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SUBROUTINE DLARRR( N, D, E, INFO )
*
* -- 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 N, INFO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
* ..
*
*
* Purpose
* =======
*
* Perform tests to decide whether the symmetric tridiagonal matrix T
* warrants expensive computations which guarantee high relative accuracy
* in the eigenvalues.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N > 0.
*
* D (input) DOUBLE PRECISION array, dimension (N)
* The N diagonal elements of the tridiagonal matrix T.
*
* E (input/output) 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.
*
* INFO (output) INTEGER
* INFO = 0(default) : the matrix warrants computations preserving
* relative accuracy.
* INFO = 1 : the matrix warrants computations guaranteeing
* only absolute accuracy.
*
* 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, 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 ..
*
* 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
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