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377 lines
11 KiB
377 lines
11 KiB
2 years ago
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*> \brief \b DLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download DLARRJ + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrj.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrj.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrj.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
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* RTOL, OFFSET, W, WERR, WORK, IWORK,
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* PIVMIN, SPDIAM, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER IFIRST, ILAST, INFO, N, OFFSET
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* DOUBLE PRECISION PIVMIN, RTOL, SPDIAM
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* ..
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* .. Array Arguments ..
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* INTEGER IWORK( * )
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* DOUBLE PRECISION D( * ), E2( * ), W( * ),
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* $ WERR( * ), WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> Given the initial eigenvalue approximations of T, DLARRJ
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*> does bisection to refine the eigenvalues of T,
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*> W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
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*> guesses for these eigenvalues are input in W, the corresponding estimate
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*> of the error in these guesses in WERR. During bisection, intervals
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*> [left, right] are maintained by storing their mid-points and
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*> semi-widths in the arrays W and WERR respectively.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix.
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*> \endverbatim
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*>
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*> \param[in] D
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*> \verbatim
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*> D is DOUBLE PRECISION array, dimension (N)
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*> The N diagonal elements of T.
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*> \endverbatim
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*>
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*> \param[in] E2
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*> \verbatim
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*> E2 is DOUBLE PRECISION array, dimension (N-1)
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*> The Squares of the (N-1) subdiagonal elements of T.
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*> \endverbatim
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*>
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*> \param[in] IFIRST
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*> \verbatim
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*> IFIRST is INTEGER
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*> The index of the first eigenvalue to be computed.
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*> \endverbatim
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*>
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*> \param[in] ILAST
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*> \verbatim
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*> ILAST is INTEGER
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*> The index of the last eigenvalue to be computed.
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*> \endverbatim
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*>
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*> \param[in] RTOL
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*> \verbatim
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*> RTOL is DOUBLE PRECISION
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*> Tolerance for the convergence of the bisection intervals.
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*> An interval [LEFT,RIGHT] has converged if
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*> RIGHT-LEFT < RTOL*MAX(|LEFT|,|RIGHT|).
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*> \endverbatim
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*>
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*> \param[in] OFFSET
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*> \verbatim
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*> OFFSET is INTEGER
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*> Offset for the arrays W and WERR, i.e., the IFIRST-OFFSET
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*> through ILAST-OFFSET elements of these arrays are to be used.
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*> \endverbatim
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*>
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*> \param[in,out] W
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*> \verbatim
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*> W is DOUBLE PRECISION array, dimension (N)
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*> On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
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*> estimates of the eigenvalues of L D L^T indexed IFIRST through
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*> ILAST.
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*> On output, these estimates are refined.
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*> \endverbatim
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*>
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*> \param[in,out] WERR
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*> \verbatim
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*> WERR is DOUBLE PRECISION array, dimension (N)
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*> On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
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*> the errors in the estimates of the corresponding elements in W.
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*> On output, these errors are refined.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension (2*N)
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*> Workspace.
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (2*N)
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*> Workspace.
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*> \endverbatim
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*>
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*> \param[in] PIVMIN
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*> \verbatim
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*> PIVMIN is DOUBLE PRECISION
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*> The minimum pivot in the Sturm sequence for T.
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*> \endverbatim
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*>
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*> \param[in] SPDIAM
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*> \verbatim
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*> SPDIAM is DOUBLE PRECISION
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*> The spectral diameter of T.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> Error flag.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup OTHERauxiliary
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*
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*> \par Contributors:
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* ==================
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*>
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*> Beresford Parlett, University of California, Berkeley, USA \n
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*> Jim Demmel, University of California, Berkeley, USA \n
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*> Inderjit Dhillon, University of Texas, Austin, USA \n
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*> Osni Marques, LBNL/NERSC, USA \n
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*> Christof Voemel, University of California, Berkeley, USA
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*
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* =====================================================================
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SUBROUTINE DLARRJ( N, D, E2, IFIRST, ILAST,
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$ RTOL, OFFSET, W, WERR, WORK, IWORK,
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$ PIVMIN, SPDIAM, INFO )
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*
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* -- LAPACK auxiliary routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER IFIRST, ILAST, INFO, N, OFFSET
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DOUBLE PRECISION PIVMIN, RTOL, SPDIAM
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* ..
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* .. Array Arguments ..
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INTEGER IWORK( * )
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DOUBLE PRECISION D( * ), E2( * ), W( * ),
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$ WERR( * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE, TWO, HALF
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
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$ HALF = 0.5D0 )
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INTEGER MAXITR
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* ..
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* .. Local Scalars ..
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INTEGER CNT, I, I1, I2, II, ITER, J, K, NEXT, NINT,
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$ OLNINT, P, PREV, SAVI1
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DOUBLE PRECISION DPLUS, FAC, LEFT, MID, RIGHT, S, TMP, WIDTH
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*
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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*
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* Quick return if possible
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*
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IF( N.LE.0 ) THEN
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RETURN
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END IF
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*
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MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
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$ LOG( TWO ) ) + 2
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*
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* Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
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* The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
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* Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
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* for an unconverged interval is set to the index of the next unconverged
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* interval, and is -1 or 0 for a converged interval. Thus a linked
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* list of unconverged intervals is set up.
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*
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I1 = IFIRST
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I2 = ILAST
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* The number of unconverged intervals
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NINT = 0
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* The last unconverged interval found
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PREV = 0
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DO 75 I = I1, I2
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K = 2*I
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II = I - OFFSET
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LEFT = W( II ) - WERR( II )
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MID = W(II)
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RIGHT = W( II ) + WERR( II )
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WIDTH = RIGHT - MID
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TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
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* The following test prevents the test of converged intervals
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IF( WIDTH.LT.RTOL*TMP ) THEN
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* This interval has already converged and does not need refinement.
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* (Note that the gaps might change through refining the
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* eigenvalues, however, they can only get bigger.)
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* Remove it from the list.
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IWORK( K-1 ) = -1
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* Make sure that I1 always points to the first unconverged interval
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IF((I.EQ.I1).AND.(I.LT.I2)) I1 = I + 1
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IF((PREV.GE.I1).AND.(I.LE.I2)) IWORK( 2*PREV-1 ) = I + 1
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ELSE
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* unconverged interval found
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PREV = I
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* Make sure that [LEFT,RIGHT] contains the desired eigenvalue
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*
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* Do while( CNT(LEFT).GT.I-1 )
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*
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FAC = ONE
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20 CONTINUE
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CNT = 0
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S = LEFT
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DPLUS = D( 1 ) - S
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IF( DPLUS.LT.ZERO ) CNT = CNT + 1
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DO 30 J = 2, N
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DPLUS = D( J ) - S - E2( J-1 )/DPLUS
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IF( DPLUS.LT.ZERO ) CNT = CNT + 1
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30 CONTINUE
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IF( CNT.GT.I-1 ) THEN
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LEFT = LEFT - WERR( II )*FAC
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FAC = TWO*FAC
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GO TO 20
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END IF
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*
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* Do while( CNT(RIGHT).LT.I )
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*
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FAC = ONE
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50 CONTINUE
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CNT = 0
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S = RIGHT
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DPLUS = D( 1 ) - S
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IF( DPLUS.LT.ZERO ) CNT = CNT + 1
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DO 60 J = 2, N
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DPLUS = D( J ) - S - E2( J-1 )/DPLUS
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IF( DPLUS.LT.ZERO ) CNT = CNT + 1
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60 CONTINUE
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IF( CNT.LT.I ) THEN
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RIGHT = RIGHT + WERR( II )*FAC
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FAC = TWO*FAC
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GO TO 50
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END IF
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NINT = NINT + 1
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IWORK( K-1 ) = I + 1
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IWORK( K ) = CNT
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END IF
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WORK( K-1 ) = LEFT
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WORK( K ) = RIGHT
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75 CONTINUE
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SAVI1 = I1
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*
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* Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
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* and while (ITER.LT.MAXITR)
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*
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ITER = 0
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80 CONTINUE
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PREV = I1 - 1
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I = I1
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OLNINT = NINT
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DO 100 P = 1, OLNINT
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K = 2*I
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II = I - OFFSET
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NEXT = IWORK( K-1 )
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LEFT = WORK( K-1 )
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RIGHT = WORK( K )
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MID = HALF*( LEFT + RIGHT )
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* semiwidth of interval
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WIDTH = RIGHT - MID
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TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
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IF( ( WIDTH.LT.RTOL*TMP ) .OR.
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$ (ITER.EQ.MAXITR) )THEN
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* reduce number of unconverged intervals
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NINT = NINT - 1
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* Mark interval as converged.
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IWORK( K-1 ) = 0
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IF( I1.EQ.I ) THEN
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I1 = NEXT
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ELSE
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* Prev holds the last unconverged interval previously examined
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IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
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END IF
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I = NEXT
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GO TO 100
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END IF
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PREV = I
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*
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* Perform one bisection step
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*
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CNT = 0
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S = MID
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DPLUS = D( 1 ) - S
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IF( DPLUS.LT.ZERO ) CNT = CNT + 1
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DO 90 J = 2, N
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DPLUS = D( J ) - S - E2( J-1 )/DPLUS
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IF( DPLUS.LT.ZERO ) CNT = CNT + 1
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90 CONTINUE
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IF( CNT.LE.I-1 ) THEN
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WORK( K-1 ) = MID
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ELSE
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WORK( K ) = MID
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END IF
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I = NEXT
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100 CONTINUE
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ITER = ITER + 1
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* do another loop if there are still unconverged intervals
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* However, in the last iteration, all intervals are accepted
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* since this is the best we can do.
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IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
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*
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*
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* At this point, all the intervals have converged
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DO 110 I = SAVI1, ILAST
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K = 2*I
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II = I - OFFSET
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* All intervals marked by '0' have been refined.
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IF( IWORK( K-1 ).EQ.0 ) THEN
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W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
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WERR( II ) = WORK( K ) - W( II )
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END IF
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110 CONTINUE
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|
|
*
|
||
|
|
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of DLARRJ
|
||
|
|
*
|
||
|
|
END
|