, GU, W, WERR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E2( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the tridiagonal matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] IW
*> \verbatim
*> IW is INTEGER
*> The index of the eigenvalues to be returned.
*> \endverbatim
*>
*> \param[in] GL
*> \verbatim
*> GL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] GU
*> \verbatim
*> GU is DOUBLE PRECISION
*> An upper and a lower bound on the eigenvalue.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E2
*> \verbatim
*> E2 is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot allowed in the Sturm sequence for T.
*> \endverbatim
*>
*> \param[in] RELTOL
*> \verbatim
*> RELTOL is 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.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION
*> The error bound on the corresponding eigenvalue approximation
*> in W.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: Eigenvalue converged
*> = -1: Eigenvalue did NOT converge
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> FUDGE DOUBLE PRECISION, default = 2
*> A "fudge factor" to widen the Gershgorin intervals.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLARRK( N, IW, GL, GU,
$ D, E2, PIVMIN, RELTOL, W, WERR, INFO)
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, IW, N
DOUBLE PRECISION PIVMIN, RELTOL, GL, GU, W, WERR
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E2( * )
* ..
*
* =====================================================================
*
* .. 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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