*> \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 *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * 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. * *> \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 -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. 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