SRC/dlarrd.f(3)

NAME
SRC/dlarrd.f

SYNOPSIS

  Functions/Subroutines
subroutine dlarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info)
DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Function/Subroutine Documentation

  subroutine dlarrd (character range, character order, integer n, double precision vl, double precision vu, integer il, integer iu, double precision, dimension( * ) gers, double precision reltol, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) e2, double precision pivmin, integer nsplit, integer, dimension( * ) isplit, integer m, double precision, dimension( * ) w, double precision, dimension( * ) werr, double precision wl, double precision wu, integer, dimension( * ) iblock, integer, dimension( * ) indexw, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Purpose:

     DLARRD computes the eigenvalues of a symmetric tridiagonal
     matrix T to suitable accuracy. This is an auxiliary code to be
     called from DSTEMR.
     The user may ask for all eigenvalues, all eigenvalues
     in the half-open interval (VL, VU], or the IL-th through IU-th
     eigenvalues.
     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.

Parameters
    RANGE

              RANGE is CHARACTER*1
              = 'A': ('All')   all eigenvalues will be found.
              = 'V': ('Value') all eigenvalues in the half-open interval
                               (VL, VU] will be found.
              = 'I': ('Index') the IL-th through IU-th eigenvalues (of the
                               entire matrix) will be found.

    ORDER

              ORDER is CHARACTER*1
              = 'B': ('By Block') the eigenvalues will be grouped by
                                  split-off block (see IBLOCK, ISPLIT) and
                                  ordered from smallest to largest within
                                  the block.
              = 'E': ('Entire matrix')
                                  the eigenvalues for the entire matrix
                                  will be ordered from smallest to
                                  largest.

    N

              N is INTEGER
              The order of the tridiagonal matrix T.  N >= 0.

    VL

              VL is DOUBLE PRECISION
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues.  Eigenvalues less than or equal
              to VL, or greater than VU, will not be returned.  VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is DOUBLE PRECISION
              If RANGE='V', the upper bound of the interval to
              be searched for eigenvalues.  Eigenvalues less than or equal
              to VL, or greater than VU, will not be returned.  VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
              Not referenced if RANGE = 'A' or 'V'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
              Not referenced if RANGE = 'A' or 'V'.

    GERS

              GERS is DOUBLE PRECISION array, dimension (2*N)
              The N Gerschgorin intervals (the i-th Gerschgorin interval
              is (GERS(2*i-1), GERS(2*i)).

    RELTOL

              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.

    D

              D is DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the tridiagonal matrix T.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              The (n-1) off-diagonal elements of the tridiagonal matrix T.

    E2

              E2 is DOUBLE PRECISION array, dimension (N-1)
              The (n-1) squared off-diagonal elements of the tridiagonal matrix T.

    PIVMIN

              PIVMIN is DOUBLE PRECISION
              The minimum pivot allowed in the Sturm sequence for T.

    NSPLIT

              NSPLIT is INTEGER
              The number of diagonal blocks in the matrix T.
              1 <= NSPLIT <= N.

    ISPLIT

              ISPLIT is INTEGER array, dimension (N)
              The splitting points, at which T breaks up into submatrices.
              The first submatrix consists of rows/columns 1 to ISPLIT(1),
              the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
              etc., and the NSPLIT-th consists of rows/columns
              ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
              (Only the first NSPLIT elements will actually be used, but
              since the user cannot know a priori what value NSPLIT will
              have, N words must be reserved for ISPLIT.)

    M

              M is INTEGER
              The actual number of eigenvalues found. 0 <= M <= N.
              (See also the description of INFO=2,3.)

    W

              W is DOUBLE PRECISION array, dimension (N)
              On exit, the first M elements of W will contain the
              eigenvalue approximations. DLARRD computes an interval
              I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
              approximation is given as the interval midpoint
              W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
              WERR(j) = abs( a_j - b_j)/2

    WERR

              WERR is DOUBLE PRECISION array, dimension (N)
              The error bound on the corresponding eigenvalue approximation
              in W.

    WL

              WL is DOUBLE PRECISION

    WU

              WU is DOUBLE PRECISION
              The interval (WL, WU] contains all the wanted eigenvalues.
              If RANGE='V', then WL=VL and WU=VU.
              If RANGE='A', then WL and WU are the global Gerschgorin bounds
                            on the spectrum.
              If RANGE='I', then WL and WU are computed by DLAEBZ from the
                            index range specified.

    IBLOCK

              IBLOCK is INTEGER array, dimension (N)
              At each row/column j where E(j) is zero or small, the
              matrix T is considered to split into a block diagonal
              matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
              block (from 1 to the number of blocks) the eigenvalue W(i)
              belongs.  (DLARRD may use the remaining N-M elements as
              workspace.)

    INDEXW

              INDEXW is INTEGER array, dimension (N)
              The indices of the eigenvalues within each block (submatrix);
              for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
              i-th eigenvalue W(i) is the j-th eigenvalue in block k.

    WORK

              WORK is DOUBLE PRECISION array, dimension (4*N)

    IWORK

              IWORK is INTEGER array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  some or all of the eigenvalues failed to converge or
                    were not computed:
                    =1 or 3: Bisection failed to converge for some
                            eigenvalues; these eigenvalues are flagged by a
                            negative block number.  The effect is that the
                            eigenvalues may not be as accurate as the
                            absolute and relative tolerances.  This is
                            generally caused by unexpectedly inaccurate
                            arithmetic.
                    =2 or 3: RANGE='I' only: Not all of the eigenvalues
                            IL:IU were found.
                            Effect: M < IU+1-IL
                            Cause:  non-monotonic arithmetic, causing the
                                    Sturm sequence to be non-monotonic.
                            Cure:   recalculate, using RANGE='A', and pick
                                    out eigenvalues IL:IU.  In some cases,
                                    increasing the PARAMETER 'FUDGE' may
                                    make things work.
                    = 4:    RANGE='I', and the Gershgorin interval
                            initially used was too small.  No eigenvalues
                            were computed.
                            Probable cause: your machine has sloppy
                                            floating-point arithmetic.
                            Cure: Increase the PARAMETER 'FUDGE',
                                  recompile, and try again.

Internal Parameters:

      FUDGE   DOUBLE PRECISION, default = 2
              A 'fudge factor' to widen the Gershgorin intervals.  Ideally,
              a value of 1 should work, but on machines with sloppy
              arithmetic, this needs to be larger.  The default for
              publicly released versions should be large enough to handle
              the worst machine around.  Note that this has no effect
              on accuracy of the solution.

Contributors:
    W. Kahan, University of California, Berkeley, USA
     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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 325 of file dlarrd.f.

Author

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