*> \brief \b DLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRR + dependencies
*>
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*
* Definition:
* ===========
*
* SUBROUTINE DLARRR( N, D, E, INFO )
*
* .. Scalar Arguments ..
* INTEGER N, INFO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * )
* ..
*
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Perform tests to decide whether the symmetric tridiagonal matrix T
*> warrants expensive computations which guarantee high relative accuracy
*> in the eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N > 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N)
*> On entry, the first (N-1) entries contain the subdiagonal
*> elements of the tridiagonal matrix T; E(N) is set to ZERO.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> INFO = 0(default) : the matrix warrants computations preserving
*> relative accuracy.
*> INFO = 1 : the matrix warrants computations guaranteeing
*> only absolute accuracy.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRR( N, D, E, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2017
*
* .. Scalar Arguments ..
INTEGER N, INFO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, RELCOND
PARAMETER ( ZERO = 0.0D0,
$ RELCOND = 0.999D0 )
* ..
* .. Local Scalars ..
INTEGER I
LOGICAL YESREL
DOUBLE PRECISION EPS, SAFMIN, SMLNUM, RMIN, TMP, TMP2,
$ OFFDIG, OFFDIG2
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
INFO = 0
RETURN
END IF
*
* As a default, do NOT go for relative-accuracy preserving computations.
INFO = 1
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
RMIN = SQRT( SMLNUM )
* Tests for relative accuracy
*
* Test for scaled diagonal dominance
* Scale the diagonal entries to one and check whether the sum of the
* off-diagonals is less than one
*
* The sdd relative error bounds have a 1/(1- 2*x) factor in them,
* x = max(OFFDIG + OFFDIG2), so when x is close to 1/2, no relative
* accuracy is promised. In the notation of the code fragment below,
* 1/(1 - (OFFDIG + OFFDIG2)) is the condition number.
* We don't think it is worth going into "sdd mode" unless the relative
* condition number is reasonable, not 1/macheps.
* The threshold should be compatible with other thresholds used in the
* code. We set OFFDIG + OFFDIG2 <= .999 =: RELCOND, it corresponds
* to losing at most 3 decimal digits: 1 / (1 - (OFFDIG + OFFDIG2)) <= 1000
* instead of the current OFFDIG + OFFDIG2 < 1
*
YESREL = .TRUE.
OFFDIG = ZERO
TMP = SQRT(ABS(D(1)))
IF (TMP.LT.RMIN) YESREL = .FALSE.
IF(.NOT.YESREL) GOTO 11
DO 10 I = 2, N
TMP2 = SQRT(ABS(D(I)))
IF (TMP2.LT.RMIN) YESREL = .FALSE.
IF(.NOT.YESREL) GOTO 11
OFFDIG2 = ABS(E(I-1))/(TMP*TMP2)
IF(OFFDIG+OFFDIG2.GE.RELCOND) YESREL = .FALSE.
IF(.NOT.YESREL) GOTO 11
TMP = TMP2
OFFDIG = OFFDIG2
10 CONTINUE
11 CONTINUE
IF( YESREL ) THEN
INFO = 0
RETURN
ELSE
ENDIF
*
*
* *** MORE TO BE IMPLEMENTED ***
*
*
* Test if the lower bidiagonal matrix L from T = L D L^T
* (zero shift facto) is well conditioned
*
*
* Test if the upper bidiagonal matrix U from T = U D U^T
* (zero shift facto) is well conditioned.
* In this case, the matrix needs to be flipped and, at the end
* of the eigenvector computation, the flip needs to be applied
* to the computed eigenvectors (and the support)
*
*
RETURN
*
* END OF DLARRR
*
END